Ément Cl

Erik Dupont

Brent Panzer
email: brent.pym@mcgill.ca

Pym

REGULARIZED INTEGRALS AND MANIFOLDS WITH LOG CORNERS

We introduce a natural geometric framework for the study of logarithmically divergent integrals on manifolds with corners and algebraic varieties, using the techniques of logarithmic geometry. Key to the construction is a new notion of morphism in logarithmic geometry itself, which allows us to interpret the ubiquitous rule of thumb "lim ε→0 log ε := 0" as the natural restriction to a submanifold. Via a version of de Rham's theorem with logarithmic divergences, we obtain a functorial characterization of the classical theory of "regularized integration": it is the unique way to extend the ordinary integral to the logarithmically divergent context, while respecting the basic laws of calculus (change of variables, Fubini's theorem, and Stokes' formula.)

and formally discards the term that diverges as ε → 0, to obtain the "regularized" value

I 1 = a 0 dx x := log(a) -✟ ✟ ✟ ✯ 0 log ε = log(a) (1)
The cost of assigning a finite value to such a divergent integral is a dependence on a choice of coordinate: if we make a change of variables x = g(x), the regularized integral changes to log(a) -log(g ′ (0)), i.e. it depends implicitly on the choice of a nonzero tangent vector at x = 0, which Deligne calls a "tangential basepoint" [START_REF] Deligne | Le groupe fondamental de la droite projective moins trois points, Galois groups over Q[END_REF]. The latter notion is a powerful tool, but it poses a basic challenge, in that it gives a notion of "basepoint in X" that does not actually correspond to a point in X. As a result, it becomes difficult to interpret the regularized integral in a cohomological fashion, that is, as the result of the natural integration pairing between classes in de Rham cohomology and singular homology. This is reflected, e.g. in Deligne and Goncharov's indirect construction of some special cases of "motivic fundamental groups with tangential basepoints" [START_REF] Deligne | Groupes fondamentaux motiviques de Tate mixte[END_REF].

In this paper, we will explain how to view a tangential basepoint as an "actual point" in a suitable sense, and use this framework to give a precise cohomological meaning to regularized integrals such as I 1 , in any dimension.

In a similar vein, consider the integral

I 2 := P 1 (C) dz z -a - dz z -1 ∧ dz z
where z is the standard holomorphic coordinate on the Riemann sphere P 1 (C). (We have borrowed this example from [START_REF] Brown | Single-valued integration and double copy[END_REF].) This integral presents us with a subtly different problem. Namely, despite the apparent singularities of the integrand

ω := dz z -a - dz z -1 ∧ dz z
at the points 0, 1, a, ∞ ∈ P 1 (C), the integral I 2 is absolutely convergent, as can be seen by working in polar coordinates centred at each of these points. More invariantly, we may pass to the real oriented blowup at these points, which is the compact oriented surface with boundary Σ obtained from P 1 (C) by replacing each point 0, 1, a, ∞ with a boundary circle; then ω extends to a smooth form on Σ. But an issue arises when we try to compute this integral. To do so, we can observe that the form α := -log |z| 2 dz z -a -dz z -1 is a primitive for ω, and attempt to apply Stokes' formula

I 2 = Σ ω = ∂ Σ i * α
where i : ∂ Σ → Σ is the inclusion of the boundary. However, the right-hand side does not makes sense as written, because α has a logarithmic pole along ∂ Σ. Indeed, if we switch to polar coordinates z -a = re iθ , we find that α = -log |a| 2 dr r + i dθ + O(1) dr + O(r) dθ as r → 0 so that α has a logarithmic pole dr r = d log(r) at r = 0 and its restriction is ill-defined. So, once again, we introduce a cutoff parameter ε, excise a tubular neighbourhood of width ε around the boundary, compute the integral over the boundary of the resulting surface Σ ε , and take the limit as ε → 0. With a bit of care, one sees that this boils down to computing the residue of α at z = a. Consequently, the same result can be achieved without introducing a cutoff parameter, by instead defining a "regularized restriction" reg i * , in which we formally set dr r to zero on the boundary, giving

P 1 (C) ω = ∂ Σ reg i * α = -log |a| 2 • ∂ Σ ✟ ✟ ✟ ✟ ✟ ✯ 0 reg i * dr r + i dθ = 2πi log |a| 2 .
(The disappearance of the minus sign is explained by the orientation of ∂Σ.) In this paper, we will explain how this ad hoc construction of reg i * α can be interpreted, in a precise sense, as the pullback of α along a morphism, resulting in a "regularized Stokes theorem". Once again, the form reg i * α itself depends on tangential datathis time, a trivialization of the normal bundle of ∂ Σ-but in this case, the value of the integral is ultimately independent. At first glance, the dependence on choices seems more like a mild nuisance than a serious issue, but the situation becomes more critical when one has many integrals and wants to prove nontrivial relations between them using the basic laws of integration: change of variables, Fubini's theorem, and Stokes' formula. These integrals may diverge (as in the case of I 1 above), but even if they converge, one may wish to make use of divergent forms in the calculations (as in the case of I 2 ). For instance, our own interest in the problem stems from Feynman-style integrals in quantum field theory, and related structures in quantum algebra, where one has a whole infinite collection of integrals indexed by graphs; they satisfy many relations amongst themselves, and logarithmic divergences abound.

Going further, one may seek to reformulate such identities between integrals in purely cohomological terms, which grants access to the powerful tools of algebraic geometry and Hodge theory, and reveals a hidden symmetry in the form of the action of a motivic Galois group. In other words, one may wish to establish such identities in the ring of "(motivic) periods" [START_REF] Kontsevich | Periods, Mathematics unlimited-2001 and beyond[END_REF][START_REF] Brown | Notes on motivic periods[END_REF] (or a logarithmic variant thereof). The formalism we develop below in this paper provides exactly such an interpretation. In future work, we will apply it to the integrals appearing in deformation quantization [START_REF] Kontsevich | Operads and motives in deformation quantization[END_REF][START_REF] Kontsevich | Deformation quantization of Poisson manifolds[END_REF][START_REF] Alekseev | Logarithms and deformation quantization[END_REF], thus realizing Kontsevich's vision of a motivic Galois action in this context. 1.2. Approach and results. In this paper, we solve the problems above by introducing a new class of geometric objects, called manifolds with log corners, which serve as the natural domains of integration for logarithmically divergent forms, in the same way that ordinary manifolds with corners are used for the integration of smooth forms. In particular, there is a natural geometric notion of "regularization" for manifolds with log corners, which serves as a replacement for the ad hoc introduction of cutoffs and tubular neighbourhoods in the usual approach, retaining only the essential geometric data (such as tangential basepoints) that are needed to regulate the integral.

In the rest of the introduction, we shall give an overview of the main definitions and ideas. As a preview of what is to come, let us summarize the main results of the paper as follows:

Theorem 1.1. The following statements hold:

(1) Manifolds with log corners carry functorial sheaves of functions and differential forms with logarithmic divergences, satisfying versions of the Poincaré lemma and de Rham isomorphism.

(2) When such manifolds are equipped with the additional data of an orientation and a regularization, logarithmically divergent forms can be naturally integrated, in a way that respects the basic laws of calculus: change of variables, Fubini's theorem and Stokes' formula. (3) A natural source of manifolds with log corners is provided by the "Kato-Nakayama" spaces of a class of logarithmic algebraic varieties, which we call "varieties with log corners". The corresponding regularized integrals provide a cohomological theory of periods for varieties with log corners that is naturally compatible with Deligne's tangential basepoints.

1.2.1. Manifolds with log corners. The most basic example of a manifold with log corners is provided by a manifold with corners in the classical sense. More generally, we may consider boundary faces of a manifold with corners, along with the natural data they carry on their normal bundles-specifically, the collection of "positive" normal vectors, i.e. those which point into the interior. These data play a crucial role in regularizing integrals. While it is possible to describe manifolds with log corners in purely classical terms (see Section 3.3.3), such a description is, in our experience, cumbersome. For this reason, we adopt the framework of "positive log geometry" of Gillam-Molcho [START_REF] Gillam | Logarithmic differentiable spaces and manifolds with corners[END_REF]-a differential geometry version of the logarithmic algebraic geometry of Fontaine-Illusie-Kato [START_REF] Kato | Logarithmic structures of Fontaine-Illusie, Algebraic analysis, geometry, and number theory[END_REF][START_REF] Illusie | Logarithmic Spaces[END_REF][START_REF] Ogus | Lectures on logarithmic algebraic geometry[END_REF]. In other words, we consider a manifold with corners Σ endowed with the additional datum of a sheaf of monoids M Σ mapping to the sheaf of non-negative smooth functions. In the basic case of a manifold with corners, M Σ is simply the sheaf of non-negative smooth functions that are locally monomial in coordinates near the boundary. For a boundary face, we include additional elements that are "phantoms" of such functions on the normal bundle. See Section 3 for the general definition. For the introduction, it will suffice to have the following examples in mind.

The first is the "standard interval" [0, ∞), which we equip with the sheaf M [0,∞) of functions of the form g(r)r j where r is the standard coordinate, g is a positive smooth function, and j ∈ N; this is a multiplicative submonoid of the sheaf C ∞, 0 [0,∞) of non-negative smooth functions.

The second is its boundary, the "standard end" [0) := ∂[0, ∞), consisting of the point 0 equipped with the monoid M [0) of monomials λt j where λ ∈ R >0 is a positive constant and t is a formal coordinate corresponding to the derivative of r at zero; this coordinate is not really a function, but rather a phantom thereof, in the sense that its "value" is defined by formally setting t = 0.

A general manifold with log corners is then locally isomorphic to an open set in a product [0, ∞) n × [0) k , and thus is covered by local charts consisting of "basic coordinates" r 1 , . . . , r n on the underlying manifold with corners, and "phantom coordinates" t 1 , . . . , t k that keep track of normal directions. 1.2.2. Log morphisms and tangential basepoints. Log geometry gives a natural notion of morphism between manifolds with log corners, expressed in terms of sheaves of monoids. For the basic case of manifolds with corners, such morphisms have a very concrete description: they are the maps that, when written in coordinates, are locally monomial near the boundary. The importance of such maps in differential geometry and geometric analysis was articulated by Melrose, who called them "interior b-maps" [START_REF] Melrose | Calculus of conormal distributions on manifolds with corners[END_REF]. A basic feature of these maps, as the name suggests, is that they preserve interiors of manifolds with corners; in particular, morphisms * → Σ, where * is a single point, are in bijection with points in the interior Σ \ ∂ Σ.

One of our key discoveries is that by a simple weakening of the axioms for a morphism in log geometry, we obtain a more flexible notion of "weak morphism" which also allows points to land in the boundary; however, when they do so, they automatically come decorated with positive normal vectors-a C ∞ counterpart of Deligne's tangential basepoints. In fact, tangential basepoints are the same thing as weak morphisms from a point: Proposition 1.2 (see Proposition 4.8). For Σ a manifold with corners viewed as a manifold with log corners, weak morphisms * → Σ are in bijection with tangential basepoints of Σ.

The category of manifolds with log corners and weak morphisms is the natural venue for our theory of integration. Interestingly, even in the context of logarithmic algebraic geometry, the notion of weak morphism seems to be new. We will explain in the forthcoming article [START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF] that it retains many good functoriality properties.

1.2.3. Functions and forms. In Section 5 we develop a theory of functions with logarithmic singularities. We construct, for each manifold with log corners Σ, a sheaf C ∞,log Σ of "logarithmic functions". This is done purely algebraically, by a simple generators-and-relations presentation, in which we formally adjoin logarithms for the elements of M Σ , subject to two natural relations. The first relation is the obvious identity log(f g) = log(f ) + log(g). The second relation identifies a formal symbol f log(g) with the corresponding function on Σ, provided that the latter is everywhere smooth; this ensures that we do not "overcount" the smooth functions.

This presentation makes it easy to show that the sheaf C ∞,log is functorial for weak morphisms, which will be crucial for our geometric interpretation of regularized integrals. However, it obscures its analytic meaning. To this end, we prove that this sheaf has the following classical description: Theorem 1.3 (see Theorem 5.10). The sheaf C ∞,log Σ is identified with the algebra of functions that in any system of basic coordinates r 1 , . . . , r n and phantom coordinates t 1 , . . . , t k on Σ, admit finite expansions of the form

f = I,J f I,J (r) log I (r) log J (t)
where I, J are multi-indices and f I,J are smooth functions.

The crucial subtlety here-which is why there is something nontrivial to proveis that the expansion in log(r) is not unique, due to the possibility of smooth function coefficients that are infinitely flat at the boundary. The theorem shows that this ambiguity is completely captured by the elementary relation defining C ∞,log Σ , so that we can manipute the formal expansions as functions in the obvious way.

In the absence of phantom coordinates, the sections of C ∞,log

Σ

are examples of polyhomogeneous functions in the sense of Melrose [START_REF] Melrose | Calculus of conormal distributions on manifolds with corners[END_REF]. Meanwhile the symbols log J (t) are monomials in the formal logarithms in the phantom coordinates, which are related to Melrose's polyhomogeneous symbols; they keep track of the singular terms that arise when attempting to restrict polyhomogeneous functions to strata.

Even though functions may diverge on the boundary, we can assign finite values at "points", simply by pulling them back along weak morphisms * → Σ. As we explain in Section 5, this simple prescription exactly encapsulates the classical approach of choosing a tangential basepoint, and using it to define the regularized limit by discarding divergent terms as in (1) above. 1.2.4. Differential forms and de Rham cohomology. In a similar way, we obtain a functorial sheaf of differential forms A •,log Σ with logarithmic singularities; it is the smallest extension of the dg algebra of smooth forms which contains all logarithmic functions A 0,log Σ := C ∞,log Σ , and hence also their differentials. In local coordinates, the latter have a basis given by the elements d log(r i ) = dri ri and d log(t j ) = dtj tj . We prove the following logarithmic version of the Poincaré lemma and de Rham isomorphism, giving a topological interpretation for the cohomology of A •,log Σ . Theorem 1.4 (see Sections 6.3 and 6.4). The natural map R Σ ֒→ A •,log Σ is a quasiisomorphism of complexes of sheaves, and hence we have canonical isomorphisms

H • sing (Σ; R) ∼ = H • dR (Σ) where H • dR (Σ)
is the cohomology of the complex of global logarithmic forms. Similar isomorphisms hold more generally for relative cohomology, with or without compact supports. In particular, logarithmic de Rham cohomology is homotopy invariant and satisfies the Künneth formula.

The case of relative cohomology is particularly important, since in practice, domains of integration often have a boundary. 1.2.5. Integration. In Section 7 we explain how our de Rham isomorphism above can be used to construct a natural and cohomologically meaningful theory of integration on manifolds with log corners.

Integration of forms in ordinary differential geometry requires an orientation to sort out the signs. In the logarithmic setting, we require the additional data of a regularization of the manifold with log corners to control the divergences. In classical terms, this amounts to the data s of a mutually compatible collection of non-negative, locally monomial sections of the normal bundles of the faces, which allows for quite a lot of variation in the qualitative behaviour. We package this succinctly as a collection of weak morphisms respecting the natural combinatorics of the boundary (that of a "symmetric semi-simplicial set"); see Definition 7.1. It is then immediate that if (Σ, s) is a regularized manifold with log corners, its boundary ∂Σ comes with a canonical regularization ∂s. We establish the following.

Theorem/Definition 1.5 (see Section 7). There is a unique collection of functionals on compactly supported log forms

(Σ,s) : A n,log c (Σ) → R,
one for each oriented and regularized manifold with log corners (Σ, s) whose underlying manifold has dimension n 0, which reduces to the ordinary integral whenever the latter converges absolutely, and which satisfies the regularized Stokes formula

(Σ,s) dα = (∂ Σ,∂s) α.
We then show by a straightforward calculation in coordinates that (Σ,s) reduces to the classical regularized integral, defined in terms of regularized limits of asymptotic expansions relative to tangential basepoints. Calculations such as the integrals I 1 and I 2 (from Section 1.1 above) then become direct applications of the definition and Stokes' formula, as desired.

By construction, the regularized integral descends to logarithmic de Rham cohomology, where it induces the Alexander-Lefschetz-Poincaré duality pairing between absolute cohomology, and cohomology relative to the boundary. Note that this pairing is purely topological, and hence independent of the choice of regularization; this is reflected in the fact that the relative cohomology can be computed using logarithmic forms that vanish on the boundary. Such forms are absolutely integrable, even if they are not smooth, so that their integral is independent of regularization.

1.2.6. Regularized periods in algebraic geometry. We conclude in Section 8 by connecting our setup with the study of periods in algebraic geometry, i.e. integrals of algebraic differential forms over topological cycles.

There are natural algebro-geometric analogues of manifold with log corners, which we call varieties with log corners; these are the logarithmic algebraic varieties over C that arise as strata of normal crossing divisors in smooth varieties. The connection between these two worlds is obtained via Kato-Nakayama's construction [START_REF] Kato | Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C[END_REF] of a topological space KN(X) associated to a log scheme X over the complex numbers. In op. cit., KN(X) is viewed as a topological space with no additional structure, but as explained by Gillam-Molcho [START_REF] Gillam | Logarithmic differentiable spaces and manifolds with corners[END_REF], it actually comes equipped with a natural positive log structure. We explain in [START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF] that it is functorial for our new notion of weak morphisms.

For the basic case in which X = Y log D is the log scheme associated to a smooth variety Y equipped with a normal crossing divisor D, the resulting space KN(X) is the real-oriented blowup of (the analytification of) Y along D. The result is a manifold with corners whose boundary components are circle bundles over the irreducible components of D, viewed as a basic example of a manifold with log corners with no phantoms.

In general, one may also have phantom directions in a variety with log corners; the prototype is the "standard log point" given by the induced log structure on the origin in A 1 log {0}. Its Kato-Nakayama space is the manifold with log corners S 1 × [0), with basic and phantom coordinates corresponding to the pullback of angular and radial coordinates on the complex line A 1 (C) = C, respectively. We have the following.

Theorem 1.6 (see Sections 8.2 and 8.3). The Kato-Nakayama space construction defines a functor X → KN(X) from the category of varieties with log corners over C and weak morphisms, to the category of manifolds with log corners and weak morphisms. If, in addition, X is defined over R, then its set of real points lifts to a natural embedded submanifold with log corners KN R (X) ⊂ KN(X).

As an immediate consequence, there is a natural isomorphism

H • dR (X) → H • dR (KN(X)) ⊗ R C
where H • dR (X) is the algebraic de Rham cohomology of Kato-Nakayama [KN99] defined using logarithmic algebraic differential forms, and H • dR (KN(X)) is the cohomology of the forms with logarithmic singularities on the manifold with log corners Σ = KN(X) as above.

More generally, we may consider the cohomology of a diagram X • of varieties with log corners and weak morphisms-for instance the diagram of boundary inclusions of a normal crossing divisor, which gives rise to a relative cohomology group. Using this one could define an algebra of regularized periods as the numbers obtained by the natural pairing between singular (aka Betti) homology and algebraic de Rham cohomology for diagrams of varieties with log corners defined over Q. We expect that this algebra (and its "motivic" counterpart) equals the ring of ordinary (non-regularized) periods defined by Kontsevich-Zagier [START_REF] Kontsevich | Periods, Mathematics unlimited-2001 and beyond[END_REF].

For example, as we explain in Section 8.7, the integrals I 1 and I 2 above can both be viewed as periods of varieties with log corners X and Y where the domains of integration are cells of the real loci KN R (X) and KN R (Y ). We further explain an alternative approach to the single-valued integration and the double-copy formula from [START_REF] Brown | Single-valued integration and double copy[END_REF], which gives a complementary viewpoint on the resulting motivic relation between log a and 2πi log |a| 2 .

In future work, we will explain how the construction of real Kato-Nakayama spaces from log structures on the moduli space of stable genus zero curves can be used to give motivic meaning to the integrals appearing in deformation quantization, and their evaluation in terms of multiple zeta values [START_REF] Banks | Multiple zeta values in deformation quantization[END_REF]. In this way, we will explain that the subtle "weight drop" phenomenon observed analytically in op. cit. is, in fact, a consequence of the geometry (via mixed Hodge theory), and realize the aforementioned motivic Galois action.

1.3. Relation to other work. Many authors have approached regularized integration from various points of view. For instance, Felder-Kazhdan [START_REF] Felder | Divergent integrals, residues of Dolbeault forms, and asymptotic Riemann mappings[END_REF][START_REF]Regularization of divergent integrals[END_REF] and Li-Zhou [START_REF] Li | Regularized integrals on Riemann surfaces and modular forms[END_REF][START_REF]Regularized integrals on elliptic curves and holomorphic anomaly equations[END_REF] have studied such integrals in the presence of a suitable conformal structure, by introducing cutoff functions and examining the asympotics of the integral as the cutoff tends to zero; they show explicitly that the "finite part" of the integral only depends on a trivialization of the outward-pointing normal bundle, via residues. Our regularization procedure gives the same definition and dependence on choices without the need for cutoff functions or direct asymptotic analysis of the integral. (The only place where asymptotics appear in our approach is to prove Theorem 1.3, which establishes the structure of the sheaf C ∞,log of logarithmic functions.)

In [START_REF] Brown | Multiple zeta values and periods of moduli spaces M 0,n[END_REF], Brown studied period integrals on the moduli space of genus zero curves. In the process, he proved a version of Stokes' theorem for certain forms with logarithmic singularities; it corresponds to the special case of our regularized Stokes' theorem in which the forms have no poles, and directly inspired our approach.

In [START_REF] Alekseev | Logarithms and deformation quantization[END_REF], Alekseev-Rossi-Torossian-Willwacher formulated a regularized version of Stokes' theorem for manifolds with corners, equipped with suitable torus actions near the boundary strata. This can be understood as the special case of our regularized Stokes formula, in which the regularization of Σ is chosen to be torusinvariant, and the form satisfies their "regularizability" criterion. They then applied their approach to the integrals appearing in deformation quantization mentioned above; as we will explain in [START_REF] Dupont | Motivic aspects of deformation quantization[END_REF], the torus action in these examples is the natural phase rotation on the corresponding Kato-Nakayama spaces.

While our paper was in preparation, Kato-Nakayama-Usui posted an interesting preprint [START_REF] Kato | Classifying spaces of degenerating mixed Hodge structures, VI: log real analytic functions and log C ∞ functions[END_REF], in which they study C ∞ logarithmic functions on log complex analytic spaces by combining the Kato-Nakayama spaces with an additional "space of ratios". While there are some formal similarities (e.g. a Poincaré lemma), the aims, results, and approach appear to be quite different.

Finally, our logarithmic de Rham theorem (Theorem 1.4) has subtly different analogues in other contexts. Firstly, there is an analogous result for holomorphic logarithmic de Rham complexes, due to Kato-Nakayama [START_REF] Kato | Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C[END_REF]. Their proof is essentially algebraic, relying on the fact that log(z) is algebraically independent from holomorphic functions, which fails in the C ∞ context. In contrast, our proof of Theorem 1.4 is quite close to the classical argument in differential geometry via contracting homotopies; this is made possible by our notion of weak morphisms. Secondly, in a different direction, Mazzeo-Melrose [Mel93, §2.16] computed the cohomology of smooth forms on a manifold with corners with log poles on the boundary (but no log divergences in the coefficient functions). This gives a different answer, since absent the function log r, the form dr r is not exact near the boundary.

Conventions and notation. All monoids are implicitly commutative and with a monoid law written multiplicatively. The main exception is the set N of natural numbers (i.e. non-negative integers, including zero), which we view as a monoid under addition.

Manifolds with corners

There are several approaches to manifolds with corners in the literature; in this section we review the basic definitions and conventions used in this paper.

2.1. Definitions. For subsets A ⊂ R m and B ⊂ R n we say that a map φ :

A → B is smooth if it extends to a smooth R m -valued function on an open neighbourhood of A ⊂ R n .
A diffeomorphism is a smooth map φ : A → B that admits a smooth inverse.

A chart (with corners) on a topological space W is a pair (U, φ) where U ⊂ W is an open set and φ : U → [0, ∞) n is a continuous map that induces a homeomorphism from U to an open subset of [0, ∞) n , for some n ∈ N. We typically denote the coordinate functions in such a chart by φ = (r 1 , . . . , r n ), as they measure the distance from the origin in each direction.

Two charts (U, φ) and (V, ψ) are said to be compatible if the transition function ψ • φ -1 : φ(U ∩ V ) → ψ(U ∩ V ) is a diffeomorphism. An atlas (with corners) on W is a set of charts that are pairwise compatible and whose union is W . Definition 2.1. A manifold with corners is a second countable Hausdorff space equipped with a maximal atlas (with corners).

For any point x in a manifold with corners W , there is a unique j 0 such that x lies in a chart (U, φ) for which φ(x) ∈ {0} j × (0, ∞) n-j , i.e.

r 1 (x) = • • • = r j (x) = 0 and r j+1 (x), . . . , r n (x) > 0. (2)
We call this integer j the depth of x.

The interior is the open set W • ⊂ W consisting of points of depth zero, i.e. for which all coordinates are positive. It is a smooth manifold (without corners).

Remark 2.2. Every manifold with corners is covered by open sets diffeomorphic to [0, ∞) j × R n-j for some 0 j n, since [0, ∞) n has a basis of such open sets. ♦ Definition 2.3. A smooth map between manifolds with corners is a map that is smooth in every chart. We denote by C ∞ W the sheaf of smooth R-valued functions on W , and by

C ∞,>0 W ⊂ C ∞, 0 W ⊂ C ∞
W the subsheaves of functions whose values are strictly positive, and non-negative, respectively.

Remark 2.4. There are different notions of smooth maps between manifolds with corners. (For instance, the notion we use here is called weakly smooth in [START_REF] Joyce | On manifolds with corners[END_REF].) By a theorem of Seeley [START_REF] Seeley | Extension of C ∞ functions defined in a half space[END_REF] (see also [START_REF] Melrose | Differential analysis on manifolds with corners[END_REF]), a function f : [0, ∞) n → R is smooth in the present sense if and only if it is smooth in the interior (0, ∞) n , with all partial derivatives continuous on [0, ∞) n . Equipped with the Fréchet topology, the set of such functions is a complete, locally convex topological vector space. ♦ 2.2. Tangent structure. For a manifold with corners W and a point x ∈ W , the tangent space T x W , the cotangent space T ∨

x W and the differential df | x ∈ T ∨ x W of a smooth function are defined in the usual way, via derivations of C ∞ W . On the boundary, only some vectors actually point into W ; we single them out as follows.

The non-negative tangent space of W at x is the closed subset

T 0 x W := v ∈ T x W v, df | x 0 for all f ∈ C ∞, 0 W,x such that f (x) = 0
Its boundary is a union of hyperplanes, called the boundary tangent hyperplanes at x. Its interior is the positive tangent space

T >0 x W ⊂ T 0 x W . In coordinates (r 1 , . . . , r n ) satisfying (2), a vector v = a 1 ∂ r1 | x + • • • + a n ∂ rn | x ∈ T x W
is nonnegative (resp. positive) if and only if the first j coefficients a 1 , . . . , a j are nonnegative (resp. positive), so that the coordinate basis gives identifications

T x W ∼ = R n , T 0 x W ∼ = [0, ∞) j × R n-j , T >0 x W ∼ = (0, ∞) j × R n-j .
The boundary tangent hyperplanes are identified with the first j coordinate hyperplanes of R n , i.e. the vanishing sets of the linear functionals dr 1 | x , . . . , dr j | x ∈ T ∨

x W . The tangent face of W at x is the intersection of all boundary tangent hypersurfaces at x, given by

F x W = v ∈ T x W v, df | x = 0 for all f ∈ C ∞, 0 W,x such that f (x) = 0 (a) [0, ∞) 2 (b) ∂[0, ∞) 2 Figure 1. The boundary of the quadrant [0, ∞) 2 is the disjoint union of two copies of [0, ∞).
The quotient N x W := T x W/F x W is called the normal space of W at x. The positive and non-negative normal spaces N >0

x W ⊂ N 0 x W ⊂ N x W are the projections of the corresponding subsets of T x W . The coordinate basis above gives

F x W ∼ = {0} j × R n-j
and isomorphisms

N x W ∼ = R j , N 0 x W ∼ = [0, ∞) j , and N >0 x W ∼ = (0, ∞) j . Note that when x ∈ W • is an interior point, we have N x W = N 0 x = N >0 x = {0}
, so that in this case only, the zero vector is "positive". 2.3. Boundary. The topological boundary of a manifold with corners W is the complement of the interior ∂ top W := W \ W • . It does not naturally have the structure of a manifold with corners in general.

Rather, the correct notion of boundary has the following local picture. In the orthant [0, ∞) n with coordinates (r 1 , . . . , r n ), we define the boundary ∂[0, ∞) n as the disjoint union of the n coordinate suborthants

{r j = 0} ∼ = [0, ∞) n-1 : ∂[0, ∞) n = n j=1 [0, ∞) j-1 × {0} × [0, ∞) n-j . The natural map i : ∂[0, ∞) n → [0, ∞) n
is an immersion of manifolds with corners that fails to be injective if n > 0, since the origin of [0, ∞) n has n preimages by i; see Figure 1.

The boundary ∂W of a general manifold with corners is obtained by glueing this local construction. It is a manifold with corners equipped with an immersion i : ∂W → W.

One can view ∂W as the set of all pairs (x, b) where x ∈ W and b is a boundary tangent hyperplane of W at x. Note that i need not be injective on connected components of ∂W , as shown by the example of the "teardrop manifold" in Figure 2.

Iterating, we obtain for each k > 0 a manifold with corners ∂ k W := ∂(∂ k-1 W ), whose points are pairs (x, (b 1 , . . . , b k )) consisting of a point x ∈ W and an ordered list (b 1 , . . . , b k ) of pairwise distinct boundary tangent hyperplanes of W at x. There is a free action of the symmetric group S k on ∂ k W which permutes the boundary tangent hyperplanes b j , and there are k smooth maps ∂ k W → ∂ k-1 W which forget one of the b j . This structure makes the collection ∂ • W into a symmetric ∆-object (in the sense of [START_REF] Chan | Tropical curves, graph complexes, and top weight cohomology of Mg[END_REF]) in the category of manifolds with corners.

, . . . , b k }) ∈ (∂ k W ) • with the tangent face F x W ⊂ T x W .
3. Manifolds with log corners 3.1. Positive log structures.

3.1.1. Definitions. Let Σ be a manifold with corners. Note that the subsheaves of positive and non-negative functions

C ∞,>0 Σ ⊂ C ∞, 0 Σ ⊂ C ∞
Σ are sheaves of submonoids with respect to the operation of multiplication, and that C ∞,>0 Σ is the subgroup of invertible sections of the monoid C ∞, 0 Σ . Definition 3.1. A positive pre-logarithmic (or pre-log ) structure on Σ is a sheaf of monoids M Σ on Σ along with a morphism of sheaves of monoids

α : M Σ → C ∞, 0 Σ . It is called a positive logarithmic (or log ) structure if the induced morphism α -1 (C ∞,>0 Σ ) → C ∞,>0
Σ is an isomorphism.

We will often abuse notation and simply write Σ for a triple (Σ, M Σ , α) consisting of a manifold with corners equipped with a (pre-)log structure. When we want to distinguish between such an object and the underlying manifold with corners, we will use the notation Σ for the latter.

Example 3.2. If Σ is a manifold with corners, then taking

M Σ = C ∞,>0 Σ with α : M Σ ֒→ C ∞, 0 Σ
the inclusion, we obtain a positive log structure on Σ, which we call the trivial positive log structure. ♦ Example 3.3. On the manifold with corners [0, ∞) with standard coordinate r, we define a sheaf of submonoids ) as the functions which can be written as a positive smooth function times a non-negative integer power of r. One readily checks that the inclusion

M [0,∞) := C ∞,>0 [0,∞) r N ⊂ C ∞, 0 [0,∞
α : C ∞,>0 [0,∞) r N ֒→ C ∞, 0 [0,∞)
defines a positive log structure (M [0,∞) , α) on [0, ∞). We refer to [0, ∞) with this positive log structure as the standard half-open interval . ♦

For a positive log structure, we will tacitly identify C ∞,>0

Σ with the submonoid

α -1 (C ∞,>0 Σ ) ⊂ M Σ , viewing α as a factorization of the inclusion C ∞,>0 Σ ֒→ C ∞, 0 Σ : C ∞,>0 Σ ֒→ M Σ α -→ C ∞, 0 Σ .
Every positive pre-log structure has an associated positive log structure M log Σ defined as the pushout

M log Σ := M Σ ⊔ α -1 (C ∞,>0 Σ ) C ∞,>0
Σ with the induced morphism

α log : M log Σ → C ∞, 0 Σ
; see [Ogu18, §I.1.1 and §III.1.1.3] for the analogous construction in the algebro-geometric setting. This will be useful for the constructions below.

3.1.2. Pullback and restriction. Let Ψ = (Ψ, M Ψ , α Ψ ) be a manifold with corners equipped with a positive log structure, let Σ be a manifold with corners, and let φ : Σ → Ψ be a smooth map. The composition

φ -1 M Ψ φ -1 C ∞, 0 Ψ C ∞, 0 Σ αΨ φ *
defines a positive pre-log structure on Σ, where the second map is the pullback of smooth functions along φ. Its associated log structure is denoted by φ * (M Ψ , α Ψ ) and called the pullback of the positive log structure. When φ is an immersion, we will refer to this operation as restriction. We will implicitly equip any open subset U ⊂ Ψ with the restricted positive log structure.

Example 3.4. The point {0} ⊂ [0, ∞), equipped with the restriction of the positive log structure from Example 3.3, is called the standard end and denoted by the

symbol [0) = ({0}, M [0) , α [0) ). It behaves somewhat like a tubular neighbourhood of 0 ∈ [0, ∞). Let t := r| [0)] ∈ M [0)
denote the restriction of the standard coordinate on [0, ∞). Then we have M [0) = R >0 t N , the product of R >0 with the free monoid generated by t, and α [0) is the morphism of monoids defined by "evaluation at t = 0":

α [0) : R >0 t N → R 0 λt j → λ0 j := λ if j = 0 0 if j > 0. ♦ 3.1.3. Products. Let Σ = (Σ, M Σ , α Σ ) and Ψ = (Ψ, M Ψ , α Ψ )
be manifolds with corners equipped with positive log structures. Their product Σ × Ψ is the manifold with corners Σ × Ψ is equipped with the positive log structure associated to the positive pre-log structure

p -1 Σ M Σ × p -1 Ψ M Ψ -→ C ∞, 0 Σ×Ψ , (f, g) → p * Σ α Σ (f ) • p * Ψ α Ψ (g)
, where p Σ and p Ψ denote the projections from Σ × Ψ to Σ and Ψ respectively.

Example 3.5. Let n, k ∈ N. The standard log corner of dimension (n, k) is the product [0, ∞) n × [0) k . Explicitly, the positive log structure M [0,∞) n ×[0) k is
defined as follows. Let r 1 , . . . , r n denote the standard coordinates on [0, ∞) n , and let t 1 , . . . , t k be additional formal variables. We define

M [0,∞) n ×[0) k := C ∞,>0 [0,∞) n r N • t N
as the product of the constant monoid t

N := t N 1 • • • t N k of monomials in the variables t 1 , . . . , t k with the subsheaf of monoids of C ∞, 0
[0,∞) n consisting of functions which can be written as a positive smooth function times a monomial in r

N := r N 1 • • • r N n . The morphism of sheaves of monoids α : C ∞,>0 [0,∞) n r N • t N -→ C ∞, 0 [0,∞) n
is defined as the identity on functions of r, and sends each t i to 0. ♦ 3.2. Manifolds with log corners. We now introduce our main objects of study.

3.2.1. Definition. Let Σ = (Σ, M Σ , α) be a manifold with corners equipped with a positive log structure.

Definition 3.6. A chart of dimension (n, k) on Σ is an isomorphism from an open set U ⊂ Σ to an open set in the standard log corner [0, ∞) n × [0) k .
We say that Σ is a manifold with log corners if every point x ∈ Σ is contained in the domain of a chart of dimension (n, k) for some n, k ∈ N (which may depend on x).

If all charts have the same dimension (n, k), which is automatic if Σ is connected, we say that Σ itself has dimension (n, k).

Thus a manifold with log corners of dimension (n, k) is covered by charts consisting of basic coordinates r 1 , . . . , r n ∈ C ∞, 0 Σ on the underlying manifold with corners, and phantom coordinates t 1 , . . . , t k ∈ M Σ with α(t j ) = 0.

Example 3.7. Every connected manifold with log corners of dimension (0

, k) is isomorphic to [0) k . ♦ Example 3.8. If Σ 1 , Σ 2 are manifolds with log corners of dimensions (n 1 , k 1 ) and (n 2 , k 2 ), then their product Σ 1 × Σ 2 is a manifold with log corners of dimension (n 1 + n 2 , k 1 + k 2 ). ♦
Remark 3.9. A manifold with corners equipped with the trivial log structure (Example 3.2) is not a manifold with log corners, unless it is a manifold (i.e., without corners). Indeed, in our local model [0, ∞) n the log structure is non trivial along the boundary. ♦

(Ordinary) morphisms.

There is an obvious notion of morphism between manifolds with log corners, identical to the corresponding notion in logarithmic algebraic geometry. Since we will later need a weaker notion of morphism, we will sometimes call the usual morphisms "ordinary" to distinguish them from the more general "weak" morphisms defined below in Section 4.

Definition 3.10. Let Σ = (Σ, M Σ , α Σ ) and Ψ = (Ψ, M Ψ , α Ψ ) be manifolds with log corners. An (ordinary) morphism φ : Σ → Ψ is a pair (φ, φ * ), where φ : Σ → Ψ is a smooth map and φ * : φ -1 M Ψ → M Σ is a morphism of sheaves of monoids such the following diagram commutes, where the bottom horizontal arrow is the usual pullback of smooth functions along φ.

φ -1 M Ψ M Σ φ -1 C ∞, 0 Ψ C ∞, 0 Σ φ * αΨ αΣ φ *
Example 3.11. An example of morphism of manifolds with log corners is the in-

clusion i : [0) ֒→ [0, ∞) of the standard end (Example 3.4) inside the standard half-open interval (Example 3.3). Concretely, the smooth map i is the inclusion of {0} inside [0, ∞)
, and the morphism of sheaves of monoids is

i * : i -1 C ∞,>0 [0,∞) r N -→ R >0 t N g(r)r j -→ g(0)t j ,
which picks out the leading term in the Taylor expansion of a function at r = 0. ♦ Example 3.12. More generally, consider the standard corner [0, ∞) n with coordinates r 1 , . . . , r n . We have a morphism of manifolds with log corners

i : [0, ∞) n-1 × [0) ֒→ [0, ∞) n
whose underlying smooth map i is the inclusion of [0, ∞) n-1 as the locus {r n = 0} in [0, ∞) n and whose morphism of sheaves of monoids extracts the leading Taylor monomial in r n :

i * : i -1 C ∞,>0 [0,∞) n r N 1 • • • r N n -→ C ∞,>0 [0,∞) n-1 r N 1 • • • r N n-1 • t N g(r 1 , . . . , r n )r j1 1 • • • r jn n -→ g(r 1 , . . . , r n-1 , 0)r j1 1 • • • r jn-1 n-1 t jn .

More generally, we have inclusion morphisms of manifolds with log corners

i : [0, ∞) n × [0) k ֒→ [0, ∞) n+j × [0) k-j for 0 j k. ♦ Example 3.13. If Σ 1 , Σ 2 are manifolds with log corners, then the projections p Σ1 : Σ 1 × Σ 2 → Σ 1 and p Σ2 : Σ 1 × Σ 2 → Σ 2 are
: U → [0, ∞) n on Σ in the sense of Definition 2.1 is a chart of dimension (n, 0) in the sense of Definition 3.6. More pre- cisely, let M Σ bas ⊂ C ∞, 0 Σ
be the subsheaf of monoids consisting of functions that can be written in local coordinates (r 1 , . . . , r n ) in the form g(r)r j1 1 • • • r jn n where g ∈ C ∞,>0 Σ and j 1 , . . . , j n ∈ N, and let α Σ bas : M Σ bas → C ∞, 0 Σ be the natural inclusion.

Definition 3.14. The manifold with log corners Σ bas = (Σ, M Σ bas , α Σ bas ) is the basic manifold with log corners associated to Σ.

Note that if a smooth map φ : Σ → Ψ of manifolds with corners lifts to an ordinary morphism Σ bas → Ψ bas , it does so in at most one way; this occurs if and only if, when expressed in local coordinates, φ has the form

φ : (r 1 , . . . , r n ) -→ f 1 n i=1 r ji,1 i , . . . , f m n i=1 r ji,m i
for some strictly positive smooth functions f 1 , . . . , f m of (r 1 , . . . , r n ) and exponents (j i,l ) ∈ N n×m , i.e. it is an "interior b-map" in the sense of Melrose [START_REF] Melrose | Calculus of conormal distributions on manifolds with corners[END_REF]. Note that in this case, φ preserves interiors, in the sense that φ(Σ • ) ⊂ Ψ • .

3.2.4.

Tangent structure of a manifold with log corners. If x ∈ Σ is a point in a manifold with log corners, its non-negative tangent space is the manifold with log corners T 0

x Σ whose underlying manifold with corners is the nonnegative tangent space T 0

x Σ, and whose positive log structure is the pullback of the positive log structure on Σ via any open embedding T 0 Σ ֒→ Σ with linearization equal to the identity. This construction is evidently functorial: for φ : Σ → Ψ a morphism of manifolds with log corners, the derivative of φ gives at x gives a morphism

dφ x : T 0 x Σ → T 0 φ(x) Ψ. In particular, a chart of dimension (n, k) containing x gives an isomorphism T 0 x Σ ∼ = [0, ∞) n × [0) k of manifolds with log corners, whose coordinates correspond to the differentials dr 1 | x , . . . , dr n | x and dt 1 | x , . . . , dt k | x .
3.3. The global structure of manifolds with log corners.

3.3.1. "Basic" and "phantom" sections. For a manifold with log corners Σ, there is a fundamental dicohotomy in the behaviour of sections f ∈ M Σ , generalizing the dicohotomy between the basic coordinates r i and the phantom coordinates t j : Definition 3.15. Let Σ = (Σ, M Σ , α) be a manifold with log corners. A germ f of a section of M Σ is called basic if α(f ) = 0 and a phantom if α(f ) = 0. We denote by M bas Σ , M phan Σ ⊂ M Σ the subsheaves of basic and phantom sections.

Example 3.16. For the standard end [0) from Example 3.4, an element of M [0) has the form λt j with λ ∈ R >0 and j ∈ N. It is basic if j = 0 and phantom if j > 0, so that

M bas [0) = R >0 and M phan [0)
is the monoid ideal of M [0) generated by t. ♦

An important subtlety in the definition is that the condition α(f ) = 0 refers to the germ of the function α(f ), not the value at a point; thus it could well happen that the germs of α(f ) are everywhere nonzero, so that f is basic, even though the function α(f ) has a nonempty vanishing set.

Example 3.17. For the standard interval half-open interval [0, ∞) from Example 3.3, a section of M [0,∞) has the form g(r)r j with g a positive smooth function and j ∈ N. Even though such a function vanishes at the origin when j > 0, its germ is everywhere nonzero. Therefore,

M bas [0,∞) = M [0,∞) and M phan [0,∞) = ∅. ♦
Note that by definition, the stalk of M Σ at any point decomposes as the disjoint union of the sets of basic and phantom elements, i.e. we have a decomposition

M Σ = M bas Σ ⊔ M phan Σ
of sheaves of sets. In local coordinates (r, t), a phantom section is one that is a multiple of some phantom coordinate t j , while the basic sections are given by

M bas [0,∞) n ×[0) k = C ∞,>0 [0,∞) n r N . Therefore M bas Σ ⊂ M Σ is a sheaf of submonoids while M phan Σ ⊂ M Σ is a sheaf of monoid ideals.
For a manifold with corners Σ, the associated basic manifold with log corners Σ bas (Definition 3.14) does not have phantoms, i.e. the sheaf M phan Σ bas is empty. On the other hand, if Σ is any manifold with log corners, then it is clear from the definition of a chart that the subsheaf M bas Σ ⊂ M Σ of basic elements depends only on the underlying manifold with corners, and is canonically identified with M Σ bas via α. Hence there is a canonical ordinary morphism Σ → Σ bas to the underlying basic manifold with log corners. In local coordinates, it is simply the projection

[0, ∞) n × [0) k → [0, ∞) n . Definition 3.18. A manifold with log corners is basic if M bas Σ = M Σ , or equiva- lently the canonical map Σ → Σ bas is an isomorphism. 3.3.2.
The boundary of a manifold with log corners. Let Σ = (Σ, M Σ , α Σ ) be a manifold with log corners. Then the pullback of the positive log structure on Σ to the boundary ∂ Σ (defined as in Section 2.3) gives the latter the structure of a manifold with log corners, and makes the boundary immersion into a morphism

i : ∂Σ → Σ.
whose local picture is given by the following example. Note that if Σ has dimension (n, k), then ∂Σ has dimension (n -1, k + 1); in particular, the boundary is not basic unless it is empty.

Example 3.19. For Σ = [0, ∞) the standard half-open interval (Example 3.3), the positive log structure on ∂[0, ∞) is the standard end [0). More generally, one checks that for the standard log corner [0, ∞) n × [0) k we have an identification

∂([0, ∞) n × [0) k ) = n j=1 [0, ∞) j-1 × [0) × [0, ∞) n-j × [0) k , which is the disjoint union of n copies of [0, ∞) n-1 × [0) k+1 . ♦
Iterating this construction as in Section 2.3, we obtain manifolds with log corners

∂ k Σ and ∂ k Σ = ∂ k Σ/S k
for k 0, so that every face of Σ is also a manifold with log corners. In fact, this structure depends only on the normal bundle of ∂ k Σ, in a sense that we will now make precise.

We have canonical immersions

i : ∂ k Σ → Σ i 0 : ∂ k Σ → N 0
where N 0 := i * T 0 0 Σ/T∂ k Σ denotes the non-negative normal bundle of the immersion i, viewed as a manifold with corners equipped with the basic positive log structure, and i 0 is the zero section. We thus have a second positive log structure on ∂ k Σ, given by the pullback along i 0 :

M ∂ k Σ := i * 0 M N 0
Note that sections of M ∂ k Σ are identified with fibrewise monomial functions on the normal bundle, with coefficients in the basic functions M bas ∂ k Σ . Furthermore, note that if f ∈ M Σ is a locally monomial function on Σ, the leading term in its Taylor expansion along ∂ k Σ is exactly such a locally monomial function on the normal space. This gives rise to a canonical map of positive pre-log structures

σ : i -1 M Σ → M ∂ k Σ (3)
which we call the symbol map. Concretely, if (r 1 , . . . , r n ) are coordinates such that i is locally identified with the embedding of the locus r 1 = • • • = r k = 0, then the symbol map is given by the formula

σ f (r 1 , . . . , r n ) r j1 1 • • • r j k k = f (0, . . . , 0, r k+1 , . . . , r n )(dr 1 ) j1 • • • (dr k ) j k
where f is a germ of a strictly positive function on Σ, and the differentials dr j are viewed as non-negative linear functions on the normal bundle. From this formula we deduce the following Proposition 3.20. For Σ = Σ bas basic, the symbol map induces an isomorphism

M ∂ k Σ ∼ = M ∂ k Σ .
of positive log structures, for all k 0. Hence the log structure on ∂ k Σ depends only on the non-negative normal bundle of ∂ k Σ in Σ.

Corollary 3.21. For every x ∈ Σ, the pullback log structure M Σ | x is canonically identified with the multiplicative monoid of non-negative monomial functions on the non-negative normal space N 0

x Σ. 3.3.3. The phantom tangent bundle. Proposition 3.20 shows that the manifolds with log corners that arise as boundary faces of a basic manifold with log corners have a special form: they are determined by a vector bundle (the normal bundle) equipped with a subbundle of orthants. Our aim now is to show that in fact, every manifold with log corners has such a form, and may thus be viewed as a union of faces of a basic manifold with log corners in a canonical way. For this, we need to replace the normal bundle of the embedding with the following intrinsic notion. Definition 3.22. Let Σ be a manifold with log corners, and let x ∈ Σ be a point. A phantom tangent vector at x is a map

v : M phan Σ,x → R that is M Σ,x -linear, in the sense that v(f g) = α(f )| x • v(g) for all f ∈ M Σ,x and g ∈ M phan Σ,x . A phantom tangent vector is non-negative (resp. positive) if it takes values in R 0 (resp. R >0 ).
The set of all phantom tangent vectors on Σ is naturally a vector bundle over Σ, which we call the phantom tangent bundle T phan Σ. The positive and nonnegative vectors give subbundles

T phan,>0 Σ ⊂ T phan, 0 Σ ⊂ T phan Σ with fibres isomorphic to (0, ∞) k ⊂ [0, ∞) k ⊂ R k . Concretely, if t 1 , . . . , t k ∈ M phan Σ,x
is a set of phantom coordinates we may define phantom tangent vectors ∂ t1 , . . . , ∂ t k ∈ T phan x Σ by M Σ,x -linear extension of the formula

∂ ti (t j ) = 1 i = j 0 i = j.
Using that M phan Σ,x is freely generated over M Σ bas ,x by t 1 , . . . , t k , it is straightforward to verify that these phantom tangent vectors are well-defined, and form a local basis for T phan Σ such that the positive (resp. non-negative) vectors are the positive (resp. non-negative) linear combinations of the basis elements.

In particular, T phan, 0 Σ is a manifold with corners, which we equip with its basic positive log structure. This has the effect of turning local phantom coordinates t 1 , . . . , t k around a point x ∈ Σ into actual coordinates on the fibres of T phan, 0 Σ. Consequently, the zero section lifts canonically to an embedding Σ → T phan, 0 Σ, identifying the positive log structure on Σ with the pullback of the basic positive log structure on T phan, 0 Σ. The global structure of a manifold with log corners is thus summarized by the following statement.

Proposition 3.23. Let Σ be a manifold with log corners. Then Σ is canonically identified with the zero section in the basic manifold with log corners T phan, 0 Σ, with the induced log structure, giving a commutative diagram

Σ T phan, 0 Σ Σ bas
Note that by construction, every transition function for T phan, 0 Σ has to map a phantom coordinate t j to λt j ′ for some λ > 0 and some index j ′ . We therefore get a reduction of the structure group of the vector bundle T phan Σ from GL k (R) to the subgroup S k ⋉ R k >0 generated by permutation matrices and diagonal matrices with positive entries. This means that locally, T phan Σ has a canonical decomposition as a sum of line bundles corresponding to the phantom coordinates t 1 , . . . , t k , but this decomposition need not be globally well-defined, as the following example shows.

Example 3.24. Let Σ 0 = R × [0) 2 and consider the Z-action on Σ 0 generated by the automorphism (x, t 1 , t 2 ) → (x + 1, t 2 , t 1 ), which acts as a translation on the underlying manifold, an swaps the phantom coordinates. Then Σ := Σ 0 /Z is a manifold with log corners whose underlying manifold is the circle R/Z, and for which the pair of lines decomposing the fibres of T phan Σ are interchanged as we go around the circle. ♦ 3.4. Fixed points of group actions. In this subsection, we briefly discuss the behaviour of compact Lie group actions on manifolds with log corners.

Definition 3.25. Let Σ be a manifold with log corners, and G a Lie group, which we view as a manifold with corners equipped with the trivial positive log structure.

An action of G on Σ is a morphism G × Σ → Σ of manifolds with log corners satisfying the unit and associativity conditions in the category of manifolds with log corners.

If x ∈ Σ is a G-fixed point, then by functoriality, we obtain a linear action

G × T 0 x Σ → T 0
x Σ on the non-negative tangent space, viewed as a manifold with log corners as in Section 3.2.4. When G is compact, this gives a local model for the action, thanks to the following mild extension of Bochner's linearization theorem [START_REF] Bochner | Compact groups of differentiable transformations[END_REF].

Theorem 3.26. If G is compact, there exists a G-equivariant isomorphism of manifolds with log corners from an open neighbourhood of the fixed point x ∈ Σ to an open neighbourhood of 0 ∈ T 0

x Σ, whose derivative at x is the identity map. Proof. The proof is a straightforward adaptation of Bochner's argument [START_REF] Bochner | Compact groups of differentiable transformations[END_REF] as presented in [DK00, Section 2.2]; we will simply indicate the necessary adjustments.

By functoriality, the action of G lifts to a fibre-wise linear action on the phantom tangent bundle preserving the embedding Σ ֒→ T phan, 0 Σ from Proposition 3.23. We may thus assume without loss of generality that Σ = Σ bas is basic.

From here, the proof prooceeds as in op. cit.. By compactness of G, there exists a G-invariant open neighbourhood U of x in Σ. Choose an arbitrary open embedding χ : U ֒→ T 0

x Σ whose derivative at x is the identity. Let χ g := gχg -1 be the conjugation of φ by the given action of g on U and the induced linear action on T 0

x Σ. We may view each map χ g as a vector in the vector space of smooth maps U → T x Σ. The latter is a complete locally convex topological vector space (cf. Remark 2.4), so that we may define the average χ := g∈G φ g with respect to the Haar measure on G. As in op. cit., χ is a smooth, G-invariant map U → T x Σ whose derivative at x is the identity. In addition, since T 0

x Σ ⊂ T x Σ and all of its boundary faces are preserved by non-negative linear combinations, we conclude that the image of χ is an open neighbourhood of the origin in T 0

x Σ. Thus χ gives the desired G-equivariant isomorphism.

Corollary 3.27. If G is compact, then near any fixed point, Σ is G-equivariantly isomorphic to a product R l × [0, ∞) n × [0) k with a diagonal G-action, where G acts linearly on R l , and acts by permutations of the coordinates on [0, ∞) n × [0) k .

Proof. Endowing the tangent space with an invariant inner product, we can decompose it as an orthogonal G-invariant direct sum of the tangent space of the stratum through x and its normal directions. The former is a copy of R m with a linear G-action, and the latter are grouped into basic and phantom directions for which the action reduces to permutations by Lemma 3.28 below.

Lemma 3.28. Let G ⊂ GL n+k (R) be a group of linear transformations of R n+k that restrict to automorphisms of the manifold with log corners

[0, ∞) n ×[0) k . Then G lies in the subgroup (S n × S k ) ⋉ R n+k
>0 generated by permutations of the first n and last k coordinates, and the diagonal matrices with positive eigenvalues. If, in addition, G is compact, then G is conjugate to a subgroup of S n × S k .

Proof. Let e 1 , . . . , e n+k ∈ R n+k be the standard basis. The action of G on R n+k must permute the one-dimensional boundary faces of [0, ∞) n+k , but the latter are exactly the rays spanned by the basis vectors. Hence any g ∈ G acts by g • e i = λ i e σ -1 (i) for some λ i > 0 and σ ∈ S n+k . Furthermore, the action must preserve the grouping of the coordinates into the n basic and k phantom coordinates, so that G < (S n × S k ) ⋉ R n+k >0 as claimed.

Now suppose that G is compact; we wish to show that G is conjugate to a subgroup of S n × S k . Since S n × S k is a subgroup of S n+k it is enough to treat the case k = 0. We first treat the case where the action of G on {1, . . . , n}, induced by the map G → S n , is transitive. In this case, for all i ∈ {1, . . . , n}, there exists an element g i ∈ G such that g i • e 1 = α i e i for some α i ∈ R >0 , where we assume that g 1 = id and α 1 = 1. Performing the change of basis given by e ′ i = α i e i now gives

g i • e ′ 1 = e ′ i for all i ∈ {1, . . . , n}. Let g ∈ G and let us write g • e ′ i = λe ′ j . Now the element g i g -1 j g ∈ G sends e ′ i to λe ′ i . Since G is a compact subgroup of S n ⋉ R n >0
, this element generates a compact subgroup and therefore λ = 1. We conclude that G acts by permuting the basis elements e ′ 1 , . . . , e ′ n . In other words, the conjugate subgroup αGα -1 is contained in S n . For the general case (with k = 0), the same argument on each orbit of the action of G on {1, . . . , n} gives the claim.

Corollary 3.29. The fixed point set Σ G ⊂ Σ is an embedded submanifold with corners.

Proof. By the previous corollary, the fixed point set decomposes locally as the product of a linear subspace of R l and diagonals in [0, ∞) n . The former is an ordinary smooth manifold, and the latter are products of corners, with coordinates given by restricting subsets of the coordinates on [0, ∞) n .

We equip the manifold with corners Σ G with a positive log structure M Σ G as follows. Let i : Σ G ֒→ Σ be the inclusion, and note that the pullback log structure i * M Σ carries a residual action of G by automorphisms, for which the morphism

α : i * M Σ → C ∞, 0 Σ G is invariant. We may
α : i * M [0,∞) n -→ C ∞, 0 Σ g(r)r j1 1 • • • r jn n -→ g(r)r j1+•••+jn
where r is the coordinate on [0, ∞) and g ∈ C ∞,>0 [0,∞) . This identifies the quotient i * M [0,∞) n /S n with the standard positive log structure on [0, ∞). ♦

Example 3.32. Consider the symmetric group S k acting on the manifold with log corners [0) k by permuting the (phantom) coordinates. The inclusion i of the fixedpoint set is the identity of {0}, and

i * M [0) k is the monoid consisting of monomials λt j1 1 • • • t j k k with λ ∈ R >0
and j 1 , . . . , j k ∈ N, with the action of S k permuting the (phantom) coordinates. The quotient i * M [0) k /S k is therefore identified with the standard positive log structure on [0). ♦

The following result shows that Σ G is the categorically correct version of the fixed point set.

Theorem 3.33. Let G be a compact Lie group acting on a manifold with log corners Σ. Then M Σ G gives Σ G the structure of a manifold with log corners and i : Σ G ֒→ Σ is a morphism of manifolds with log corners. Every G-invariant morphism Ψ → Σ of manifolds with log corners factors uniquely through i.

Proof. For any fixed point, we may choose a chart on Σ as in Corollary 3.27. The action of G therefore partitions the coordinates on [0, ∞) n × [0) k into orbits, and the first claim follows from Example 3.31 and Example 3.32.

For the universal property, note that any G-invariant map of manifolds with log corners must map entirely to fixed points, and therefore factor through the pullback positive log structure i * M Σ . But the map on monoids is also G-invariant, and hence it must further factor through (i * M Σ )/G. 4. Weak morphisms, tangential basepoints, and scales 4.1. Definition and examples. The notion of (ordinary) morphism between manifolds with log corners is too restrictive for some purposes, and we will need a weaker notion, in which we only require the commutativity condition from Definition 3.10 to hold for the positive smooth functions. Definition 4.1. Let Σ = (Σ, M Σ , α Σ ) and Ψ = (Ψ, M Ψ , α Ψ ) be manifolds with log corners. A weak morphism φ : Σ → Ψ is a pair (φ, φ * ) where φ : Σ → Ψ is a smooth map and φ * : φ -1 M Ψ → M Σ is a morphism of sheaves of monoids such the following diagram commutes.

φ -1 C ∞,>0 Ψ C ∞,>0 Σ φ -1 M Ψ M Σ φ * φ *
In other words, weak morphisms are characterized by the equality

φ * f = φ * f. ( 4
)
for all positive functions f ∈ C ∞,>0 Ψ . In fact, the equality holds more generally thanks to the following proposition, whose proof we delay to the end of this subsection: Proposition 4.2. If φ is a weak morphism then (4) holds for all f ∈ M bas Ψ such that φ * f = 0. In other words, we have

φ * α Ψ (f ) = α Σ (φ * f ) for all f ∈ M Ψ such that the left-hand side φ * α Ψ (f ) is not the zero function.
With the obvious notion of composition, weak morphisms form a category. Every ordinary morphism is a weak morphism, but not conversely. The following examples illustrate the key similarities and differences between these notions. is determined by a morphism of monoids s * : R >0 t N → R >0 which acts as the identity on R >0 , but may send the generator t to an arbitrary positive real number. Thus weak morphisms s : * → [0) are in bijection with R >0 , and are all (weak) sections of the unique morphism p : [0) → * . Note, however, that there are no ordinary morphisms from * to [0). Indeed, since α(t) = 0, an ordinary morphism would have to send t to a phantom in R >0 , of which there are none. ♦ Example 4.5. A weak morphism

q : [0, ∞) → [0)
is the datum of a morphism of monoids q * : R >0 t N → Γ([0, ∞), M [0,∞) ) which acts as the identity on the basic elements R >0 , but may send the generator t to an arbitrary element g(r)r j , with g a positive smooth function on [0, ∞) and j ∈ N. If g(0) = 1 and j = 1 then f is a (weak) retract of the ordinary morphism i : [0) ֒→ [0, ∞). Note, however, that there are no ordinary morphisms from [0, ∞)

to [0) because M [0,
* (f | V ) = φ * (f | V ), or in other words (5) (φ * f )| U = (φ * f )| U ,
so it suffices to show that U = Σ. Since Σ is connected and U is nonempty and open, this reduces to showing that U is closed. Note that since U is nonempty, φ * f is necessarily a basic global section of M Σ , hence strictly positive. (Thanks to the decomposition

M Σ = M bas Σ ⊔ M phan Σ
, being basic is a property that can be checked at a point in a connected manifold with log corners.) Now take a sequence x n → x in Σ where all x n ∈ U . By (5) we have, for all n that (φ

* f )(x n ) = (φ * f )(x n ),
and passing to the limit we get (φ * f )(x) = (φ * f )(x) > 0. Hence x ∈ U , and so U is closed, as desired.

4.2. Tangential basepoints. As we saw in Example 4.4, a point, equipped with the trivial log structure, does not admit any ordinary morphism to [0). More generally, a point does not admit any ordinary morphism to a manifold with log corners that has phantoms; this includes the boundary of a basic manifold with corners. However, it admits many weak morphisms to the boundary. As we now explain, these correspond a C ∞ analogue of Deligne's notion of a tangential basepoint in x Σ is a positive normal vector at x in the sense of Section 2.2.

Recall that if x lies in the interior of Σ, the set N >0

x Σ has a unique element; hence, in the interior, a tangential basepoint is just an ordinary point. However, the notion becomes nontrivial over the boundary: at a point x ∈ Σ of depth j, there is a j-dimensional space of tangential basepoints; see Figure 3. These can be viewed as "weak points" of Σ thanks to the following result. Proposition 4.8. Let Σ be a manifold with corners and let Σ = Σ bas be the associated basic manifold with log corners. Then there is a natural bijection between the set of weak morphisms * → Σ and the set of tangential basepoints of Σ.

Proof. Every weak morphism * → Σ factors uniquely through the pullback log structure on its underlying image point x ∈ Σ. (This is the standard universal property for pullback log structures, which works equally well in the context of weak morphisms.) But by Corollary 3.21 the pullback log structure M Σ | x is naturally isomorphic to the monoid of non-negative monomial functions on the non-negative normal space N 0

x Σ. Evaluating such functions at points in N >0

x Σ gives a bijection between tangential basepoints and monoid homomorphisms M Σ | x → R >0 that act as the identity on the constants R >0 ⊂ M Σ | x , or equivalently weak morphisms

* → (x, M Σ | x , α Σ | x ), as desired.
Concretely, in local coordinates (r 1 , . . . , r n ) such that r 1 (x) = • • • = r j (x) = 0 and r i (x) > 0 for i > j we may write a tangential basepoint as

v = v 1 ∂ r1 | x + • • • + v j ∂ rj | x
where v 1 , . . . , v j ∈ R >0 . The corresponding weak morphism to * → Σ bas is thus determined by the positive constants

s * r i = v i 1 i j r i (x) j < i n.
In light of this result we shall often denote the corresponding morphism s : * → Σ bas simply by

v i ∂ ri | x .
Remark 4.9. In the literature, it is common to consider paths between tangential basepoints v, w on Σ: these are usually defined as smooth maps γ : [0, 1] → Σ that send the interior (0, 1) to Σ • , and whose initial and final velocities in the normal directions are given by (6) γ ′ (0) = v and γ ′ (1) = -w.

This can be phrased in the language of weak morphisms as follows. Consider the interval [0, 1] with coordinate t, viewed as a basic manifold with corners, equipped with its two "canonical" tangential basepoints Note that (7) holds more generally if we only require the leading Taylor coefficients at 0 and 1 to be v and -w respectively. For instance, for (Σ, v, w) = ([0, 1], s 0 , s 1 ), this is the case for any smooth map γ : [0, 1] → [0, 1] sending (0, 1) to (0, 1) and satisfying γ(t) ∼ 0 t a and γ(t) ∼ 1 (1 -t) b for some a, b ∈ N * , even though only those with (a, b) = (1, 1) are classically referred to as paths between tangential basepoints. ♦ 4.3. Scales. There are natural higher-dimensional counterparts of tangential basepoints, given by sections of normal bundles of strata, or more intrinsically, the phantom tangent bundle. As we shall see, these correspond to the following notion.

Definition 4.10. A scale for a manifold with log corners Σ is a weak morphism s : Σ bas → Σ that is a section of the natural projection Σ → Σ bas . We say that s is nondegenerate if the pullback of every phantom function is strictly positive, i.e.

s * M phan Σ ⊂ C ∞,>0 Σ .
Concretely, in local coordinates (r, t), the sheaf M Σ is identified with the monoid M bas Σ t N freely generated by the phantom coordinates t 1 , . . . , t k over M bas Σ , and thus a scale is equivalent to the data of the k-tuple of nonnegative locally monomial functions

s * t 1 , . . . , s * t k ∈ M bas Σ ⊂ C ∞, 0 Σ . It is
: M phan Σ → M bas Σ ⊂ C ∞, 0 Σ
, which defines a non-negative section of the phantom tangent bundle. We thus have the following. Lemma 4.11. There is a natural bijection between scales (respectively, nondegenerate scales) on a manifold with log corners Σ, and morphisms of basic manifolds with log corners Σ bas → T phan, 0 Σ (resp. Σ bas → T phan,>0 Σ) that are sections of the natural projection.

This explains the terminology: a scale assigns a notion of "unit length vectors" in the phantom directions.

We will tacitly identify a scale with the corresponding phantom vector field, writing a scale with components s j (r) := s * t j ∈ M bas Σ in local coordinates as Proof. Recall that the bundle T phan,>0 Σ → Σ has structure group S k ⋉ R k >0 . Let P → Σ be the associated S k -bundle; its fibres are in bijection with the boundary tangent hyperplanes along the zero section in T phan, 0 Σ. Note that P is determined up to isomorphism by the monodromy representation π 1 (Σ) → S k , and hence there exists a finite cover π : Σ ′ → Σ such that π * P is trivial. The bundle π * T phan,>0 Σ then admits a further reduction of structure to the group (R k >0 , •) ∼ = (R k , +), and is thus classified by an element in the sheaf cohomology group H 1 (Σ ′ , C ∞ Σ ′ ) ⊕k = 0. Therefore π * T phan,>0 Σ is a trivial bundle with fibre (0, ∞) k , and hence it admits a section, say s ′ . Averaging s ′ over the action of S k , we may assume without loss of generality that s ′ is S k -invariant, and hence descends to a section of T phan,>0 Σ, giving the desired nondegenerate scale s. Definition 4.15. Let (Σ, s) and (Ψ, s) be manifolds with log corners equipped with scales. A weak morphism φ : Σ → Ψ is scale-preserving if there exists a weak morphism φ ′ : Σ bas → Ψ bas making the following diagram commute:

s = s j (
(9) Σ Ψ Σ bas Ψ bas . φ φ ′ s s
Clearly compositions of scale-preserving weak morphisms are scale-preserving. Note that the morphism φ ′ , if its exists, is unique, being given by the formula

φ ′ = p • φ • s where p : Ψ → Ψ bas is the canonical projection. Therefore φ is scale-preserving if and only if s • p • φ • s = φ • s.
Remark 4.16. Note that for a weak morphism φ : Σ → Ψ there rarely exists a weak morphism φ ′ : Σ bas → Ψ bas that fits into a commutative diagram with φ and the natural projections Σ → Σ bas and Ψ → Ψ bas . For instance, there is no weak morphism i ′ that makes the following diagram commute, where i is the inclusion and p the projection.

[0) [0, ∞) {0} [0, ∞) i p i ′
Indeed, i * sends the basic coordinate r to the phantom coordinate t whereas p * is the inclusion of R >0 inside R >0 t N and does not contain t in its image. ♦

Functions with logarithmic singularities

Let Σ be a manifold with log corners. In this section, we construct a natural sheaf C ∞,log Σ of functions with logarithmic singularities on Σ, by adding formal logarithms log(f ) for every f ∈ M Σ , in such a way that log(f ) agrees with the usual logarithm of the function α(f ) whenever the latter is not identically zero. To make the functoriality of the construction clear, we will start by defining C ∞,log Σ abstractly using generators and relations; we then spell out what this means concretely in local coordinates.

Throughout the present Section 5 we will denote formal logarithms by log(f ), reserving log(f ) for the logarithm of an actual function, in order to avoid confusion. This notation is temporary and from Section 6.1 onwards we shall simply write log(f ) for both the formal and actual logarithm; Theorem 5.10 below will guarantee that this does not introduce any ambiguity. 5.1. Construction and functoriality. Our sheaf of functions is defined as follows.

Definition 5.1. The sheaf of logarithmic functions on Σ is the sheaf C ∞,log Σ of C ∞ Σ -algebras generated by formal symbols log(f ) where f ∈ M Σ , subject to the relations:

(1) For every f 1 , f 2 ∈ M Σ , we have log(f 1 f 2 ) = log(f 1 ) + log(f 2 ).

(2) If f ∈ M bas Σ and g ∈ C ∞ Σ are such that g log(α(f )) is smooth on Σ, then g log(f ) = g log(α(f )),

where log(α(f )) denotes the real-valued logarithm of the nonnegative function α(f ).

Remark 5.2. By "g log(α(f )) is smooth on Σ", we mean the following: since f is basic, it is expressed in local coordiantes as f = g(r)r j1 1 • • • r jn n for some integers j i and a positive smooth function g. Therefore, g log α(f ) is a well-defined smooth function on the interior of Σ, and the condition is that this function extends as a smooth function on Σ. The right-hand side of the second relation above refers to this extension, as a section of C ∞ Σ (or rather its image in C ∞,log

Σ

). ♦

The meaning of the first relation is self-evident. The second relation ensures that we do not overcount the smooth functions. For instance, taking g = 1 and identifying a positive function f with the corresponding section of M Σ , we have the following.

Lemma 5.3. If f ∈ C ∞,>0 Σ , then log(f ) = log(f ) ∈ C ∞,log Σ .
The second relation in Definition 5.1 is more subtle when α(f ) has zeros, since then log(α(f )) is not smooth. For instance, we have the following useful property.

Lemma 5.4. Suppose that f ∈ M bas Σ and g ∈ C ∞ Σ are such that the function g log(α(f )) is continuous. If x ∈ Σ is any point such that the function α(f ) vanishes at x, then g and g log(α(f )) also vanish at x.

Proof. The problem is local, so we may work in a chart with coordinates (r, t). Then f has the form f = f 0 (r)r J where f 0 (r) is a positive smooth function and J = (j 1 , . . . , j n ) is a multi-index. In particular the vanishing set of f is the union of the boundary facets r i = 0 for indices i such that j i > 0, so it suffices to show that g also vanishes there. But the function

g log(α(f )) = g • log(f 0 ) + n i=1 j i log(r i ) .
is continuous, hence bounded, which implies that g → 0 as r i → 0, to compensate for the divergence of log(r i ). Hence, by Hadamard's lemma, g is divisible by r i in the algebra of smooth functions. Since r i log r i → 0 as r i → 0, it follows that g log α(f ) also vanishes in this limit.

Remark 5.5. If g log α(f ) is not just continuous, but actually smooth, then g must vanish to infinite order on the vanishing set of α(f ); see Proposition 5.12 below. ♦

The sheaves C ∞,log are functorial with respect to weak morphisms, in the following sense.

Lemma 5.6. If φ : Σ → Ψ is a weak morphism, then the formula

φ * ( log(f )) := log(φ * (f )) (f ∈ φ -1 M Ψ ) uniquely defines a morphism φ * : φ -1 C ∞,log Ψ → C ∞,log Σ of sheaves of φ -1 C ∞ Ψ -algebras.
Proof. Uniqueness is clear because the monoid elements generate C ∞,log over C ∞ . It remains to show that the pullback is well-defined, that is, we must show that the map φ * : φ -1 M Ψ → M Σ preserves the ideals generated by the two types of relations in Definition 5.1. Compatibility with the first relation is immediate since φ * is a monoid homomorphism. For the second relation, let x ∈ Σ be a point, let f ∈ φ -1 (M bas Ψ ) x = M bas Ψ,φ(x) be a germ of a section and let g ∈ C ∞ Ψ,φ(x) be a germ of a function such that g log(α Ψ (f )) is smooth on Ψ. We must show that

φ * g • log(φ * f ) = φ * (g log (α Ψ (f ))) ∈ C ∞,log Ψ,x . (10)
There are two possibilities: either the germ φ * (α Ψ (f )) is nonzero, or it is zero. If φ * (α Ψ (f )) is nonzero, then it is equal to α Σ (φ * f ) since φ is a weak morphism, and hence (10) is explicitly one of the defining relations for C ∞,log Σ,x . Otherwise, φ * (α Ψ (f )) = 0 says that φ maps a neighbourhood of x to the vanishing set of α Ψ (f ). Hence by Lemma 5.4, we have φ * g = φ * (g log α Ψ (f )) = 0, so that both sides of (10) are identically zero.

Example 5.7. As an illustration of the subtle point in the proof of the lemma, consider the weak morphism i : {0} → [0, ∞) corresponding to a tangential basepoint c ∂ r at 0, i.e. defined by i * (r) = c. Let g(r) be a smooth function on [0, ∞) such that g(r) log(r) is smooth. Then we have the relation

g(r) log(r) = g(r) log(r) in C ∞,log
[0,∞) and we need to prove that the relation

g(0) log(c) ? = g(r) log(r) r=0 is satisfied in C ∞,log {0} = R.
The fact that g(r) log(r) is smooth implies, thanks to Lemma 5.4, that both g(r) and g(r) log(r) vanish at r = 0, and therefore the latter relation reads 0 = 0. ♦ 5.2. Local structure of logarithmic functions. Our goal now is to prove Theorem 1.3 from the introduction, which describes the local structure of C ∞,log Σ . This follows from Lemma 5.9 and Theorem 5.10 below which treat the contributions from phantom and basic directions, respectively. 5.2.1. Phantom logarithms. After the logarithms of positive functions-which are just smooth functions-the next simplest elements of C ∞,log to understand are those of the form log f , where f is a phantom.

Example 5.8. Let Σ = [0) k with phantom coordinates t 1 , . . . , t k . Since M Σ = R >0 t N is freely generated over the positive constants by t 1 , . . . , t k , the logarithmic functions form a free commutative algebra generated by log t 1 , . . . , log t k , i.e.

C ∞,log Σ ∼ = R[ log t 1 , . . . , log t k ],
is a polynomial ring in the formal logarithms of the phantom coordinates. ♦ More generally, note that the projection p : Σ → Σ bas gives a canonical map

p * : C ∞,log Σ bas → C ∞,log Σ of C ∞ Σ -algebras.
We have the following: Lemma 5.9. If t 1 , . . . , t k is a system of phantom coordinates in a neighbourhood of any point x ∈ Σ, then p * gives a canonical isomorphism

C ∞,log Σ bas ,x [ log t 1 , . . . , log t k ] ∼ -→ C ∞,log Σ,x
of C ∞ Σ,x -algebras. Proof. By definition, M Σ is freely generated over M bas Σ by the phantom coordinates, hence the map is surjective. Since the second relation in Definition 5.1 only involves basic elements, the only relations involving t 1 , . . . , t k , are those of the first type, which define the monoid algebra of M Σ . The result follows since the monoid algebra of a free monoid is a polynomial ring. 5.2.2. Structure of the basic logarithms. In light of Lemma 5.9, it remains to understand the structure of the subalgebra C ∞,log Σ bas ⊆ C ∞,log Σ generated by basic elements. Therefore, we now assume without loss of generality that Σ = Σ bas is basic.

Let r 1 , . . . , r n be coordinates in a neighbourhood of x ∈ Σ, so that denotes the corresponding monomial in the formal logarithms of the coordinates. Our aim now is to show that the relations defining C ∞,log Σ bas amount to the statement that the formal symbols log r i can be identified with the actual logarithm functions log r i , so that the expressions f as above can be manipulated like ordinary functions of r; this is the content of Theorem 5.10 below but to formulate the statement we need to set some notation. Let j : Σ • ֒→ Σ be the inclusion of the interior, where all coordinates r i are positive. By functoriality, we have a canonical map of sheaves of algebras

M Σ,x = C ∞,>0 Σ,x
j * : C ∞,log Σ → j * C ∞,log Σ • ,
given by restriction of functions to the interior. Since Σ is assumed basic, we have

M Σ • = C ∞,>0 Σ •
, so we deduce from Lemma 5.3 that C ∞,log obtained by replacing each formal logarithm log r i with the corresponding smooth function log r i defined in the interior.

Σ • = C ∞ Σ • ,
Theorem 5.10. For a basic manifold with log corners Σ = Σ bas , the map j * is injective, and identifies C ∞,log Σ bas with the sheaf of smooth functions in the interior with at worst polynomially logarithmic divergences along the boundary, i.e. those which have the form (12) in some (and hence any) coordinate chart.

The rest of this subsection is devoted to the proof of Theorem 5.10. In fact we will formulate and prove a stronger statement (Proposition 5.12 below) that gives more precise control over the behaviour of the coefficients, allowing us to prove the theorem by induction on the number of boundary components.

The statement is local, so it suffices to prove it for the stalk at the origin in the manifolds with log corners

Σ m,n := R m × [0, ∞) n
for m, n 0, with coordinates r 1 , . . . , r n on the factor [0, ∞) n .

To formulate the stronger statement, we adopt the following terminology. If I = (I 1 , . . . , I n ) is a multi-index, its support is the collection of variables r l such that I l = 0, and its vanishing set V (I) = {I 1 r 1 = • • • = I n r n = 0} is the locus where all coordinates in the support vanish. If h ∈ C ∞ Σm,n,0 is a germ of a smooth function, we say that h is I-flat if we have

∂ a1 r1 • • • ∂ an rn h V (I) = 0
for every multi-index A = (a 1 , . . . , a n ) ∈ N n whose support is contained in that of I. This always includes A = 0, where the statement amounts to h| V (I) = 0. The condition to be I-flat becomes stronger for smaller support of I. The strongest case is I = 0 (empty support), where h is I-flat if and only if h = 0, because V (I) = Σ m,n . The weakest condition arises when I has full support {r 1 , . . . , r n }; then an I-flat germ h is only constrained near the codimension n corner {r 1 = • • • = r n = 0} of Σ m,n . Note, however, that for any I, being I-flat implies that the Taylor expansion of h at the origin is identically zero.

Lemma 5.11. Suppose that I is a multi-index with support {r 1 , . . . , r j } for some j 0, and h is an I-flat smooth function. Then there exist smooth functions h 1 , . . . , h j such that h = h 1 + • • • + h j and h i is r i -flat for all i. In particular, h i log(r i ) is smooth and r i -flat.

Proof. We proceed by induction on j and show that all functions h i can in fact be chosen to vanish wherever h does. Let ψ ∈ C ∞ (R) denote a smooth function of one variable, with compact support, that is equal to 1 in a neighbourhood of 0. Note that a partition of unity h = ψ(z)h + (1 -ψ(z))h, with z = r i or z = x i , preserves I-flatness. We can thus assume that h has compact support.

In the cases j = 0 (h = 0) and j = 1 (h = h 1 ) there is nothing to prove. For j > 1, let a n (r 1 , . . . , r j-1 ) = 1 n! (∂ n rj h) rj =0 be the nth Taylor coefficient of h with respect to r j ; it is a smooth function of all remaining variables and it is flat with respect to {r 1 , . . . , r j-1 } by assumption. Hence by induction we may write

a n = a n,1 + • • • + a n,j-1
where a n,i is r i -flat for 1 i < j. Following a proof of Borel's lemma, we may find a sequence of positive numbers C n such that for all i < j, the sum

h i := n r n j a n,i ψ(r j C n )
and all its partial derivatives are absolutely convergent. It follows that the derivative of h i with respect to r i can be computed termwise, and hence the r i -flatness of a n,i implies that

h i is r i -flat. Now let h j = h -h 1 -• • • -h j-1 .
Then by construction, the Taylor expansion of h j along r j = 0 is identically zero. Hence h 1 , . . . , h j give the desired functions.

Finally, observe that since h i is r i -flat, all partial derivatives of h i log(r i ) exist and are smooth, even r i -flat.

The following is a strengthened version of Theorem 5.10. . Proposition 5.12. Suppose that f I ∈ C ∞ Σm,n,0 are germs of smooth functions at the origin, set

f := I f I (x, r) log I (r) ∈ C ∞,log
Σm,n,0 and denote the germ of the corresponding function on the interior as

f := j * f = I f I (x, r) log I (r) ∈ j * C ∞ Σ • m,n 0 .
We write I > J to denote that I k J k for all indices 1 k n, with at least one of these inequalities being strict (I k > J k ).

If f = 0 then the following statements hold:

(α) We have f = 0 ∈ C ∞,log Σm,n,0 . (β) If J is such that f I = 0 for all I with I > J, then the function f J is J-flat.
Proof. We first show that (α) follows from (β). Namely, note that for a multiindex J as in (β), since f J is J-flat, we may write f J = f J,j1 + • • • + f J,j k where the indices j l range over the support of J and f J,j l is r j l -flat. Therefore, using the second defining relation for C ∞,log from Definition 5.1, we may replace f J log J (r) in the expression for f by a sum of monomials of strictly smaller degree in the symbols log r i . By induction over the degree in log(r), we conclude that f has a representative without logarithms. But then, f = f 0 is 0-flat by (β), and hence identically zero.

We now prove (β) by induction over the support of J. The base case is J = 0 ∈ N n with empty support, so we must show that f 0 = 0. Since we have I > J for all I = 0, and thus f I = 0, we have f 0 | Σ • m,n = f = 0. By continuity, f 0 = 0 and so f 0 is indeed J-flat.

From now on we will assume that we are given J as in (β), with non-empty support, and that (β) holds in all cases with smaller support. Label one of the coordinates in the support of J as r n , and consider the embedding

i : Σ m,n-1 × [0) ֒→ Σ m,n
of the vanishing set of r n . Furthermore, we denote the inclusion of the relative interior of this facet as

j n : Σ • m,n-1 ֒→ Σ m,n-1 .
We denote the r n -derivaties of the coefficients of log

l (r n ) in f as (13) g (k) l := I : In=l ∂ k rn f I rn=0 log I ′ (r) ∈ C ∞,log Σm,n-1,0 ,
where we write I ′ = (I 1 , . . . , I n-1 ). We will now show, by induction on k, that

j * n g (k) l = lim rn=0 ∂ k rn I : In=l j * f I log I ′ (r) = 0
for all k and l. Namely, by Hadamard's lemma there are smooth functions h (k) I for each I and k such that

f I = k a=0 r a n a! ∂ a rn f I rn=0 + r k+1 n h (k) I Therefore, once j * n ( g (0) l ) = • • • = j * n ( g (k-1) l ) = 0 is established, it follows from 0 = f r k n = 1 k! l j * n g (k) l log l (r n ) + l r n log l (r n ) I : In=l h (k) I log I ′ (r)
by considering the limit r n → 0 that we must have, for all l, that j * n g (k) l = 0. Hence, by the induction hypothesis, we conclude in particular that for l = J n and for every k, the coefficient

∂ k rn f J | rn=0 of log J ′ (r) in g (k) l
is J ′ -flat, where J ′ = (J 1 , . . . , J n-1 ). Note that J ′ fulfils the condition (β) for g

(k) l , since the coefficient of any multi-index I ′ > J ′ in g (k) l is ∂ k rn f I | rn=0
where I = (I ′ , l) > J = (J ′ , l) and thus f I = 0. We have thus proved that f J is J-flat (recall that l = J n > 0 since r n is in the support of J).

Regularized limits.

Using scales, we may assign finite values to divergent limits of logarithmic functions, by the following general recipe.

First, note that if Ψ is a manifold with log corners, and s : Ψ bas → Ψ is a scale on Ψ, given in local coordinates by s = i s i (r)∂ ti , then the pullback s * :

C ∞,log Ψ → C ∞,log
Ψ bas acts as the identity on the subsheaf of basic logarithms, and sends the phantom logarithms to the functions

s * log(t i ) = log(s i (r)) ∈ C ∞,log Σ bas .
Then if φ : Ψ → Σ is any weak morphism, and g ∈ C ∞,log Σ is a logarithmic function, we may define its regularized pullback to be the function

reg s φ * (f ) := s * φ * f ∈ C ∞,log
Ψ bas , which is smooth function in the interior of Ψ but may have logarithmic divergences on the boundary.

In particular, in the case in which φ is the natural immersion ∂ Σ → Σ of the boundary of a basic manifold with log corners, we obtain a regularized restriction

f | reg s ∂ Σ bas ∈ C ∞,log ∂ Σ bas
This construction recovers the usual restriction when the latter makes sense: Proof. For smooth functions the statement is vacuous. In general, if we have an expansion f = f I log I (r), and f is continuous, then arguing as in Lemma 5.4, we see that the coefficient f I must be divisible by r i whenever r i is in the support of I. But r i log(r i )| reg s ri=0 := s * (0 • log(t i )) = 0, so the non-smooth terms in f do not contribute to the regularized restriction.

Lemma 5.13. If f ∈ C ∞,log
The regularized restriction defined in this way recovers the classical notion of regularized limit, as follows.

Example 5.14. Let Σ = [0, ∞) and Ψ = [0) be equipped with the standard coordinates r and t = r| [0) , respectively. Define a scale on [0) by s = c ∂ t for some c > 0.

If f = f j (r) log j (r) ∈ C ∞,log
[0,∞) , its regularized restriction to zero is given by

f | reg s 0 = s * (f | [0) ) = s *   j f j (0) log j (t)   = j f j (0) log j (c)
When c = 1 is the unit scale, we get

f | reg s 0 = f 0 (0) =: reglim ε→0 j f j (ε) log j (ε),
which is the standard definition of the regularized limit. ♦ The same argument as in classical differential geometry shows that logarithmic vector fields on Σ are local operators, so that they form a sheaf. Definition 6.2. We denote by T log Σ the sheaf of logarithmic vector fields on Σ.

Concretely, in local coordinates (r, t), define derivations

r 1 ∂ r1 , . . . , r n ∂ rn , t 1 ∂ t1 , . . . , t k ∂ t k ∈ T log Σ (14)
by the "obvious" formulae, as follows. Suppose that f ∈ C ∞,log Σ . Then by Lemma 5.9 and Theorem 5.10 we may write f uniquely in the form

f = J f J (log t) J where f J ∈ C ∞,log
Σ bas is a smooth function in the interior that can be written in the form f J = I f I,J (log r) I for some smooth functions f I,J . We then set

r i ∂ ri f := J (r i ∂ ri f J )(log t) J
and define t i ∂ ti be the unique C ∞,log Σ bas -linear derivation such that

t i ∂ ti (log t j ) = 1 i = j 0 i = j
It is straightforward to check that these operations are well-defined derivations.

Proposition 6.3. The derivations (14) form a local basis for

T log Σ as a C ∞,log Σ - module. Hence if Σ is a manifold with log corners of dimension (n, k), the sheaf T log Σ is a locally free C ∞,log Σ -module of rank n + k.
Proof of Proposition 6.3. By Lemma 5.9, the algebra C ∞,log Σ is freely generated over C ∞,log Σ bas by log t 1 , . . . , log t j . Thus the operators t 1 ∂ t1 , . . . , t m ∂ tm form a basis for the C ∞,log Σ bas -linear derivations, and hence it suffices to show that r 1 ∂ r1 , . . . , r l ∂ r l form a basis for the derivations of C ∞,log Σ bas . We first claim that any derivation Z of C ∞,log Σ bas is uniquely determined by its action on smooth functions. Indeed, let h ∈ C ∞ Σ be any smooth function that is nonvanishing in the interior and such that h log r i is smooth. Then

Z(h log r i ) = Z(h) log r i + hZ(log r i ) so that Z(log r i ) = h -1 • (Z(h log r i ) -Z(h) log r i ).
gives a formula for Z(log r i ) in terms of the action of Z on smooth functions, for every i. By linearity and the Leibniz rule, this determines the action of Z on any element of C ∞,log Σ from the action of Z on smooth functions. But the action of Z on smooth functions is a derivation from C ∞ Σ to C ∞,log Σ bas , and must therefore have the form

Z = n i=1 Z i ∂ ri for some coefficient functions Z i ∈ C ∞,log
Σ bas . It remains to show that such a derivation extends to C ∞,log Σ bas if and only if Z i is divisible by r i for all i. Equivalently, we must show that the expression for Z(log r i ) above is a logarithmic function if and only if Z i is divisible by r i . Let h = h(r i ) be any function of r i such that h log r i is smooth. Then Z acts as Z i ∂ ri on both h log r i and h. Hence we have

Z(log r i ) = h -1 • (Z(h log r i ) -Z(h) log r i ) = Z i ∂ ri log r i = Z i r i which lies in C ∞,log
Σ bas if and only if Z i is divisible by r i .

6.2. The de Rham complex. Let Σ be a manifold with log corners. The logarithmic cotangent sheaf T ∨,log

Σ is the dual C ∞,log Σ -module of T log Σ . There is thus a natural derivation d : C ∞,log Σ → T ∨,log
Σ sending a logarithmic function f to the function df : Z → Z(f ) on logarithmic vector fields. If (r 1 , . . . , r n , t 1 , . . . , t k ) is a local system of coordinates on Σ, then by Proposition 6.3 the elements d log(r i ) = dr i r i and d log(t j ) =: dt j t j , for 1 i n and 1 j k, form a local basis of T ∨,log Σ . Definition 6.4. Let Σ be a manifold with log corners. The sheaf of logarithmic j-forms on Σ is

A j,log Σ := j C ∞,log Σ T ∨,log Σ .
The differential d extends uniquely to a graded derivation

d : A •,log Σ → A •+1,log
Σ such that d 2 = 0, given by the usual formula for the de Rham differential.

Lemma 6.5. The sheaf (A •,log Σ , d) of commutative differential graded algebras is called the de Rham complex of Σ.

Note that A j,log Σ is a sheaf of C ∞ Σ -modules for all j 0, hence soft. The hypercohomology of (A •,log Σ , d) thus reduces to the cohomology of the complex A •,log (Σ) := Γ(Σ, A •,log Σ ) of global sections, and similarly for the hypercohomology with compact supports. Definition 6.6. The de Rham cohomology of a manifold with log corners Σ is the cohomology of the complex of forms with logarithmic singularities, denoted

H • dR (Σ) := H • A •,log (Σ), d .
The compactly supported de Rham cohomology of Σ is the cohomology

H • dR,c (Σ) := H • A •,log c (Σ) , d of the complex of compactly supported sections of A •,log Σ .
Given a weak morphism φ : Σ → Ψ, the pullback on logarithmic functions extends uniquely to a map φ * : φ

-1 A •,log Ψ → A •,log
Σ of sheaves of commutative differential graded algebas, so that the de Rham complex and its cohomology are functorial for weak morphisms, and the compactly supported versions are functorial for weak morphisms whose underlying map of manifolds is proper. 6.3. Homotopy invariance and the log de Rham theorem. We now show that our de Rham cohomology agrees with the ordinary de Rham cohomology defined using the smooth forms A • Σ , and hence inherits its usual topological properties. The key is to directly prove the "homotopy invariance"

H • dR (Σ) ∼ = H • dR (Σ × [0)) ∼ = H • dR (Σ × [0, ∞))
in the spirit of the classical argument, e.g. as presented in [BT82, §I.4], for which we make essential use of weak morphisms.

Let Σ be a manifold with log corners. Let r and t be the standard coordinates on [0, ∞) and [0), respectively. Consider the natural maps

(15) Σ × [0) Σ × [0, ∞) Σ i p p
induced by the projection to a point and the canonical embedding [0) → [0, ∞).

Each of these (ordinary) morphisms is canonically split by a weak morphism: we have a commutative diagram

(16) Σ × [0) Σ × [0, ∞) Σ q s s
where s is defined by s * (t) = 1, q is defined by q * (t) = r, and s := is. We thus have the compositions

qi = id Σ×[0) ps = ps = id Σ .
so that the maps in (16) are one-sided inverses for the maps in (15). We claim that on the level of cohomology, these maps becomes two-sided inverses. Indeed, we have the following stronger statement.

Theorem 6.7. The three operators

p * s * ∈ End A •,log (Σ × [0)) and p * s * , q * i * ∈ End A •,log (Σ × [0, ∞))
are canonically cochain homotopic to the identity, and hence the morphisms (15) and (16) give a commutative diagram of mutually inverse isomorphisms

H • dR (Σ × [0)) H • dR (Σ × [0, ∞)) H • dR (Σ) q * s * i * s * p * p *
Before proving the theorem, let us remark on some immediate consequences of this result.

First, by repeated application of the theorem, we deduce that the cohomology of Σ × [0, ∞) n × [0) k reduces to that of Σ. In particular, taking Σ = R m we obtain the following logarithmic version of the Poincaré lemma.

Corollary 6.8. The log de Rham cohomology of

R m × [0, ∞) n × [0) k is given by H j dR (R m × [0, ∞) n × [0) k ) ∼ = R j = 0 0 j = 0
Since every point in a manifold with log corners has a basis of neighbourhoods isomorphic to R m ×[0, ∞) n ×[0) k for some n, k, we deduce the following logarithmic counterpart of de Rham's theorem, with or without compact suport.

Corollary 6.9. The inclusions

R Σ ֒→ A • Σ ֒→ A •,log Σ bas ֒→ A •,log
Σ are quasi-isomorphisms. Hence they induce natural isomorphisms

H • sing (Σ; R) ∼ = H • dR (Σ) ∼ = H • dR (Σ bas ) ∼ = H • dR (Σ)
. of graded commutative algebras, and natural isomorphisms

H • sing,c (Σ; R) ∼ = H • dR,c (Σ) ∼ = H • dR,c (Σ bas ) ∼ = H • dR,c (
Σ). of their graded modules.

The classical Künneth formula then gives the following. Corollary 6.10. If Σ is of finite type (e.g. compact), then the natural map

H • dR (Σ) ⊗ R H • dR (Ψ) → H • dR (Σ × Ψ) is an isomorphism of graded commutative algebras.
We now turn to the proof of Theorem 6.7. We will deal with each of the three operators in the statement in Section 6.3.1 through Section 6.3.3 below. 6.3.1. Contracting homotopy for p and s. We have an isomorphism of commutative differential graded A •,log (Σ)-algebras:

A •,log (Σ × [0)) ∼ = A •,log (Σ) [log(t), d log(t)] .
The graded A •,log (Σ)-linear operator

h : A •,log (Σ × [0)) → A •-1,log (Σ × [0)) , log j (t) → 0 log j (t) d log(t) → 1 j+1 log j+1 (
t) is easily seen to satisfy dh + hd = id -p * s * since s * sends log(t) and d log(t) to zero. 6.3.2. Contracting homotopy for q and i. Since i * q * = id, we have a splitting

A •,log (Σ × [0, ∞)) ∼ = A •,log (Σ × [0)) ⊕ ker i * where ker i * ⊂ A •,log (Σ × [0, ∞))
is the subcomplex of forms that vanish on [0). Under the splitting, the operator q * i * corresponds to the projection onto A •,log (Σ × [0)) and hence it suffices to produce a contracting homotopy for the complex ker i * .

To this end, note that since i * (d log(r)) = d log(t), any form ω ∈ ker i * has no poles in r, i.e. it can be written as

ω = ω 0 + ω 1 ∧ dr
where ω 0 and ω 1 are logarithmic forms that do not involve dr, although their coefficient functions may depend smoothly on r and polynomially on log r. Since log j (r)dr is absolutely integrable near r = 0 for all j 0, we may define an operator

h ′ : ker i * → ker i * [-1] by the formula h ′ ω := r 0 ω 1 (r ′ )dr ′ so that (dh ′ + h ′ d)ω = ω, as desired.
6.3.3. Contracting homotopy for p, s. Since s = is and p = pq, we may compose the homotopy equivalences from the previous to subsections to obtain a homotopy between p * s * and the identity. This completes the proof of Theorem 6.7. 6.4. Relative de Rham cohomology. We may also define a version of de Rham cohomology relative to the boundary. Here, as for classical manifolds with corners, a complication arises: since the topological boundary is not itself a submanifold, we need to replace it with its natural simplicial resolution to obtain a sensible de Rham complex. 6.4.1. Symmetric semi-simplicial objects. As explained in [START_REF] Chan | Tropical curves, graph complexes, and top weight cohomology of Mg[END_REF] in the algebrogeometric context, the combinatorics of boundary strata are most naturally organized using a variant of semi-simplicial sets, which they refer to as symmetric ∆-complexes. We recall the basics here to set our notation and terminology, which differs somewhat from op. cit..

Let I + denote the category of finite sets and injective maps, and I ⊂ I + the full subcategory of nonempty sets. For a category C , a symmetric semi-simplicial object in C is a functor Y : I op → C . From such a functor we may extract the objects Y n := Y ({1, . . . , n}), which by functoriality carry an action of the symmetric groups S n , and a collection of morphisms d n j : Y n → Y n-1 for j ∈ {1, . . . , n}, induced by the unique increasing map {1, . . . , n -1} → {1, . . . , n} whose image omits the element j. Since all injective maps are conjugate to increasing maps by permutations, this data determines Y up to isomorphism. Thus, as a shorthand, we denote the I -object by

Y • = Y 1 Y 2 Y 3 • • •
where the names of the maps d n j and symmetric group actions are left implicit. Similarly, functors (I + ) op → C are called augmented symmetric semisimplicial objects and are determined by the data

(X, Y • ) = X Y 1 Y 2 Y 3 • • •
where X is the image of the empty set, Y • encodes the I -object defined by the restriction of the functor to I ⊂ I + , and the morphism Y 1 → X, which could be denoted d 1 1 , is obtained by functoriality from the inclusion of the empy set in {1}. Remark 6.11. Our indexing differs from the standard indexing of semi-simplicial sets, in that we start from one instead of zero. This convention is chosen to better match the indexing of boundary strata. ♦ 6.4.2. Symmetric semi-simplicial manifolds with log corners. Specializing to the case in which C is the category of manifolds with log corners with weak morphisms, we have the following definition.

Definition 6.12. An I -manifold with log corners is a symmetric semi-simplicial object Ψ • in the category of manifolds with log corners and weak morphisms.

Similarly, an I + -manifold with log corners is an augmented symmetric semisimplicial object (Σ, Ψ • ) in the category of manifolds with log corners and weak morphisms.

We emphasize that the structure maps of an I -or I + -manifold with log corners are weak morphisms by default in this definition. When we want to explicitly say that the morphisms are weak (resp. ordinary) we will talk about weak (resp. ordinary) I -manifolds with log corners and similarly for the augmented case.

Underlying every I -manifold with log corners Ψ • is an I -manifold with corners Ψ • , and similarly in the augmented case. Our primary source of (ordinary) I +manifolds with log corners is given by the following construction.

Example 6.13. If Σ is a manifold with log corners then its boundary strata assemble into an ordinary I -manifold with log corners, which we denote by

∂ • Σ := ∂ Σ ∂ 2 Σ ∂ 3 Σ • • •
and an ordinary I + -manifold with log corners

(Σ, ∂ • Σ) = Σ ∂ Σ ∂ 2 Σ ∂ 3 Σ • • •
where the augmentation is the natural immersion ∂ Σ → Σ of the boundary. ♦ 6.4.3. The relative de Rham complex. If Ψ • is an I -manifold with log corners, the logarithmic de Rham complexes of its components assemble into a symmetric semicosimplicial dg algebra, and we may define the logarithmic de Rham complex of Ψ • to be its totalization, given by the homotopy limit

A •,log (Ψ • ) := holim I A •,log (Ψ • ) .
In concrete terms, we have a canonical quasi-isomorphism

A •,log (Ψ • ) ∼ = n 1 A •,log (Ψ n ) ⊗ sgn n Sn [1 -n]
where [-] is the usual degree shift functor, the differential is the alternating sum of the de Rham differential and the pullbacks along the face maps, sgn n denotes the sign representation of S n , and (-) Sn denotes S n -invariants.

Similarly for a weak I + -manifold with log corners (Σ, Ψ • ) with augmentation map denoted i : Ψ 1 → Σ, we define its relative logarithmic de Rham complex

A •,log (Σ, Ψ • ) := fibre i * : A •,log (Ψ • ) → A •,log (Σ)
Concretely, we have

A •,log (Σ, Ψ • ) ∼ = A •,log (Σ) ⊕ n 1 (A •,log (Ψ n ) ⊗ sgn n ) Sn [-n]
where once again the differential is the sum of the de Rham and Čech differentials.

We also have the de Rham complex A • (Ψ • ) and A • (Σ, Ψ • ) of the underlying symmetric semi-simplicial manifolds with corners, and the inclusion of smooth forms into logarithmic forms give quasi-isomorphisms

A • (Ψ • ) A •,log (Ψ • ) A • (Σ, Ψ • ) A •,log (Σ, Ψ • ) ∼ ∼
which also induce quasi-isomorphisms of the compactly supported forms. In the special case in which Ψ • = ∂ • Σ is the semi-simplicial boundary, a standard inclusion/exclusion argument gives the following. Proposition 6.14. The de Rham cohomology of ∂ • Σ with/without compact supports is naturally isomorphic to the corresponding singular cohomology groups of the topological boundary ∂ top Σ:

H • dR (∂ • Σ) ∼ = H • sing (∂ top Σ; R) H • dR,c (∂ • Σ) ∼ = H • sing,c (∂ top Σ; R)
. and similarly for the relative cohomology

H • dR (Σ, ∂ • Σ) ∼ = H • sing (Σ, ∂ top Σ; R) H • dR,c (Σ, ∂ • Σ) ∼ = H • sing,c (Σ, ∂ top Σ; R).
Remark 6.15. When Σ = Σ bas basic, there is also a smaller model for the relative cohomology, namely the subcomplex

A •,log (Σ, ∂ • Σ) 0 := ker(i * : A •,log (Σ) → A •,log (∂ Σ))
consisting of forms that vanish identically on the boundary. Its inclusion in the total complex A •,log (Σ, ∂ • Σ) is a quasi-isomorphism. ♦ 7. Regularized integration 7.1. Regularization via scales. In the classical theory of integration, one needs an orientation to integrate a volume form on a manifold. In our setting, this is not sufficient, as the integral may diverge due to singularities on the boundary. To overcome this, we need additional structure to regulate the divergences, which we encapsulate in the following.

Definition 7.1. Let Σ be a manifold with log corners. A regularization of Σ is a tuple s = (s ∂ j Σ ) j 0 consisting of a scale s ∂ j Σ on ∂ j Σ for each j 0, which is invariant under the natural action of S j and such that the natural morphisms are scale-preserving. It is nondegenerate if all of the scales s ∂ j Σ are nondegenerate. A regularized manifold with log corners is a pair (Σ, s) where Σ is a manifold with log corners and s is a regularization for Σ.

Σ ∂ Σ ∂ 2 Σ ∂ 3 Σ • • •
By the discussion after Definition 4.15, a regularization of Σ gives rise to (and is equivalent to the data of) an I + -manifold with log corners (Σ bas , ∂ • Σ bas ; s) whose structure morphisms are induced by the scales, and a morphism of I + -manifolds with log corners s

• : (Σ bas , ∂ • Σ bas ; s) → (Σ, ∂ • Σ)
In particular, note that the boundary of a regularized manifold with log corners admits a unique regularization such that the canonical maps ∂ j ∂ Σ → ∂ j+1 Σ are scale-preserving. Similarly, if (Σ, s) and (Σ ′ , s ′ ) are regularized manifolds with log corners, then the product inherits a canonical regularization which we denote by (Σ × Σ ′ , s × s ′ ).

(Σ, ∂ • Σ) : [0, ∞) [0) ∅ • • • (Σ bas , ∂ • Σ bas ; s) : [0, ∞) {0} ∅ • • • i s• i•s s ♦
When unpacked in an example, the compatibility conditions relating the scales on the various boundary faces exhibit some subtleties; these are visible already in the case of a quadrant, as follows.

Example 7.3. Consider the standard corner Σ := [0, ∞) 2 with coordinates (r 1 , r 2 ). Let t 1 , t 2 be the phantom coordinates obtained by restricting r 1 , r 2 to their vanishing loci. The boundary inclusions have the form

[0, ∞) 2 r1,r2 [0) t1 × [0, ∞) r2 ⊔ [0, ∞) r1 × [0) t2 [0) 2 t1,t2 ⊔ [0) 2 t2,t1
∅ where we have used a subscript to denote the coordinates on each space. Then as illustrated in Figure 4, a regularization of Σ consists of the following data:

• A scale on Σ; this must be the identity map since Σ = Σ bas is basic.

• A scale on ∂ Σ = [0) t1 × [0, ∞) r2 ⊔ [0, ∞) r1 × [0) t2 , given by s ∂ Σ = f 2 (r 2 )r a2 2 ∂ t1 on [0) t1 × [0, ∞) r2 f 1 (r 1 )r a1 1 ∂ t2 on [0, ∞) r1 × [0) t2
where f 1 , f 2 are strictly positive functions on [0, ∞) and a 1 , a 2 ∈ N. Thus s * ∂ Σ t 1 = f 2 (r 2 )r a2 2 and s * ∂ Σ t 2 = f 1 (r 1 )r a1 1 on the respective components.

• A scale on ∂ 2 Σ = [0) 2 t1,t2 ⊔ [0) 2 t2,t1
that is invariant under the S 2 -action induced by the involution that identifies the two components. It is thus given on both components by

s ∂ 2 Σ = λ 1 ∂ t1 + λ 2 ∂ t2
where λ 1 , λ 2 > 0, so that s * ∂ 2 Σ t j = λ j . These data must satisfy the consistency condition that the boundary inclusions are scale-preserving. For the map ∂ Σ → Σ this is vacuous, but for the maps i : ∂ 2 Σ → ∂ Σ this is a nontrivial condition; we require i * s * ∂ Σ t j = s * ∂ 2 Σ i * t j for j = 1, 2. This is equivalent to the following equation of phantom tangent vectors:

f 1 (0)λ a1 1 ∂ t2 + f 2 (0)λ a2 2 ∂ t1 = λ 1 ∂ t1 + λ 2 ∂ t2 , By linear independence of ∂ t1 , ∂ t2 , this is equivalent to the equations f 1 (0) = λ 2 λ a1 1 f 2 (0) = λ 1 λ a2
2 of positive real numbers, or equivalently to the linear system

log f 1 (0) log f 2 (0) = -a 1 1 1 -a 2 log λ 1 log λ 2
for their logarithms. Since ∂ 3 Σ = ∅, there are no further constraints. Note that if (a 1 , a 2 ) = (1, 1), the constants λ 1 , λ 2 are uniquely determined by f 1 (0), f 2 (0), for any values of the latter. This applies, in particular, to the case a 1 = a 2 = 0 in which the regularization is nondegenerate (Figure 4a); in this case, s ∂ Σ is an arbitrary positive trivialization of the normal bundle of ∂ Σ, and it completely determines the regularization.

On the other hand, if a 1 = a 2 = 1, the equations have a solution if and only if f 1 (0) = f 2 (0) -1 ; when this condition holds, the constant λ 1 can be chosen arbitrarily, and λ 2 = λ -1 1 . Hence, in this case, the scale on ∂ Σ is not arbitrary, and does not completely determined the regularization. Geometrically, the scales bisect the quadrant into triangles and the tangential basepoint at zero points along the "diagonal" edge; see Figure 4b. ♦

The (non)degeneracy of a regularization will not play any role in our main results below. However, the above example illustrates a useful aspect of nondegerate regularizations: they are specified by the scales assigned to faces of codimension 1, making them relatively easy to construct, as follows.

Proposition 7.4. Let Σ be a manifold with log corners, and let N be denote the normal line bundle of the immersion ∂ Σ → Σ. Then there is a natural bijection between nondegenerate regularizations of Σ and pairs (s, s ′ ) where s is a nondegenerate scale on Σ and s ′ is a positive section of N .

Proof. Given a pair (s, s ′ ) we construct a scale on all iterated boundaries ∂ k Σ as follows. Note that the phantom tangent bundle is given by

T phan ∂ k Σ ∼ = i * T phan Σ ⊕ i * 1 N ⊕ • • • ⊕ i * k N i : ∂ k Σ → Σ and i 1 , . . . , i k : ∂ k Σ → ∂ Σ are the canonical immersions. The section s k := i * s + i * 1 s ′ + • • • + i * k s ′
then gives a nondegenerate scale on ∂ k Σ such that the structure maps of the diagram (Σ, ∂ • Σ) are scale-preserving, so that the tuple (s 0 , s 1 , s 2 , . . .) defines a regularization. It is, moreover, the unique regularization having s 0 = s and s 1 = i * s + s ′ as the scales on Σ and ∂ • Σ, respectively.

Corollary 7.5. Every manifold with log corners admits a nondegenerate regularization.

7.2. The regularized integral. Let (Σ, s) be a regularized manifold with log corners; we make no assumption about the (non)degeneracy of s. We have an induced morphism of I + -manifolds with log corners

s • : (Σ bas , ∂ • Σ bas ; s) → (Σ, ∂ • Σ)
which is the identity on the underlying I + -manifold with corners (Σ, ∂ • Σ), so that the natural maps

A • c (Σ, ∂ • Σ) A •,log c (Σ, ∂ • Σ) A •,log c Σ bas , ∂ • Σ bas ; s s * • are quasi-isomorphisms.
Furthermore, if n = dim Σ denotes the dimension of the underlying manifold with corners, then the complex A •,log c Σ bas , ∂ • Σ bas ; s is concentrated in degrees [0, n], so that we have a canonical projection A n,log c Σ bas , ∂ • Σ bas ; s ։ H n dR,c (Σ bas , ∂ • Σ bas ; s). Definition 7.6. Let (Σ, s) be an oriented and regularized manifold with log corners whose underlying manifold with corners Σ has dimension n. The regularized integral is the linear functional

(Σ,s) : A n,log c (Σ) → R
defined by the composition of the following canonical maps of vector spaces:

A n,log c (Σ) A n,log c (Σ, ∂ • Σ) A n,log c Σ bas , ∂ • Σ bas ; s H n dR,c (Σ bas , ∂ • Σ bas ; s) H n dR,c (Σ, ∂ • Σ) R s * • ∼ Σ
If ω is a form of degree different from n, we set (Σ,s) ω := 0 as usual.

Note that by passing through the isomorphism between logarithmic and smooth de Rham cohomology, the definition reduces the problem of integrating an arbitrary logarithmic form to that of integrating a smooth form. More precisely, we have the following result, which is immediate from the definition. Since the isomorphism between ordinary and logarithmic cohomology is natural and compatible with products, we immediately deduce that the basic identities of integration remain true for our regularized integral.

: A •,log c (Σ) → R[-n]
If, in addition, Σ is connected, this gives an isomorphism H n dR,c (Σ) ∼ = R. Example 7.12. Let a > 0, let r be the standard coordinate on Σ = [0, a], and consider a logarithmic one-form of the form

ω = f (r) dr r = f (0) dr r + f (r)dr
where f (r) = f (0) + r f (r) for a smooth function f . If f (0) = 0, the form ω is smooth and we have

(Σ,s) ω = a 0 f (r) dr
for any choice of regularization s of Σ.

On the other hand, if f (0) = 0, the integral is divergent and will depend on the regularization. The latter is determined by a scale on the boundary, or equivalently weak morphisms s 0 = λ ∂ r | 0 : {0} → [0, a] and s a = -µ ∂ r : {a} → [0, a] for some λ, µ > 0. We then have the regularized pullbacks s * 0 log(r) = log(λ), s * a (log(r)) = log(a), where the second of these is independent of the scale since log(r) is smooth at a. The divergent part of the regularized integral can be computed using the regularized Stokes formula has no classical meaning due to the phantom u; rather, the definition of the regularized integral in this context requires us to first convert the phantoms to functions using a scale on Σ, as follows.

A regularization of Σ consists of a scale on Σ and a compatible scale on ∂ Σ; these must have the form

s Σ = g(r)r j ∂ u s ∂ Σ = λ∂ t + g(0)λ j ∂ u
where g > 0 is a positive smooth function, j ∈ N, and λ > 0 is a constant. By definition, the regularized integral is given by

(Σ,s) f (r) du u = (Σ bas ,s bas ) s * Σ f du u = ([0,∞),s) f (r) j dr r + dg(r) g(r)
The rightmost integral is a classical regularized integral, and can be computed as in Example 7.12, taking the length of the interval to ∞. ♦ 7.3. Non-smooth convergent integrals. As we have just seen, the regularized integral reduces to the usual integral when the integrand is a smooth form. We will now prove a stronger statement: it reduces to the ordinary integral whenever the latter converges absolutely.

Proposition 7.14. Let Σ = Σ bas be a basic, oriented manifold with log corners of dimension n, and let ω ∈ A n,log c (Σ) be a top-degree form. Let j : Σ • → Σ be the inclusion of the interior (a smooth manifold), and let i : ∂ Σ → Σ be the canonical immersion of the boundary. Then the following statements are equivalent:

(1) The integral Σ • j * ω converges absolutely.

(2) The form ω has no poles on ∂ Σ.

(3) We have i * ω = 0 ∈ A n,log (∂ Σ). Moreover, under these conditions, we have

(Σ,s) ω = Σ • j * ω for any regularization s of Σ.
Proof. The equivalence of (1) and ( 2) is the content of [Bro09, Lemma 4.9]; the statement in op. cit. refers to analytic forms, but the argument only uses the existence of an expansion in powers of logarithms, and corresponding bounds on the integral, and hence it applies equally well in our setting. To see the equivalence with (3), note that ω can be written in local coordinates (r 1 , . . . , r n ) as

ω = I ω I (r) log I (r) dr 1 r 1 ∧ • • • ∧ dr n r n
for some smooth functions ω I . Then ω is free of poles if and only if each ω I is divisible by r 1 • • • r n . But i * ω is computed along the boundary stratum r l = 0 by setting r l = 0 in each coefficient function ω I , and making the substitutions log(r l ) → log(t l ) and dr l r l → dt l t l where t l = i * r l | r l =0 is the corresponding phantom. Hence i * ω = 0 if and only if ω I is divisible by r l for all I and l, as desired. Now suppose that the equivalent conditions (1)-(3) hold. By Remark 6.15, it follows that there exists a form η ∈ A n-1,log c (Σ) such that i * η = 0 ∈ A n-1,log c (∂ Σ) and ω -ω = dη. We will deduce the equality (Σ,s) ω = Σ • j * ω by computing both sides using different versions of Stokes' formula.

One the one hand, by our regularized Stokes formula (Corollary 7.10) we have

(Σ,s) ω = (Σ,s) ω + ∂(Σ,s) i * η = Σ ω + 0 = Σ • j * ω, (17)
where in the last step we use that the boundary has measure zero.

One the other hand, we claim that η extends continuously to ∂ Σ and the resulting continuous form i * η on ∂ Σ is zero. Indeed, in local coordinates, η has the form

η = η I,k (r) log I (r) dr 1 r 1 ∧ • • • ∧ dr k r k ∧ • • • ∧ dr n r n
for some smooth functions η I,k , and the condition i * η = 0 is equivalent to the condition that each function η I,k (r) vanishes on the boundary, from which the claim follows immediately. It then follows by applying the classical Stokes formula to the continuous form η as in [Bro09, Theorem 4.11], that the ordinary integral is given by

Σ • j * ω = Σ • j * ω + ∂ Σ i * η = Σ • j * ω + 0 = Σ • j * ω,
and hence it agrees with the regularized integral by (17).

Example 7.15. For a > 0, let Σ = [0, a] with coordinate r and consider the logarithmic one-form ω = log(r)dr, which is absolutely integrable despite the singularity at r = 0; the classical argument is to introduce a cutoff around zero, compute the integral a ε log(r)dr using the fundamental theorem of calculus, and take the limit as ε → 0. Hence for any choice of regularization of Σ we have

(Σ,s) ω = 1 0 log(r)dr = a log(a) -a
Alternatively, this can be derived using our regularized Stokes's formula. Indeed, the logarithmic function η := r log(r) -r ∈ C ∞,log Σ is a primitive for ω. Adopting the notation of Example 7.12, the regularized Stokes formula gives

(Σ,s) ω = ∂(Σ,s) η = s * a (r log(r) -r) -s * 0 (r log(r) -r) = (a log(a) -a) -(0 log(λ) -0) = a log(a) -a,
explicitly exhibiting the independence of from the choice of regularization. ♦

Periods of logarithmic varieties

We now turn to the application of our results to the study of period integrals on logarithmic algebraic varieties. In this section we assume basic familiarity with logarithmic algebraic geometry, as treated for instance in [Kat89, KN99, Ogu18]. 8.1. Varieties with log corners. Here and throughout, by a variety , we mean separated scheme of finite type over the field K = R or C. A log variety is a tuple X = (X, M X , α) where X is a variety, M X is a sheaf of monoids in the étale topology on X, and α : M X → O X is a morphism of sheaves of monoids that identifies α -1 (O × X ) with O × X . We denote by X(K) the set of K-points of X, equipped with the classical analytic topology.

If Y is a variety and D ⊂ Y is a divisor, we denote by X = Y log D the divisorial log variety , for which X = Y and α : M Y log D → O Y is the inclusion of the subsheaf of regular functions on Y that are invertible on Y \ D. If Z → Y is a locally closed immersion (which may have components in common with D) we endow it with the restricted log structure, giving a log variety we denote by Z log D.

Example 8.1. Let z be the standard coordinate on A 1 . The log structure on A 1 log {0} is given by the monoid

M A 1 log {0} = O × A 1 z N ⊂ O A 1 of
monomials in z with invertible coefficients. Its restriction to the origin gives the log variety {0} log {0} with M {0} log {0} = K × w N where w = z| {0} log {0} is a phantom; this log variety is often called the "standard log point". ♦ Definition 8.2. A variety with log corners is a log variety X that is étalelocally isomorphic to a Zariski open log subvariety of (A 1 log {0}) n × ({0} log {0}) k where n, k ∈ N.

Thus a variety with log corners is covered by charts consisting of functions z 1 , . . . , z n ∈ O X cutting out a normal crossing D ⊂ X, together with phantom elements w 1 , . . . , w k ∈ M X . Globally, a variety with log corners is a log variety that is isomorphic to one of the form X = Z log D where Z → Y is the immersion of a union of strata of a normal crossing divisor D in a smooth variety Y .

Ordinary and weak morphisms X → Y of varieties with log corners are defined in the same fashion as for manifolds with log corners: they consist of a map φ : X → Y of varieties over K, together with a pullback morphism of monoids φ * : φ -1 M Y → M X , which we require to commute with α on all of M Y in the ordinary case, and commute with α on the submonoid O × Y ⊂ M Y in the weak case. See [START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF] for more details on weak morphisms in logarithmic algebraic geometry.

Every variety with log corners X has a boundary ∂ X, given by the pullback of the log structure along the map D → X, where D is the normalization of D. It comes equipped with a canonical morphism ∂ X → X, so that the iterated boundaries ∂ • X form an (ordinary) I + -variety with log corners. 8.2. Kato-Nakayama spaces. In [KN99], Kato-Nakayama associate topological spaces to a class of log varieties over C, which includes all varieties with log corners. In [START_REF] Gillam | Logarithmic differentiable spaces and manifolds with corners[END_REF], Gillam-Molcho explained how to endow these spaces with differentiable positive log structures. As we explain in [START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF], this construction is functorial for weak morphisms. The prototype is the following example.

Example 8.3. The Kato-Nakayama space of A 1 log {0} is the real-oriented blowup of A 1 (C) = C at the origin; it is a manifold with boundary equipped with its basic log structure. If z is the standard coordinate on A 1 , then the polar coordinates r = |z| and θ = arg z give an isomorphism of manifolds with log corners

KN(A 1 log {0}) ∼ = [0, ∞) × S 1 . The inclusion {0} log {0} ֒→ A 1 log {0} then gives an isomorphism KN({0} log {0}) ∼ = ∂ KN(A 1 log {0}) ∼ = [0) × S 1
of manifolds with log corners. ♦

In general, if X is a variety with log corners and z 1 , . . . , z n , w 1 , . . . , w k are coordinates identifying an analytic open set U in X with an analytic open set in (A 1 log {0}) n ×({0} log {0}) k , then KN(U ) is identified with the corresponding open set in the manifold with log corners ([0, ∞) × S 1 ) n × ([0) × S 1 ) k . The functions r i = |z i |, θ i = arg(z i ) and φ i = arg(w i ) give basic coordinates, while t i := |w i | are phantom coordinates on the factor [0) k . We therefore have the following. Proposition 8.4. If X is a variety with log corners of dimension (n, k), then KN(X) is a manifold with log corners of dimension (2n + k, k), and the natural map KN(X) → X(C) is a morphism of manifolds with log corners, where X(C) is viewed as a smooth manifold with the trivial positive log structure. 8.3. Real points. We now discuss a version of Kato-Nakayama's construction for varieties over R. If X is a variety with log corners defined over R, then we have an inclusion X(R) ⊂ X(C) expressing the real points as the fixed locus of the antiholomorphic involution given by complex conjugation. This lifts to an involution σ : KN(X) → KN(X) of manifolds with log corners. Theorem 3.33 then allows us to make the following construction. Definition 8.5. Let X be a variety with log corners over R. The real Kato-Nakayama space of X is the manifold with log corners KN R (X) ⊂ KN(X) defined as the fixed locus of the complex conjugation involution. Concretely, supppose that z i , w j are coordinates on X defined over R, and (r i , θ i , t j , φ j ) are the induced coordinates on KN(X) as above, with the angles θ i and φ j defined modulo 2π. Then the conjugation is given by θ i → -θ i and φ i → -φ i , so that the fixed locus is given by θ i , φ j ∈ Zπ, with coordinates induced by the basic radial coordinates r i and the phantom radial coordinates t j . From this we deduce the following.

Corollary 8.6. If X is a variety with log corners of dimension (n, k) over R, then KN R (X) is a manifold with log corners of dimension (n, k).

Note that the proper morphism KN(X) → X(C) restricts to a finite morphism of manifolds with log corners KN R (X) → X(R), whose fibers are of the form (S 0 ) r .

Example 8.7. The log variety X = A 1 log {0} is defined over R. Its real Kato-Nakayama space is the fixed locus of complex conjugation on the real oriented blowup of C at the origin. It therefore consists of two semi-infinite intervals, which map to the non-negative and non-positive real axes, giving an isomorphism

KN R A 1 log {0} ∼ = (-∞, 0] ⊔ [0, ∞)
of manifolds with log corners, with boundary given by two standard ends

∂ KN R A 1 log {0} ∼ = KN R ({0} log {0}) ∼ = (0] ⊔ [0).
See Figure 5c A similar correspondence holds for varieties with log corners defined over R, replacing Spec(C) with Spec(R) and KN(Y log D) with KN R (Y log D).

Remark 8.9. More generally, one may consider log schemes over a base ring K ⊂ C, in which case the resulting tangential basepoints must be defined over K as well, which may greatly rigidify the geometry. For instance, if X = A 1 log {0} with coordinate z, defined over K = Z, the only tangential basepoints at 0 are ±∂ z ; this gives a natural notion of a tangential basepoint having "unit length". ♦ 8.5. Betti and de Rham cohomology. For a class of log varieties over C, Kato-Nakayama [KN99] defined the Betti cohomology of X to be the singular cohomology of the Kato-Nakayama space H • B (X) := H • sing (KN(X); Z), and the (algebraic) de Rham cohomology to be the hypercohomology (in the Zariski topology) of the algebraic logarithmic de Rham complex (Ω • X , d), generated by the logarithmic derivatives of elements of M X :

H • dR (X) := H • (X, (Ω • X , d)) .
Furthermore, they established a comparison isomorphism

H • dR (X) ∼ -→ H • B (X) ⊗ Z C, (18)
under certain assumptions on X that are satisfied in the case of a variety with log corners. In the special case of varieties with log corners, this boils down to the classical fact that the log complex Ω • Y (log D) of a normal crossing divisor D ⊂ Y computes the cohomology of Y \ D. We prove in [START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF] that these cohomology groups are functorial for weak morphisms, and the comparison isomorphism is natural.

Concretely, the real and imaginary parts of every algebraic log form ω ∈ Ω • X define C ∞ log forms ℜω, ℑω ∈ A •,log KN(X) on the Kato-Nakayama space, viewed as a manifold with log corners. This gives a canonical map H • dR (X) → H • dR (KN(X)) ⊗ R C which induces the isomorphism (18) via our log de Rham theorem (Corollary 6.9). 8.6. Logarithmic periods. We now turn to the definition and cohomological interpretation of regularized period integrals in logarithmic algebraic geometry.

By a (weak) I + -variety with log corners we mean an I + -object

(X, Y • ) = X Y 1 Y 2 Y 3 • • •
in the category of weak morphisms of varieties with log corners over C. By the discussion above, the Betti and de Rham cohomology of log schemes extends immediately to such objects by totalizing the relevant symmetric coaugmented cosimplicial complexes as in Section 6. Note that this construction includes the absolute cohomology H • (X) as the special case Y j = ∅ for all j > 0, and the relative cohomology H • (X, Y ) of a morphism Y → X as the special case Y 1 = Y and Y j = ∅ for j > 1. Proof. This follows from the functoriality of the Betti-de Rham comparison for I +manifolds with log corners (Proposition 6.14) and the definition of the regularized integral.

8.7. Examples of logarithmic periods. We now explain how the constructions in this section recover and unify some classical examples of periods. 8.7.1. The residue theorem with "radius zero". Let z be the standard coordinate on X = A 1 log {0}. The class of the logarithmic form dz z is a basis of the first de Rham cohomology group H 1 dR (X). The Kato-Nakayama space of X is KN(X) ∼ = [0, ∞) × S 1 with radial coordinate r = |z| and angular coordinate θ = arg z. For ε 0, let γ ε : S 1 -→ KN(X) ∼ = [0, ∞) × S 1 e iθ -→ (ε, e iθ ) denote the circle of radius ε in the Kato-Nakayama space, oriented counterclockwise. The homology classes of these cycles are all equal and form a basis of H B 1 (X). To compute the period pairing H B 1 (X) ⊗ Z H 1 dR (X) → C one can therefore assume ε > 0 and we get the usual result However, this computation does not make sense classically for ε = 0 since dz z = dr r + i dθ is ill defined at r = 0. This issue is solved using our formalism by lifting each γ ε to a logarithmic cycle (Definition 8.10) with domain Σ = S 1 , i.e. a weak morphism

γ ε : S 1 → KN(X) ∼ = [0, ∞) × S 1 .
For ε > 0 there is nothing to add to the datum of γ ε , but for ε = 0, since γ 0 lands in the boundary {0} × S 1 , one also needs to specify the pullback by γ 0 of the coordinate r, which may be any positive smooth function λ(e iθ ) on S 1 . (The choice of the constant function λ = 1 is somewhat canonical, but our formalism allows more flexibility.) This choice can be thought of as a family of tangential basepoints at 0 on [0, ∞) indexed by S 1 , or alternatively as a scale for the manifold with log corners ∂KN(X) ∼ = [0) × S 1 . Using our notation for tangential basepoints from Section 4.2 we write γ 0 : S 1 -→ KN(X) ∼ = [0, ∞) × S 1 e iθ -→ ( λ(e iθ ) ∂ r | 0 , e iθ ).

The pullback of dz z via γ 0 is now a well-defined smooth 1-form on S 1 , γ * 0 (dlog(z)) = γ * 0 (dlog(r) + i dθ) = dlog(λ(e iθ )) + i dθ, and Proposition 8.11 implies that the pairing between [γ 0 ] and [ dz z ] is equal to which determines the Betti-de Rham pairing completely.

In the limit ε → 0, the integral I(0, a) = I 1 is the divergent integral discussed at the beginning of the paper, which must be regularized by choosing a tangential basepoint of A 1 at 0. For simplicity, we choose a tangential basepoint that points in the positive real direction, i.e. a tangent vector v = λ ∂ z | z=0 ∈ (T 0 A 1 ) × with λ > 0, and view it as a weak morphism {0} → A 1 log {0} by Proposition 8.8. Combined with the ordinary inclusion of a ∈ A 1 \ {0} we obtain a diagram (20)

A 1 log {0} {0} ⊔ {a} which we view as a (weak) I + -variety with log corners. Its relative cohomology

H 1 A 1 log {0}, { v, a}
deserves the name "regularized Kummer motive". The classes of dz and dz z still form a basis of relative de Rham cohomology. In order to describe a basis of Betti homology, we equip the interval [0, a] with the regularization s given by the tangential basepoint λ ∂ r | r=0 at 0, and any tangential basepoint at a. This gives the following logarithmic cycle (Definition 8.10) for (20), denoted by η 0 . 8.8. Single-valued integration and the double-copy formula. The periods we have considered so far involve the integration of algebraic log forms on a complex variety with log corners X over subspaces of X(C). In many applications, one is interested in integrals of products of holomorphic and antiholomorphic forms over X(C) itself, like the integral I 2 from the introduction. Such integrals can be reduced to holomorphic periods of X by way of the "double copy" formula for single-valued integration from [START_REF] Brown | Single-valued integration and double copy[END_REF]. We now explain how that recipe can be recovered using our formalism.

8.8.1. Doubling and the twisted diagonal. The key point is that the integrals in question can be thought of in purely holomorphic/algebraic terms, as the integral of holomorphic forms on X × X over the diagonal copy of X, where X denotes the complex conjugate of X. Thus X is given by the same underlying log variety, but with the conjugate complex structure. Equivalently, we replace the structure map X → Spec(C) with its complex conjugate. Note that X is not a complex subvariety of X × X; rather it is the fixed locus of the antiholomorphic involution of X ×X which interchanges the factors, and is thus totally real. Note further that if X is disconnected, the product X × X will have connected components that do not intersect the diagonal; these may be ignored for the purposes of studying such integrals. This motivates the following definition be dual bases. The cycle class of the twisted diagonal in the relative homology H B 2n (X A × X B , ∂ • (X A × X B )) is then given, under the Künneth decomposition, by [KN(X D )] = γ i ⊗ γ ∨ i , from which we deduce the following. Example 8.17. We now explain how to treat the integral I 2 from the introduction as a logarithmic period. Let Y = P 1 , suppose that a ∈ P 1 \ {0, 1, ∞} and consider the divisors

A = {1, a} B = {0, ∞} D = A ∪ B = {0, 1, a, ∞}.
Setting ω = dz z-a -dz z-1 and ν = dz z , we have

Y (C) ω ∧ ν = P 1 (C) dz z -a - dz z -1 ∧ dz z = I 2
As explained in the introduction, we can compute its value by applying the regularized Stokes formula on Σ = KN(Y log D) (with respect to any choice of regularization), yielding I 2 = 2πi log |a| 2 . On the other hand in [BD21, Example 1.6], the same result is obtained by the double copy formula, which reduces the computation to the periods of the Kummer motives from Section 8.7.2. ♦

  cutoff parameter ε > 0, computes the integral a ε dx x = log(a) -log(ε)

  Figure 2. The "teardrop" manifold W is a compact surface with a single corner and a single boundary component. Its double boundary consists of two points that are interchanged by the action of S 2 , and correspondingly have the same image in W .

  therefore take the quotient of the monoid by conguence generated by the G-action (see [Ogu18, §I.1.1]) to obtain the following. Definition 3.30. The fixed locus Σ G is the fixed point set equipped with the positive log structure M Σ G := (i * M Σ )/G. Example 3.31. Consider the symmetric group S n acting on the manifold with log corners [0, ∞) n by permuting the coordinates r 1 , . . . , r n . The fixed-point set is the image of the diagonal i : [0, ∞) ֒→ [0, ∞) n . The pullback log structure is given by

Example 4 .

 4 3. Let * denote the point equipped with the trivial positive log structure. Then for any Σ, the projection Σ → * is a (ordinary or weak) morphism in a unique way. Hence * is a terminal object in the category of either ordinary or weak morphisms. ♦ Example 4.4. Let * be the point as in the previous example, and let [0) be the standard end, given by the point with the positive log structure M [0) = R >0 t N from Example 3.4. There is a unique map of the underlying manifolds * ∼ = [0) since both consist of a single point. A weak morphism s : * → [0)

  ∞) has no phantoms. ♦ Example 4.6. More generally, let Σ = (Σ, M Σ , α) be a manifold with log corners. A weak morphism f : Σ → [0) is the datum of a morphism of monoids f * : R >0 t N → M (Σ) which acts as the identity on R >0 , where M (Σ) := Γ(Σ, M Σ ) denotes the monoid of global sections of M Σ . Thus weak morphisms Σ → [0) are in bijection with M (Σ). In contrast, ordinary morphisms Σ → [0) are in bijection with M phan (Σ) := Γ(Σ, M phan Σ ). ♦ Proof of Proposition 4.2. The statement being local, we can assume that Σ is connected, and up to shrinking Ψ, that f is a global section of M bas Ψ . Furthermore, since the restriction to the interior M (Σ) → M (Σ • ) is injective, we may assume without loss of generality that ∂ Σ = ∅, and hence M bas Σ = C ∞,>0 Σ . Consider the open sets V = {f > 0} ⊂ Ψ and U = φ -1 (V ) = {φ * f > 0} ⊂ Σ, which are non-empty by assumption. By definition of a weak morphism we have φ

Figure 3 .

 3 Figure 3. Some tangential basepoints on the teardrop manifold.

  s 0 := ∂ t | 0 : * → [0, 1] and s 1 := -∂ t | 1 : * → [0, 1]. Then (6) implies the equalities (7) γ • s 0 = v and γ • s 1 = w of weak morphisms * → Σ. In other words, we have the following commutative diagram of weak morphisms. (8) * ⊔ * [0, 1] * ⊔ * Σ

  r N . Using the first relation in Definition 5.1, we may write every f ∈ C ∞,log Σ locally as a finite sum f = I f I (r) log I (r) (11) over multi-indices I = (I 1 , . . . , I n ) ∈ N n where f I (r) are smooth functions on the underyling manifold with corners, and log I (r) := log I1 (r 1 ) • • • log In (r n )

  and j * sends the expression f from (11) to the smooth function (12) j * f := I f I (r) log I (r)

  Σ bas is continuous up to the boundary, then the regularized restriction f | reg s ∂ Σ bas agrees with the ordinary restriction as a function.

Example 5 .

 5 15. Let Σ = [0, ∞) 2 with basic coordinates (r 1 , r 2 ) = (x, y). Let u and v be the corresponding phantom coordinates on the boundary components x = 0 and y = 0, respectively. A scale on the boundary then has the forms = e g(x) x a ∂ v on y = 0 e h(y) y b ∂ u on x = 0where g, h are smooth functions on [0, ∞) and a, b ∈ N. If f = log(x) log 2 (y), then the regularized restriction of f to the boundary is given byf | reg s ∂ Σ = log(x)(g + a log(x)) 2 on y = 0 (h + b log(y)) log 2 (y) on x = 0 exhibiting the explicit dependence on all components of the scale. ♦ 6. de Rham theory for manifolds with log corners 6.1. Vector fields. The natural notion of vector field in the context of functions with logarithmic singularities is as follows.Definition 6.1. A logarithmic vector field on Σ is a derivation of the R-algebra C ∞,log (Σ) of global logarithmic functions on Σ.

Figure 4 .

 4 Figure 4. Regularizations of the quadrant Σ = [0, ∞) 2 , showing the scale on ∂ Σ in blue, and the scale on ∂ 2 Σ in red.

  Example 7.2. For Σ = [0, ∞), since Σ = Σ bas and ∂ j Σ = ∅ for j 2, a regularization for Σ is the same thing as a scale for ∂Σ = [0), i.e. a weak morphism s : {0} → [0), i.e. a tangential basepoint c ∂ r | 0 by Proposition 4.8. It gives rise to the following morphism of I + -manifolds with log corners.

Proposition 7. 7 .

 7 If ω ∈ A n,log c (Σ) is a compactly supported logarithmic form, then (Σ,s) ω = Σ ω where ω ∈ A n c (Σ) is any compactly supported smooth form such that ω = s * ω + dη for some η ∈ A n,log c Σ bas whose restriction to the boundary is zero.

Corollary 7. 8 ((

 8 Change of variables). For an open embeddingj : Ψ ֒→ Σ and a form ω ∈ A •,log c (Ψ), let j * ω ∈ A •,log c (Σ) be the extension by zero of ω. Then (Ψ,j * s) ω = (Σ,s) j * ω.Corollary 7.9 (Fubini's formula). Suppose that (Σ, s) and (Σ ′ , s ′ ) are oriented, regularized manifolds with log corners, ω ∈ A •,log c (Σ) and ω ′ ∈ A •,log c (Σ ′ ). Then(Σ×Σ ′ ,s×s ′ ) ω ∧ ω ′ = (Σ,s) ω • (Σ ′ ,s ′ ) ω ′Corollary 7.10 (Stokes' formula). If η ∈ A •,log c Corollary 7.11. The regularized integral defines a map of cochain complexes (Σ,s)

0 f

 0 ) = s * a log(r) -s * 0 log(r) = log(a/λ), so that by linearity of the integral we have(Σ,s) ω = f (0) (Σ,s) dr r + (Σ,s) f (r) dr = f (0) log(a/λ) + a (r) drClassically, the same result would be obtained by introducing a cutoff parameter ε > 0, computing the convergent integrala-µε λε dr r = log(a/λ) -log(ε) + O(ε),and formally discarding log(ε) in the limit as ε → 0. Either way, the result depends on the parameter λ determining the scale at r = 0, but is independent of the parameter µ determining the scale at r = a. The reason is that ω has a pole at r = 0 but is smooth at r = a. ♦ Example 7.13. Let Σ = [0, ∞) × [0) with coordinates (r, u) and let (t, u) be the induced phantom coordinates on ∂ Σ = [0) 2 . For a smooth function f (r) with compact support, an integral of the form Σ f (r) du u

Figure 5 .

 5 Figure 5. Geometry of the log scheme X = A 1 log {0}, which is defined over any ring K. The portions of the diagram shown in blue correspond to the log subscheme ∂ X = {0} log {0}, which gives the boundary of the Kato-Nakayama space.

(a) A 1 Figure 6 .

 16 Figure 6. Algebraic vs. C ∞ tangential basepoints at the origin in A 1 .

  Definition 8.10. A logarithmic cycle in (X, Y • ) is the data of a compact oriented regularized manifold with log corners (Σ, s) and a morphism of I + -manifolds with log corners φ :(Σ bas , ∂ • Σ bas ; s) → KN(X, Y • ).A logarithmic cycle has an underlying map of symmetric semi-simplicial spacesφ : (Σ, ∂ • Σ; s) → KN(X, Y • ) inducing a morphism in homology φ * : H sing • (Σ, ∂ top Σ) → H B • (X, Y • ) Since Σ is compactand oriented of dimension n, the image of its fundamental class gives the cycle class, which we denote simply by [φ] := φ * [Σ] ∈ H B n (X, Y • ). They naturally arise as integration domains in many interesting situations. As for integrands, let ω ∈ Γ(X, Ω n X ) be a global closed logarithmic n-form on X whose pullback to Y • is zero. It defines a class [ω] ∈ H n dR (X, Y • ), and we denote by φ ω := [φ], [ω] ∈ C the corresponding period, induced by the Betti-de Rham pairing (19). It can be computed as a regularized integral as in Section 7, as the following result shows. Proposition 8.11. The pairing (19) between the cycle class [φ] ∈ H B n (X, Y • ) and the class [ω] ∈ H n dR (X, Y • ) equals the regularized integral φ

[

  e iθ )) + i dθ = 2πi, which shows that [γ 0 ] is independent of the choice of the function λ(e iθ ). 8.7.2. The logarithm as a regularized Kummer period. For real numbers 0 < ε < a, consider the integralI(ε, a) = a ε dz z = log(a) -log(ε) = log(a/ε).It is classically interpreted as a period of the "Kummer motive" H 1 A 1 \ {0}, {ε, a} , whose algebraic de Rham cohomology and Betti homology are given byH 1 dR (A 1 \ {0}, {ε, a}) = C • [dz], dz z and H B 1 (A 1 \ {0}, {ε, a}) = Z • {[γ ε ], [η ε ]} respectively,where γ ε is the circle of radius ε as before and η ε is the interval [ε, a]. We then have I(ε, a) = ηε dz z , and more generally we have the period matrix

KNFigure 7 .

 7 Figure7. The geometry of the "regularized Kummer motive"H 1 A 1 log {0}, { v, a}

  Proposition 8.16 (Double copy formula, c.f. [BD21, Corollary 1.5]).If ω ∈ Γ(Y, Ω n Y log A ) and ν ∈ Γ(Y, Ω n Y log B) are global logarithmic forms, then ω ∧ ν is polar smooth, and its period over the twisted diagonal is given by[KN(X D )], [ω ∧ ν] =

  Motivation. The need to regulate logarithmic divergences of integrals is a recurring theme in geometry, number theory, and mathematical physics. For instance, to make sense of the divergent integral

	1. Introduction	1
	2. Manifolds with corners	9
	3. Manifolds with log corners	12
	4. Weak morphisms, tangential basepoints, and scales	22
	5. Functions with logarithmic singularities	27
	6. de Rham theory for manifolds with log corners	34
	7. Regularized integration	40
	8. Periods of logarithmic varieties	47
	References	57
	1. Introduction
	1.1.

  {b 1 , . . . , b k }), where x ∈ W and {b 1 , . . . , b k } is a set of pairwise distinct boundary tangent hyperplanes of W at x. Its connected components are called the (boundary) faces of W . These objects are less practical to work with than the ∂ k W , because of the impossibility to define maps ∂

		"teardrop" manifold W is a compact surface with a
	single corner and a single boundary component. Its double bound-
	ary consists of two points that are interchanged by the action of
	S 2 , and correspondingly have the same image in W .
	The quotient ∂	k W := ∂ k W/S k is a manifold with corners, whose points are
	pairs (x, k+1 W → ∂	k W for k > 0 in a
	choice-free manner. There is, however, a natural immersion ∂	k W → W for each k,
	which identifies the interior (∂	k W ) • with the set of depth-k points in W , and the
	tangent space at (x, {b 1

  nondegenerate in this chart if and only if the functions s * t 1 , . . . , s * t k are strictly positive. Invariantly, a scale is uniquely determined by its restriction to the phantoms, giving a morphism s| M phan Σ

  r)∂ tj . Example 4.12. By Example 4.4, a scale for [0) with phantom coordinate t is the same thing as a positive number λ := s * t ∈ R >0 ; we write s = λ∂ t . Note that s is automatically nondegenerate. ♦ Example 4.13. A scale for [0, ∞) × [0) with coordinates (r, t) is the same thing as a function s * (t) = g(r)r j , where g is a positive smooth function on [0, ∞) and some j ∈ N; we write s = g(r)r j ∂ t . Then s is nondegenerate if and only if j = 0. ♦ Proposition 4.14. Every manifold with log corners admits a nondegenerate scale.

  for an illustration. ♦ 8.4. Tangential basepoints: algebraic vs. differential geometry. Let Y be a smooth variety over C and D ⊂ Y a normal crossing divisor. If p ∈ Y is a point, a tangential basepoint at p, in the sense of Deligne [Del89, §15], is a choice of a nonzero normal vector for each local irreducible component of D passing through p. We refer to such basepoints as algebraic to distinguish them from the C

∞ tangential basepoints for manifolds with corners in this paper. They correspond to weak morphisms from a point to the log scheme Y log D via the algebro-geometric analogue of Proposition 4.8; see

[START_REF] Dupont | Logarithmic morphisms and tangential basepoints[END_REF]

.

  4.3, and the comparison isomorphism gives a canonical pairing (19)-, -:H B • (X, Y • ) ⊗ Z H • dR (X, Y • ) → Cwhich becomes nondegenerate after tensoring with C.

Acknowledgements. We thank Federico Binda, Damien Calaque, Danny Gillam, Samouil Molcho, Sam Payne and Matt Satriano for helpful discussions and correspondence. This work grew out of discussions held by the authors during a Research in Pairs activity at the Mathematisches Forschungsinstitut Oberwolfach (MFO). We thank the MFO for providing excellent hospitality and working conditions. C. D. was supported by the Agence Nationale de la Recherche through grants ANR-18-CE40-0017 PERGAMO and ANR-20-CE40-0016 HighAGT. E. P. was funded as a Royal Society University Research Fellow through grant URF\R1\201473. B. P. was supported by Discovery Grant RGPIN-2020-05191 from the Natural Sciences and Engineering Research Council of Canada (NSERC), Établissement de la relève professorale grant 313101 from the Fonds de recherche du Québec -Nature et technologies (FRQNT), and a startup grant at McGill University.

Definition 8.12. Let X be a complex variety with log corners whose connected components are denoted by X i . The double of X is the complex variety with log corners defined as the disjoint union i X i × X i ⊂ X × X.

The diagonal X(C) → X(C)× X(C) lifts canonically to a morphism of manifolds with log corners KN(X) → KN(Dbl(X)) ⊂ KN(X) × KN(X), which we call the twisted diagonal ; it identifies KN(X) with fixed points of the involution that swaps the factors in the product.

Remark 8.13. More abstractly, we may consider the Weil restriction W (X); it is a variety with log corners over R whose R-points are in bijection with the C-points of X. We have W (X) × R C ∼ = X × X, so that the twisted diagonal is the inclusion KN R (W (X)) → KN(W (X)) of the real Kato-Nakayama space in the sense of Section 8.3. ♦ 8.8.2. Polar smooth forms. We now restrict to the case in which X = Y log D is the variety with log corners associated to a connected smooth proper complex variety Y of dimension n and a normal crossing divisor D. Applying the doubling construction to X and its iterated boundaries, we obtain an ordinary I + -variety with log corners (Dbl(X), Dbl(∂ • X)). The twisted diagonal then gives a morphism of I + -manifolds with log corners

whose class in Betti homology we denote by

To understand the periods we must examine the differential forms on the open subscheme Dbl(X) ⊂ X × X relative to Dbl(∂X) ⊂ ∂ X × ∂X. For this note that since the restriction map Ω • X×X → Ω • ∂X×∂X is surjective, the relative de Rham complex is modelled by its kernel

.

Following [START_REF] Brown | Single-valued integration and double copy[END_REF], we use the following terminology: Proof. The problem is local and invariant under taking products with smooth varieties, so we may assume without loss of generality that X = (A 1 log {0}) n with coordinates z j , so that Ω X×X thus has the form

The log variety Dbl(∂X) has n connected components Z 1 , . . . , Z j , each of codimension two, identified with the loci z j = z j = 0 for 1 j n. The restriction of ω to such a component is given by

where t j , t j are the phantom coordinates corresponding to z j , z j . From this we deduce that ω| Zj = 0 if and only if f vanishes on the linear subspace z j = z j = 0 in A n or equivalently the double residue Res zj=0 Res z j =0 ω is zero. This gives the equivalence of conditions (1) and ( 4).

On the other hand, the pullback to KN(X) is computed by converting to polar coordinates z j = r j e iθj and z j = r j e -iθj , giving

Considering the behaviour of ω as r j → 0, we see that ω is smooth on KN(X) if and only if the pullback of f vanishes on the boundary component r j = 0 for every j, but this is evidently equivalent to the vanishing of f when z j = z j = 0, and also to the absolute integrability, as desired.

8.8.3. The double copy formula. Now suppose further that the divisor D ⊂ Y is decomposed as union D = A ∪ B, where A and B have no common irreducible components. We have canonical morphisms of varieties with log corners

induced by the inclusions of divisors. On the level of forms, these maps induce the inclusion of forms with logarithmic poles on A or B into the forms with poles on D = A ∪ B. Applying the doubling construction, we obtain a morphism of I + -varieties with log corners

We also have maps

induced by the pullback of log structures along the normalization maps of A and B. Since D = A ∪ B, Alexander-Lefschetz-Poincaré duality implies that the intersection pairing