1 on any q-dimensional cycle in X and that the corresponding functions obtained by integration on an analytic family of q-cycles are locally bounded and locally continuous on the complement of closed analytic subset.

AMS classification. 32 C 15 -32 C 30 -32 S xx -32 S 45.

Key words. Meromorphic differential forms on a singular space-Universal pullback property-Normalized Nash transform-Integral dependence equation for differential forms.

The case X := {(x, y, z) ∈ C 3 / x.y = z k }, k ≥ 2 . . . . . . . . . . . . 27 6.3 The case X := {(x, y, z) ∈ C 3 / x 3 + y 3 + z 3 = 0} . . . . . . . . . . . 30 6.4 The case X := {(x, y, u, v) ∈ C 4 / x.y = u.v} . . . . . . . . . . . . . 32 6.5 The case X := {(x, y, z, t) ∈ C 4 / x.y.z = t 3 } . . . . . . . . . . . . . . 34

Abstract. We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf ω • X which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms Ω • X /torsion. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf Ω • X /torsion. This sheaf α • X is also closely related to the normalized Nash transform. We also show that these q-meromorphic differential forms are locally square-integrable

Introduction

In this article we discuss the following question: given a reduced complex space X the normalization of X consists in building a proper modification ν : X → X such that meromorphic locally bounded functions on X becomes holomorphic after pull-back to X. Moreover this process gives a desingularization process for curves, that is to say for X of pure dimension 1. It seems then natural to define an analogous process for meromorphic locally bounded differential forms. The main trouble is to define what means a "locally bounded" for a meromorphic differential form of positive degree on a reduced complex space. To define this notion is the purpose of this paper. Of course, this does not lead to a simple proof of a desingularization process for a reduced complex space, but we will show that the natural process associated to "normalization of differential meromorphic forms" is simply the classical normalized Nash transform, and it is an old (an probably very difficult) conjecture that this process leads to a desingularization. We hope that the introduction of this "new sheaf" α • X will be useful in that direction. But in fact, the main reason to introduce this sheaf is the look for the "universal pull-back property" which means to define a coherent sheaf of meromorphic differential forms which admits a natural pull-back for any holomorphic map between reduced complex spaces and which is "maximal" with this property. Note that if we only consider complex manifolds the sheaf Ω • X has this property, but we will show that this is no longer maximal when X admits singularities. Our main result is the theorem 3.0.1 (and its precise formulation 3.0.2) giving the "universal pull-back property" for these sheaves. We obtain also two other results which may be useful:

• The fact that for any section α of the sheaf α q X the form α ∧ ᾱ is locally integrable on any holomorphic cycle of dimension q and also the local boundness and the "generic" continuity of such an integral when the q-cycle moves in an analytic family (see theorem 4.0.7);

• The existence of a local integral dependence equation for a section of α q X over the symmetric algebra of the sheaf Ω q X /torsion (see proposition 5.1.1).

We conclude this article by computing some simple examples showing that the sheaf α • X may be different from other classical sheaves of meromorphic differential forms which are used on singular complex spaces.

1 Universal pull-back for Ω • X /torsion

Conventions and Notations.

• Any complex space will be reduced in the sequel.

• The graded sheaf Ω • X on the complex space X is locally defined in a local embedding of X in an open set U ⊂ C N as the quotient

Ω • C N I X .Ω • C N + dI X ∧ Ω •-1 C N
where I X is the reduced ideal of X in U.

• On a complex space X we shall consider the graded sheaf L • X which is the direct image of the sheaf of holomorphic forms on a desingularization of X, and the graded sheaf ω • X ⊂ j * j * Ω • X , where j : X \ S → X is the inclusion of the regular part of X, which is the sheaf of ∂-closed (•, 0)-currents on X modulo these currents which have support in S (see [B.78]). Then we have the natural inclusions of graded coherent sheaves

Ω • X torsion ⊂ L • X ⊂ ω • X .
Note that in the case where the codimension of the singular set in X is at least equal to 2, we have the equality ω • X = j * j * Ω • X . • When X is not irreducible, let X := ∪ i∈I X i be the decomposition of X in its irreducible components. In this case a desingularization of X will be given by the disjoint union τ : X → X of desingularizations τ i : Xi → X i ֒→ X, where X is the disjoint union of the connected complex manifolds Xi .

It is easy to see that in this situation the sheaf L • X is the direct sum of the direct images in X of the sheaves L • X i . This direct sum is locally finite on X as the irreducible components of X are locally finite on X.

It is clear that the sheaf Ω •

X on a complex space has the "universal pull-back" property because the pull-back of a holomorphic form by a holomorphic map f : Y → X is well defined and functorial in the map f .

To begin we shall prove that the sheave Ω • X /torsion still have this "universal pullback" property.

Proposition 1.0.1 Let X be a reduced complex space and consider a torsion holomorphic p-form α on X ( meaning that α vanishes at smooth points in X). Let Z be an analytic subset in X. Then the p-holomorphic form induced by α on Z is again a torsion form on Z.

Proof. Without any lost of generality we can assume that Z is irreducible. Let S the singular set of X. If Z is not contained in S the result is obvious. Also if the dimension of Z is less that p the conclusion is again obvious. So let dim Z = p + q with q ≥ 0 and let Z ′ the dense open set of smooth points x in Z for which the multiplicity of x in X is minimal. It is enough to show that the restriction of α to Z ′ vanishes. As the problem is local on Z ′ , we can assume that we have an open neighbourhood X ′ of x 0 in X and a local parametrisation π : X ′ → U on a polydisc U of C n with the following properties: i) π(x 0 ) = 0.

ii) U = V × W where V and W are polydiscs with center 0 respectively in C p+q and C n-p-q .

iii)

Z ′′ := Z ′ ∩ X ′ = π -1 (V × {0}) set theoretically and π : Z ′′ → V × {0} is an isomorphism.
Define the analytic family of (p + q)-cycles (Z w ) w∈W in X ′ parametrized by W by letting Z w := π * (V × {w}), where the pull-back by π is taken in the sense of cycles1 . Then, if k is the degree of π (which is the multiplicity in X of each point in Z ′′ ) we have Z 0 = k.Z ′′ as a cycle in X ′ . Remark that for w generic in W the intersection of the cycle Z w with the ramification set of π has no interior point in Z w which is a reduced cycle. So the restriction of the holomorphic form α to Z w for w generic is a torsion form. Now choose a non negative continuous function with compact support ρ on X ′ and a holomorphic q-form β on X ′ and define the function on

W ϕ : W → R + , w → ϕ(w) := Zw ρ.(α ∧ β) ∧ (α ∧ β).
It is a continuous function (see [B-M 1] ch.IV) and it vanishes for w generic in W as α generically vanishes on Z w for such a w. Then it vanishes for w = 0 and this shows that the restriction of α to an open dense subset of Z ′ vanishes.

Corollary 1.0.2 Consider a holomorphic map f : X → Y where X and Y are reduced complex spaces.Then, if α is a p-holomorphic form on Y which is a torsion form, the p-holomorphic form f * (α) is a torsion form on X.

Proof. It is enough to consider the case where X is a connected complex manifold. Let X ′ be the open dense subset of X where f has maximal rank. On X ′ the map f is locally a submersion on a locally closed complex sub-manifold of Y and the previous proposition applies to show that the pull-back of α on this locally closed sub-manifold vanishes. So the holomorphic form f * (α) vanishes on X ′ . Then it is a torsion form on X.

Conclusion.

• The usual pull-back for holomorphic differential forms induced a natural pullback for the sheaf Ω • X /torsion by any holomorphic map between reduced complex spaces.

Definition 1.0.3 Let f : X → Y a holomorphic map between two reduced complex spaces. We have a natural graded pull-back O X -morphism

f * : f * (Ω • Y /torsion) → Ω • X /torsion (*)
We shall denote f * * (Ω • Y ) the image of this graded sheaf morphism. We shall also denote f * * (G) for any sub-sheaf G of Ω • Y /torsion its image by the morphism f * above (or also when

G is a subs-sheaf of Ω • Y ). So, by definition, f * * (Ω • Y ) (and more generally f * * (G)) is a sub-sheaf of the sheaf Ω • X /torsion, so it has no O X -torsion.
Lemma 1.0.4 Let f : X → Y and g : Y → Z two holomorphic maps between reduced complex spaces. Then we have equality of the sub-sheaves f * * (g * * (H)) and

(g • f ) * * (H) for any sub-sheaf H of the sheaf Ω • Z /torsion.
Proof. The pull-back by g gives a morphism

g * (Ω • Z /torsion) → Ω • Y /torsion
with image g * * (Ω • Z /torsion) and the pull-back by f gives a morphism

f * (g * (Ω • Z /torsion)) → f * (Ω • Y /torsion)
which, by right-exactness of the tensor product, is surjective on f * (g * * (Ω • Z /torsion)). Then we have the following commutative diagram

f * (g * (Ω • Z /torsion)) α / / ≃ f * (g * * (Ω • Z /torsion) u / / β ) ) T T T T T T T T T T T T T T T f * (Ω • Y /torsion) v (g • f ) * (Ω • /torsion) γ / / Ω • X /torsion .
Here α is surjective and the image of β is the sub-sheaf f * * (g * * (Ω Z )) by definition. Also the image of γ is (g • f ) * * (Ω • Z ) by definition. Now the commutativity of the diagram allows to conclude.

2 Definition of the sheaf α •

X

The following result is the key of the definition of the sheaf α • X on the reduced complex space X.

Theorem 2.0.1 Let X be a reduced complex space and let S be a closed analytic subset with no interior point in X containing the singular set of X. Let α be a section on X of the sheaf ω p X . The following properties are equivalent for α:

• There exists locally on X a desingularization τ : X → X such that α extends to a section on X of the sub-sheaf τ * τ * * (Ω p X ) of ω p X .

(A)

• There exists locally on X a finite collection (ρ j ) j∈J of C ∞ functions on X \ S which are locally bounded near S and holomorphic p-forms (ω j ) j∈J on X such that α = j∈J ρ j .ω j as a (p, 0) currents on X.

(B)

Note that under the second property stated in the theorem, the (p, 0)-current on X associated to the form j∈J ρ j .ω j on X \ S is defined by

C ∞ c (X) n-p,n ∋ ϕ → X ϕ ∧ ( j∈J ρ j .ω j )
and this integral is absolutely convergent as the functions ρ j are locally bounded near a point in S. It defines a (p, 0)-current on X with order 0. The assumption that α is a section of the sheaf ω p X implies that this current is ∂-closed on X.

Proof. Let us begin by the implication (A) ⇒ (B). By definition, a section α ∈ ω p X is in the sub-sheaf τ * τ * * (Ω • X ) if, locally on X, it can be written as a linear combination of pull-back of holomorphic forms on X with holomorphic coefficients in O X . Using the properness of the desingularization τ and a C ∞ partition of the unity on X we obtain the first part of (B) because τ induces an isomorphism X \ τ -1 (S) → X \ S by hypothesis. The last property in (B), that is to say the fact that the current defined on X by the right hand-side coincides with α, is consequence of the fact that both are sections of the sheaf ω p X and are equal on X \ S. Note that the formula given describes the direct image by τ of the (p, 0)-current defined by α on X. To prove the implication (B) ⇒ (A) consider the pull-back to X \ τ -1 (S) of the form j∈J ρ j .ω J . We obtain a holomorphic form on X \ τ -1 (S) which has locally bounded coefficients along τ -1 (S) when we compute it in a local chart of X near a point of τ -1 (S). So it extends to a holomorphic form α in Ω p X and then defines a section τ * ( α) of the sheaf L p X . But τ * ( α) and α coincide on X \ S, so on X as the sheaf ω p X has no non zero section supported in S. But we have a better estimate. Consider, locally on X, a holomorphic function g ∈ O X such that g.Ω p X ⊂ τ * * (Ω p X ) (such a non zero g existe as the sheaf Ω p X τ * * (Ω p X ) has its support in τ -1 (S)), then the local boundness assumption on the functions ρ j near any point in S gives a local constant C on X such that our section α has coefficients bounded by C.|g| in a local chart of X on which Ω p X is a free O X -module. The following lemma allows to conclude that α is a section of the sheaf τ * * (Ω p X ) and then α = τ * ( α) is a section of the sheaf τ * τ * * (Ω p X ).

Lemma 2.0.2 Let M be a normal complex space and let F ⊂ O N M be a coherent sheaf such that the quotient sheaf O N M F has support in a closed analytic subset S without interior point in M. Let σ ∈ Γ(M, O N M ) such that for any local system of generators g 1 . . . , g k of the annihilator of the sheaf O N M /F in a neighbourhood of a compact set K in M there exists a constant C such that we have

σ ≤ C. sup i∈[1,k] |g i | at each point of K. Then σ is in Γ(M, F ).
Proof. Let τ : M → M be a proper modification of M with M normal, such that the sheaf τ * * (F ) ⊂ O N M becomes locally free (of rank N). Then we have, locally on M a holomorphic function γ ∈ O M such that γ.O N M = τ * * (F ). Fix a point y 0 in M and note x 0 := τ (y 0 ). Let g 1 , . . . , g k be holomorphic function in an open neighbourhood of x 0 which generates the ideal J in O M which annihilates O N M F . Then, near y 0 there exists i ∈ [1, k] and a holomorphic invertible function θ near y 0 such that γ = g i .θ near y 0 . As we have σ ≤ C i .|g i | near x 0 by hypothesis, there exists some constant C with τ * * (σ) ≤ C.|γ| near y 0 . Then the meromorphic section τ * * (σ)/γ of the sheaf O N M is holomorphic near y 0 by normality of M and then τ * * (σ) belongs to τ * * (F ). But, as τ * (τ * * ) induces the identity on the sheaf O M , σ belongs to Γ(M, F ).

Definition 2.0.3 The graduate sub-sheaf α • X := τ * (τ * * (Ω • X )) of the sheaf L • X is independent of the choice of the desingularization. So it is naturally defined and O X -coherent thanks to Grauert's direct image theorem.

Remark. A direct proof of the fact that the sub-sheaf τ * (τ * * (Ω • X /torsion) is independent of the choice of the desingularization τ is rather easy. But it does not gives the implication (B) ⇒ (A), and property (B) is a simple characterization of the sheaf α • X which does not use any desingularization of X. Nevertheless neither (A) nor (B) are easy to compute on examples (see section 6).

Corollary 2.0.4 Let X be a pure dimensional reduced complex space and let X := ∪ i∈I X i be its decomposition in irreducible components. Then the sheaf α • X has a natural injection in the locally finite direct sum of the direct images in X of the sheaves α

• X i for i ∈ I.
Proof. This is an easy consequence of the fact that a section of the sheaf L • X is a section of α • X if and only if it satisfies the condition (B) in the previous theorem, because we have an isomorphism L

• X ≃ ⊕ i∈I (j i ) * (L • X i ), where j i : X i . → X is the inclusion. Note that when X is not irreducible the injective map α • X → ⊕ i∈I (j i ) * (α • X i ) is not an isomorphism, in general, because the injective map Ω • X /torsion ֒→ ⊕ i∈I (j i ) * (Ω • X i /torsion) is not an isomorphism, in general.
Remark that, for each i ∈ I, and any point x ∈ X i , the "restriction" map

α • X,x → α • X i ,x is surjective because each restriction map Ω • X,x /torsion → Ω • X i ,x /torsion is surjec- tive.
3 Universal pull-back for α •

X

The main result of this paragraph if the following theorem.

Theorem 3.0.1 For any holomorphic map f : X → Y between reduced complex spaces, there exists a functorial2 graduate O Y -morphism

f * : f * α • Y → α • X
which is compatible with the usual pull-back of the sheaf Ω • Y /torsion. For any holomorphic maps f : X → Y and g : Y → Z between reduced complex spaces we have

f * (ĝ * (α)) = g • f * (α) ∀α ∈ α • Z . (1)
Let now give a precise formulation of this result. For that purpose let C be the category of reduced complex spaces with morphisms all holomorphic maps. We may enrich this category, using the universal pull-back property for the graded sheaf Ω • X /torsion : Let C dif f be the category whose objects are pairs (X, Ω • X /torsion) where X is an object in C and where the morphisms are given by pairs (f, f * ) where f :

X → Y is a morphism in C and f * : f * (Ω • Y /torsion) → Ω • X /torsion is the graded pull-back by f of holomorphic forms modulo torsion (see section 2). Of course the forget-full functor G 0 : C dif f → C obtained by (X, Ω • X /torsion) → X, (f, f * ) → f is an equiv- alence of category.
Then the precise content of the theorem above is the following result.

Theorem 3.0.2 [Precise formulation] There exists a category C b-dif f whose objects are pairs (X, α • X ) where X is in C and where the graded coherent sheaf α • X has been defined in section 2 for any object X in C. The morphisms are given by pairs

(f, f * ) for each f : X → Y a morphism in C where f * : f * (α • Y ) → α • X is the graded O X -
linear sheaf map defined by f . Moreover, the following properties holds:

1. For each X ∈ C we have a graded O X -linear injection η X : Ω • X /torsion → α • X .

For any morphism

f : X → Y in C we have a commutative diagram of graded O X -linear maps of sheaves f * (Ω • Y /torsion) f * / / f * (η Y ) Ω • X /torsion η X f * (α • Y ) f * / / α • X (2)
where f * is the graded O X -linear map of coherent sheaves on X associated to the holomorphic map f .

Of course the interest of this result comes from the fact that the sheaf α • X is, in general, strictly bigger that the sheaf Ω • X /torsion; see section 6. Note also that the sheaf L • X does not have such a functorial pull-back by holomorphic maps: let τ : X → X be a desingularization of X ∈ C and let x ∈ X be a point such that τ -1 (x) has dimension ≥ 1. Let ω be a holomorphic form near τ -1 (x) in X which does not induce a torsion form on an irreducible component Γ of τ -1 (x). Then, because the map τ |Γ : Γ → X factorizes by the constant map to {x} the pull-back of ω on Γ has to be zero. But this map factorizes also by the inclusion of Γ in X and τ . As the pull-back by τ is injective (by definition of L • X ), this gives a contradiction. Such an example is given in section 6.3.

Preliminaries. Consider the following situation : let Z be a connected complex manifold and consider a proper holomorphic map π : Z → X which is surjective on a reduced (irreducible) complex space X. Let q := dim Zdim X and let k be the number of connected components of the generic fibre of π. Assume that we have a kähler form ω on Z. Then, after a suitable normalization of ω, the smooth (q, q)-form w :=1 k .ω ∧q is d-closed and satisfies the condition π * (w) = 1 as a d-closed (0, 0)-current on X. This is consequence of the fact that in the Stein factorization π 0 : Z → Y, θ : Y → X of π, the reduced complex space Y is irreducible and this implies that the generic fibres of π 0 are in the same connected component of the space of q-cycles in Z. So the volume computed by ω ∧q of the connected components of the generic fibres of π is constant, and we may normalized ω in order that this volume is equal to 1. Then the d-closed (0, 0)-current π * (w) on X is equal to 1 on a dense Zariski open set in X. This implies our claim.

Assume now that the complex manifold Z has finitely many connected components Z 1 , . . . , Z r such that the restriction of π is surjective on each Z j and such that each Z j has a kähler form ω j . We can normalize each ω j in order that the form w j := 1 k j .ω ∧q j has integral equal to 1/r.k j on each connected component of the generic fibres of π j := π |Z j and then the smooth form w := r j=1 w j satisfies again the condition π * (w) = 1 and for each j any connected component of the generic fibres of π j has the same w-volume equal to 1/r.k j .

In this situation we shall say that the smooth for w on Z satisfies the condition (@).

The proof of the theorem 3.0.2 will use the following proposition.

Proposition 3.0.3 Let X = ∪ i∈I X i be the decomposition of a reduced complex space X as the union of its irreducible components. Let Z := ∪ j∈J Z j be a disjoint union of connected complex kähler manifolds. Assume that we have a map θ : J → I which is surjective and has finite fibres. Let π : Z → X be a proper holomorphic map such that for each j ∈ J it induces a surjective map

π j : Z j → X θ(j)
and let q j := dim Z jdim X θ(j) . For each i ∈ J let w j be a smooth (q j , q j )-form on Z j which is d-closed and satisfies the condition (@) relative to the restriction of π to π -1 (X i ) = ∪ θ(j)=i Z j (see preliminaries above). Note w := j∈J w j . Let β be a section on Z of the sheaf π * * (Ω p X ). Then we have:

of the sheaf π * * (Ω p X ) is stronger than the condition on each β j := β |Z j , j ∈ J to be a section of the sheaf π * * j (Ω p X θ(j) ).

2. In general, a section β ∈ Γ(Z, π * * (Ω p X )) is not equal to some π * * (α) where α is in Γ(X, Ω p X ) even in the case where π : Z → X is a desingularization of X.

Proof. Thanks to the previous remark, we may assume that X is irreducible to prove assertions 1) and 3) of the proposition.

In the case q j = 0 the map π j is generically finite and w j is a locally constant function on Z with a prescribed value on each Z j . So there is no choice for w j and the first assertion of the proposition is trivial. As the second assertion is also clear in this case (the sheaf ω p X has no torsion on X by definition), we shall assume q j ≥ 1 in the sequel. The fact that the current π * (β ∧ w) is ∂-closed on X is consequence of the fact that on each Z j the smooth (p + q j , q j ) form β ∧ w j is ∂-closed and of the holomorphy of π. Let w ′ be a smooth form on Z which is d-closed and satisfies the condition (@). We want to show that π * (β ∧ (ww ′ )) vanishes as a section of the sheaf ω p X . Let X ′ be the open and dense subset of X of smooth points for which the Stein factorization of each π j : Z j → X is a covering of degree k j . Remember that, as we assume that X is irreducible here, the set I is reduced to one point and so J is a finite set. On this open set X ′ it is enough to prove that for each j ∈ J the current (π j ) * (β ∧ (w jw ′ j )) vanishes. So we can fix j and replace locally X ′ by one sheet of the corresponding finite covering and make the proof in this case. That is to say that we may assume that Z is smooth and connected and that π : Z → X has connected fibers on X. In this case the generic fibres of π are irreducible and of dimension q. For any x ∈ X ′ there exists an open neighbourhood V (x) of π -1 (x) which is a deformation retract of π -1 (x). Then we have an isomorphism H 2q (V (x), C) → C which is given by integration on π -1 (x). But w and w ′ have the same integral on π -1 (x) by the property (@). So there exists a (2q-1) smooth form θ on V (x) such that dθ = w-w ′ by de Rham's theorem. Consider now a small open neighbourhood U of x in X ′ such that π -1 (U) ⊂ V (x). Let x 1 , . . . , x n be a local coordinate system on U. Then the sheaf π * (Ω p X ) is a free sheaf of O Z -modules on π -1 (U) with basis π * (dx L ) where L runs in all ordered sub-sets of cardinal p in [1, n]. If we write β = |L|=p g L .π * (dx L ) on U the holomorphic functions g L on π -1 (U) are constant along the fibres of π and so there exists holomorphic functions f -p,n) . By definition of the direct image we have

L , |L| = p with g L = π * (f L ) (recall that U is a smooth open set in X). This means that there exists a holomorphic p-form α on U such that β = π * (α) on π -1 (U). Let ψ ∈ C ∞ c (U) (n
π * (β ∧ dθ), ψ = π -1 (U ) β ∧ dθ ∧ π * (ψ).
But it follows from the equality β = π * (α) on π -1 (U) that the form

β ∧ π * (ψ) = π * (α ∧ ψ) is d-closed as α ∧ ψ is d-closed on U (its degree is 2n). So by Stokes formula the integral π -1 (U ) β ∧ dθ ∧ π * (ψ) = ± π -1 (U ) d β ∧ θ ∧ π * (ψ)
vanishes. This implies that the section π * (β ∧ (ww ′ )) of the sheaf ω p X vanishes on the open dense subset X ′ , so everywhere on X as the sheaf ω p X has no torsion. The assertion 3) of the proposition is clear, because the equality is obvious at the generic points in X.

Let now prove the assertion 2). We no longer assume that I has a unique point. Let τ : X → X be a desingularization of X, so X is the disjoint union of desingularizations τ i : Xi → X i for each i ∈ I, and consider the commutative diagram

X × X,str Z π τ / / Z π X τ / / X
where X × X,str Z is the strict transform, so the union of irreducible components of X × X Z which dominate some Xi . Then the p-form τ * * (β) gives, for each such component, a section of the sheaf (τ • π) * * (Ω p X ) and as the d-closed form τ * (w) satisfies the condition (@) for the map π, the ∂-closed current π * (τ * * (β) ∧ τ * (w)) is in fact a p-holomorphic form on X thanks to Dolbeault-Grothendieck's lemma. This already proved that α := π * (β ∧w) is a section of the sheaf L p X , because τ * * (π * (β ∧ w)) = π * (τ * * (β) ∧ τ * (w))) at the generic points of X, so everywhere on X. Now the map η : Ω p X → Ω p X given by γ → π * (π * * (γ) ∧ τ * (w)) is the identity map, thanks to the assertion 3). So, if π * * (γ) gives a section of the image of the sub-sheaf π * * τ * * (Ω p X )) of the sheaf Ω p X× X,str Z torsion, γ will be a section of the image of the sub-sheaf τ * * (Ω p X ) because the map π * : π * (Ω p X ) → Ω p ( X× X,str Z) is injective.

Apply this to γ := τ * * (α) = π * τ * * (β) ∧ τ * (w) which is a section of Ω p X as we already proved that α is a section in L p X ; we obtain that τ * * (α) is a section of the sheaf τ * * (Ω p X ) because, as the diagram above commutes, τ * * (β) is a section of the sheaf τ * * π * * (Ω p X ) = π * * τ * * (Ω p X thanks to the lemma 1.0.4.

Remark. If Z is not assumed to be smooth in the previous proposition, replacing Z by a projective desingularization σ : Z → Z (as before, this means that Z is the disjoint union of projective desingularizations σ j : Zj → Z j for j ∈ J), the proposition applies to the proper map σ • π and to β := σ * (β) which is a section of the sheaf (σ • π) * * (Ω p X ). Then the result is still true.

Proof of the theorem 3.0.1. The first step in proving the theorem will be the construction of f * (α) ∈ α

f / / Ỹτ X f / / Y
where X ⊂ X × Y Ỹ is the strict transform of X, that is to say the union of irreducible components of X × Y Ỹ which dominate an irreducible component of X, and where π and f are induced by the natural projections of X × Y Ỹ . Now the problem is local on X and Y and we may assume that X, Y and Ỹ are kähler. So we may assume that we have on X a smooth d-closed form w which satisfies the condition (@) for the proper map π (we may replace X by a projective desingularization to reach the precise situation of the proposition 3.0.3; see the remark above).

Then, as β is a section of τ * * (Ω p Y ), the form f * * (β) is a section of π * * (Ω p X ) because if we write locally on Ỹ

β := l g l .τ * * (ω l )
where ω l are local sections of Ω p Y and g l are holomorphic functions on Ỹ , we obtain

f * * (β) = l f * (g l ). f * * (τ * * (ω l ))
and the equality f * * (τ * * (ω l )) = π * * (f * * (ω l )) due to the commutativity of the diagram and the lemma 1.0.4 shows that f * * (β) is a section of the sheaf π * * (Ω p X ). So we can apply the proposition 3.0.3 and obtain that π * ( f * * (β) ∧ w) is a section of the sheaf α p X . This will give the definition of f * (α) when we will have proved that it is independent of the choice of the desingularization τ : Ỹ → Y . Note that the proposition 3.0.3 already gives the independence of the choice of w (assumed d-closed and satisfying (@)) in this construction. Note also that the proposition 3.0.3 gives also that for α a section of Ω p Y /torsion f * (α) is a section of Ω p X /torsion and coincides with the usual pull-back f * (α) (see section 2).

Remark now that, as the sheaf α p X has no torsion on X, to prove the independence of f * (α) on the choice of the desingularization τ , it is enough to prove it at the generic points of X. Moreover, this problem is local on X and so we may assume that X is smooth and connected. In our construction, we sum the various direct images (π j ) * ( f * (β) ∧ w j ) when j describes the various connected components of the desingularization of X. Each such component is send by f in a connected component of Ỹ and then it is enough to show the invariance of the current (π j ) * ( f * (β) ∧ w j ) if we change only one connected component of Ỹ in the given desingularization, and also if we consider only the corresponding connected components of the desingularization of X. So, in fact, it is enough to prove the following special case of our problem: Assume that X is smooth and connected and that Y is irreducible. Let τ : Ỹ → Y is a projective desingularization of Y and that θ : Ỹ → Ỹ is a proper smooth modification of Ỹ . So our new desingularization of Y is θ • τ : Ỹ → Y . Note that we can make this assumption without any lost of generality because two desingularizations can always be dominated by a third one (thanks to Hironaka). Now we shall consider the following diagram, where X is a desingularization of an irreducible component of the strict transform X × Y Ỹ and X is a desingularization of the strict transform of X × Ỹ Ỹ :

Xθ f / / Ỹθ Xτ f / / Ỹτ X f / / Y
Let q the dimension of the generic fibres of τ and k the number of connected components of its generic fibres. Let ω be a kähler form of Ỹ normalized in order that the form f * (ω ∧q ) satisfies the condition (@) for the map τ . Let q be the dimension of the generic fibre of θ and let ω a kähler form on Ỹ normalized in order that the form f * (ω ∧q ) satisfies the condition (@) for the map θ. Now consider the (q + q, q + q)-smooth form w := f * (θ * (ω ∧q )∧ ωq ) on X which is d-closed. It satisfies the condition (@) for the map θ • τ . So the definition of f * (α) using the desingularization θ • τ is given by

( θ • τ ) * ( f • θ) * (β) ∧ w ,
But, as f * * (β) is a section of the sheaf Ω p X /torsion, we have the equality

θ * ( θ * * ( f * * (β)) ∧ f * (ω q)) = f * * (β)
and the conclusion follows from the fact that

( θ • τ ) * ( f • θ) * * (β) ∧ w = τ * [ θ * θ * * ( f * * (β)) ∧ f * (ω q) ∧ f * (ω q ))].
The compatibility of this construction with the pull-back of holomorphic forms modulo torsion which is given by the last assertion of the proposition 3.0.3 obviously gives that the injective O X -linear morphism

η X : Ω • X /torsion → α • X
for each X ∈ C gives the commutative diagram (2) of the precise formulation 3.0.2 of the theorem for each morphism f : X → Y in C.

Now we have to prove the functoriality of f * . Then consider a holomorphic maps f : X → Y and g : Y → Z. We want to prove the formula (1) of the theorem. Consider the commutative diagram

X f / / θ Ỹθ X f / / τ 2 Ỹ g / / τ 1 Zτ X f / / Y g / / Z
where τ : Z → Z is a desingularization, where g : Ỹ → Z is the strict transform of g by τ , where f : X → Ỹ is the strict transform of f by τ 1 , where f : X → Ỹ is the strict transform of f by θ : Ỹ → Ỹ which is a desingularization of Ỹ . Let α be a section of α p Z , note β := τ * * (α) ∈ τ * * (Ω p Z )3 and let w 1 and w 2 be smooth d-closed forms satisfying the condition (@) of the proposition 3.0.3 for the maps τ 1 and θ respectively. We have ĝ * (β) = (τ 1 ) * (g * * (β) ∧ w 1 ) but we have also, because g * * (β) is a section of τ * * 1 (Ω p Y ) ĝ * (β) = (θ • τ 1 ) * (θ * * (g * * (β) ∧ θ * (w 1 ))).

Then we obtain

f * (ĝ * (α)) = ( θ • τ 2 ) * f * * (θ * * (g * * (β))) ∧ f * (w 1 ) ∧ w 2 .
As the square

X f •g / / τ 2 Zτ X f •g / / Z
is also the strict transform of f • g by τ we have

f • g * (α) = (τ 2 ) * ( f • g) * * (β) ∧ f * (w 1 ) .
Then the conclusion follows from the equality

θ * f * * (θ * * (g * * (β))) ∧ f * * (θ * (w 1 )) ∧ w 2 = f * * (g * * (β)) ∧ f * (w 1 )
obtained by the comparaison of both hand-sides at the generic points of X.

Our next result shows that the sheaf α • X is "maximal" in order to construct the pull-back via the method of the proposition 3.0.3. Theorem 3.0.4 Let π : Z → X be a proper surjective holomorphic map between irreducible complex spaces. Put q := dim Zdim X. Let β ∈ α p Z be equal to π * (α) for a section α of the sheaf α p X . Let also w be a smooth (q, q)-form on Z which is d-closed and satisfies the condition (@) of the proposition 3.0.3. Then the (p, 0)-current π * (β ∧ w) on X (which is ∂-closed and independent of the choice of w satisfying dw = 0 and (@); see proposition 3.0.3) is equal to the image in ω p X of the section α of the sheaf α p X . Using the "pull-back" theorem 3.0.1 the theorem above will follow from the following simple lemma.

Lemma 3.0.5 Let X be a complex space and τ : X → X a desingularization of X. Then the image of the pull-back τ * : τ

* (α • X ) → α • X = Ω • X is the subsheaf τ * * (Ω • X ) of Ω • X .
Proof. By definition, a section of this image is locally on X a O X -linear combination of holomorphic forms on X which are locally O X -linear combinations of pull-back by τ of holomorphic forms on X. So the conclusion is clear.

Proof of the theorem 3.0.4. Let π : Z → X be the strict transform of π by τ , and denote by τ : Z → Z the corresponding projection on Z which is a modification. So we have the following commutative diagram

Z π / / τ Xτ Z π / / X
The (q, q)-form τ * (w) is smooth and d-closed in Z and satisfies the condition (@) of the proposition 3.0.3 for the proper surjective holomorphic map τ • π. As we can write β = π * (α) where α is a section of α p X , we have, by functoriality of the pull-back for the sheaf α • Y and the equality τ

• π = π • τ τ * (β) = π * (τ * (α)).
But, thanks to the previous lemma, we have τ * (α) which is a section of τ * * (Ω p X ) and using the smoothness of X we have π * = π * * . Then we obtain, using the lemma 1.0.4, the fact that τ * (β) is a section of the sheaf (τ • π) * * (Ω p X ). Then the proposition 3.0.3 applies to the map τ • π : Z → X with the form τ * (w) and the section τ * (β) of the sheaf (τ • π) * * (Ω p X ) and gives that the (p, 0)-current on X given by

σ := (τ • π) * (τ * (β) ∧ τ * (w)
) is ∂-closed on X and is a section of the sheaf α p X . But the (p+q, q)-current τ * (τ * (β)∧ τ * (w)) is equal to β ∧w at least over the generic points in X, the (p, 0)-current ∂-closed in X π * (β ∧ w) is generically equal to σ and α. So α and σ are equal as sections of the sheaf α p X .

Integration on cycles

Notations. Let V be a complex manifold and h be a continuous hermitian form on V . So h is a real continuous definite positive (1, 1)-differential form on V . If ω is a continuous (p, p)-form on V , we shall consider ω as a continuous sesqui-linear form on Λ p (T V ) and we shall write

ω K ≤ C.h ∧p
where K is a subset in V and C > 0 a constant, if for any point x ∈ K and any v 1 , . . . , v p ∈ T V,x the inequality

|ω(x)[v 1 ∧ • • • ∧ v p ]| ≤ C.h ∧p (x)[v 1 ∧ • • • ∧ v p ] holds. For instance, if α, β ∈ Ω p V we shall write α ∧ β K ≤ C K .h ∧p when for any x ∈ K and any v 1 , . . . , v p ∈ T V,x we have |α(x)[v 1 ∧ • • • ∧ v p ]|.|β(x)[v 1 ∧ • • • ∧ v p ]| ≤ C K .h ∧p (x)[v 1 ∧ • • • ∧ v p ].
(1)

Remark. If f : W → V is a holomorphic map and if (1) holds then we shall have

f * (α) ∧ f * (β) f -1 (K) ≤ C K .f * (h) ∧p (2)
but, in general, f * (h) is no longer definite positive on W . Conversely if (2) holds on a set L in W then (1) is satisfied on f (L).

Proposition 4.0.1 Let X be a reduced complex space, let S be the singular set in X and let h be a continuous hermitian metric on X. Let U be a relatively compact open set in X. For all α, β ∈ α p X there exists a constant C U > 0 such that the following inequality holds at each point in

Ū \ S α ∧ β Ū \S ≤ C U .h ∧p Ū \S .
Proof. Remark that the problem is local on the compact set Ū ∩ S because near smooth points in X the assertion obviously holds. Let τ : X → X be a desingularization of X. Then we shall show that for each point y ∈ τ -1 ( Ū ∩ S) there exists an open neighbourghood W of y in X and a positive constant C W such that the inequality where x 1 , . . . , x N are local coordinates in a closed embedding of an open set U ⊂⊂ X in C N near τ (y). Our estimates is consequence of the facts that the holomorphic functions g I and h I are locally bounded and that for any (I, J) there is a constant c I,J U > 0 with dx I ∧ dx J ) U ≤ c I,J U .h ∧p because we can assume that h in induced by a continuous hermitain form on C N . Now the properness of τ allows to find a a constant C U such that the inequality

τ * * (α) ∧ τ * * (β) W ≤ C W .τ * * (h) ∧p
τ * * (α) ∧ τ * * (β) K ≤ C U .τ * * (h) ∧p
holds on the compact set K := τ -1 ( Ū ). This allows to conclude thanks to the remark above.

Corollary 4.0.2 Let X be a complex space of pure dimension n, and let α, β be sections on X of the sheaf L n X . Then, if ρ is a continuous compactly supported function on X the integral X\S ρ.α ∧ β is absolutely convergent for any closed analytic subset S containing the singular set in X and its value does not depends on the choice of S.

If, moreover, α and β are sections of the sheaf α n X , for any continuous hermitian metric h on X there is constant C > 0 depending on α, β and on the support K of ρ such that

| X\S ρ.α ∧ β| ≤ C. K |ρ|.h ∧n ≤ C.||ρ||. K h ∧n .
(3)

Proof. The first part is consequence of the fact that τ * * (α) and τ * * (β) are holomorphic n-forms on X. The estimates when α, β are sections of α n X is a direct consequence of the previous proposition.

Definition 4.0.3 For α, β sections of the sheaf L n X the common values of the absolutely convergent integrals X\S ρ.α ∧ β will be denoted simply by X ρ.α ∧ β.

Lemma 4.0.4 Let f : Y → X a proper generically finite and surjective holomorphic map between two complex spaces of pure dimension n; let k be the generic degree of π. Let α, β be sections on X of the sheaf L n X and ρ ∈ C 0 c (X). Then the holomorphic n-forms f * * (α) and f * * (β) are well defined on a dense Zariski open set in Y and extend as sections on Y of the sheaf L n Y . We have the equality

X ρ.α ∧ β = k. Y f * (ρ).f * * (α) ∧ f * * (β).
Proof. Remark that it is enough to prove the lemma for α = β. As τ * * (α) is an holomorphic n-form on X implies that α is locally L 2 on X. Let S ε be an open ε-neighbourhood of S a closed analytic subset in X such that the map f : Y \ f -1 (S) → X \ S is a finite covering between two complex manifolds. Then the usual change of variable gives, if

ρ is in C 0 c (X) X\Sε ρ.α ∧ α = k. Y \f -1 (Sε) f * (ρ).f * * (α) ∧ f * * (α).
Letting ε goes to 0 shows that f * * (α) is locally L 2 on any desingularization of Y and so f * * (α) is a section of the sheaf L n Y . The conclusion follows easily.

Definition 4.0.5 Let X be a complex space and let Y ⊂ X be an irreducible p-dimensional analytic subset in X. We shall denote j : Y → X the the inclusion. Let α, β be sections of the sheaf α p X on X and ρ be a continuous function with compact support in X. We shall define the number Y ρ.α ∧ β as the integral

Y j * (ρ). ĵ * (α) ∧ ĵ * (β).
Note that this definition makes sense because the pull-back ĵ * : j * (α p X ) → α p Y is well defined and because the inclusion α p Y ⊂ L p Y allows to use the definition 4.0.3. Remark that this definition only depends on the irreducible analytic subset Y of X. So we may extend by additivity the definition of the integral

Y ρ.α ∧ β to any p-dimensional cycle Y in X.
Remark. It is not clear that the definition 4.0.5 may be extended to sections of the sheaf L p X for 0 < p < n, for a p-cycle Y contained in the singular locus of X.

The next lemma shows that the change of variable holds for such a integral.

Lemma 4.0.6 Let f : X → Y be a holomorphic map and let α, β be sections on Y of the sheaf α p Y . Let ρ be a continuous compactly supported function on Y . Let Z be a p-cycle in X and assume that the cycle f * (Z) is defined in Y4 . Then the restriction to |Z| of the continuous function f * (ρ) has compact support and the integral Z f * (ρ). f * (α) ∧ f * (β) is well defined and we have

Z f * (ρ). f * (α) ∧ f * (β) = f * (Z) ρ.α ∧ β.
proof. First remark that any irreducible component Γ of Z which has an image of dimension at most equal to p -1 does not contribute to the right hand-side and also to the left hand-side because the forms f * (α) and f * (β) vanish on such a irreducible component : Let g : Γ → f (Γ) be the map induced by f ; by functoriality of the pull-back ĝ * factorizes through α p f (Γ) which is zero. Then the result is in fact a local statement near each point of the set |f * (Z)|. And because of our previous remark and the fact that closed analytic subsets with no interior point can be neglected in the integrals, it is enough to prove the result when Z is smooth and when f induces an isomorphism of Z on f (Z). In this case, which is not trivial because Z and f (Z) can be contained in the singular sets of X and Y , the functorial property of the pull-back and the fact that for a complex manifold V we have α p V = Ω p V allow to conclude.

Theorem 4.0.7 Let X be a complex space and let (Y t ) t∈T be an analytic family of p-cycles in X parametrized by a reduced complex space T . Let α, β be sections of the sheaf α p X on X and ρ be a continuous function with compact support in X. Then the function ϕ : T → C defined by

ϕ(t) := Yt ρ.α ∧ β
is locally bounded and for any given hermitian metric h on X there exists a constant C such the following estimate holds for each t ∈ T :

|ϕ(t)| ≤ C. Yt |ρ|.h ∧p . (E)
Moreover for each point t 0 ∈ T there exists an open neighbourhood T 0 of t 0 in T and a closed analytic subset Θ 0 with no interior point in T 0 such that ϕ is continuous on T 0 \ Θ 0 .

Proof. We shall cut this proof in several steps.

Step 1. Let ν : T → T the normalization of T . The family (Y ν( t) ) t∈ T is an analytic family of p-cycles in X parametrized by T , and if the theorem is proved for this family it implies the result for the initial family, because the function is constant on the fibres of the normalization map. So we shall assume that T is normal in the sequel.

Step 2. If the generic cycle Y t is not reduced and irreducible, the normality of T allows to write the family (Y t ) t∈T as a finite sum of analytic families of p-cycles in X parametrized by T such the sum of these families is our initial family (see [B-M 1] ch. IV theorem 3.4.1). So it is enough to prove the theorem for such a family. We shall assume that for t generic in T the cycle Y t is reduced and irreducible.

Step 3. Let G ⊂ T × X the cycle-graph of our analytic family. It is a reduced cycle and the projection π : G → T is (by definition) a geometrically flat map, that is to say that there exists an analytic family of cycles (Z t ) t∈T in G such that for each t ∈ T we have |Z t | = π -1 (t) and such that the generic cycle Z t is reduced and irreducible. Of course, here we have Z t := {t} × Y t for each t ∈ T . Note pr : G → X the projection and define on G the sections of the sheaf α p G by letting α 1 := pr * (α) and β 1 := pr * (β). Then, it is enough to prove the theorem for the function t → Zt ρ.α 1 ∧ β1 where ρ := pr * (ρ) thanks to the change variable theorem proved in lemma 4.0.6. Remark that pr induces an isomorphism of |Z t | onto |Y t | for each t ∈ T . Note also that the continuous function ρ on G has a π-proper support.

Step 4. Let τ : G → G be a desingularization of G. Define the subset

Θ := {t ∈ T / ∃y ∈ K dim y (τ • pr) -1 (t) ≥ p + 1}.
This is a locally closed analytic subset5 in T with no interior point. For t 0 ∈ T fix an open neighbourhood T 0 of t 0 small enough in order that Θ 0 := Θ ∩ T 0 is a closed analytic subset. The map

q : G ∩ (τ • pr) -1 (T 0 ) \ (τ • pr) -1 (Θ) → T 0 \ Θ 0
is p-equidimensional on a normal basis, so it is geometrically flat and we have an analytic family ( Zt ) t∈T 0 \Θ 0 of fibres of q which are p-cycles in G, and for t generic in T 0 \ Θ 0 the cycle Zt is irreducible. Note that the pull-back of α 1 and β 1 on G ∩ (τ • pr) -1 (T 0 ) are holomorphic p-forms. So, by the usual result of the continuity of integration of a continuous form on a continuous family of cycles (see [B-M 1] ch. IV prop. 2.3.1), we conclude using the lemma 4.0.6 that the function ϕ is continuous on T 0 \ Θ 0 .

Step 5. The local boundness on T of the function ϕ is given by the corollary 4.0.2 which gives the the estimate (E) by integration.

Remarks.

1. In the case of a proper family of compact cycles in X, it is easy, using results of [B-M 1] chapter IV, to prove that the function ϕ becomes continuous after a suitable modification of the complex space T .

2. Already in the case of the normalization map, if α is a locally bounded meromorphic function on X, the function x → |α(x)| 2 is not continuous on X in general.

Integral dependence equations and normalized Nash transform

We shall use here the notion of linear bundle on a reduced complex space X . Recall that there is an equivalence between coherent sheaves on X and linear bundles on X which is local and sends a O X -coherent sheaf F having a local presentation O q X M -→ O p X → F → 0 to the linear bundle F ⊂ X × C p given as the kernel of the map t M : X × C p → X × C q . The inverse correspondence associates to the linear bundle F → X the sheaf of linear bundle homomorphism U → Hom U (F, C). In particular, when F is a vector bundle, the associated coherent sheaf is the sheaf of holomorphic section of the dual vector bundle F * .

In this section we consider a normal complex space X.

Integral dependence equations

Two examples.

1. We shall show in section 6.2 that for k ≥ 2 and k -1 ≥ q ≥ k/2 the form ω q := z q .(dx/xdy/y) is a section of the sheaf α 1 S k where

S k := {(x, y, z) ∈ C 3 / x.y = z k }
which are not sections of the sheaf Ω 1 S k /torsion. But as we have dx/x + dy/y = k.dz/z on S k we obtain the equality ω 2 q = k.z 2q-2 .(dz) 2 -4z 2q-k .dx.dy; so ω 2 q is, for q ≥ k/2, a section of S 2 (Ω 1 S k ), the piece of degree 2 in the symmetric algebra of the sheaf Ω 1 S k .

2. We shall show in section 6.4 that on X := {(x, y, u, v) ∈ C 4 / x.y = u.v} the form a := u.dv ∧ dx/x is a section of the sheaf α 2 X which is not in Ω 2 X /torsion. But using the following identities on X: u.dv ∧ dx/x + u.dv ∧ dy/y = dv ∧ du u.dv ∧ dy/y + v.du ∧ dy/y = dx ∧ dy we obtain that a 2 + a.(du ∧ dv + dx ∧ dy) -(dv ∧ dx).(du ∧ dy) = 0 which is a homogeneous integral dependence equation for a on the symmetric algebra of the sheaf Ω 2 X /torsion.

The following proposition shows that these examples are special cases of a general phenomenon.

Proposition 5.1.1 Let X be a normal complex space. Then for each integer q the sheaf α q X is the sub-sheaf of meromorphic sections of the sheaf Ω q X /torsion which satisfy a homogeneous integral dependence equation over the sheaf S • (Ω q X ), the symmetric algebra of the sheaf Ω q X /torsion.

Proof. Let T q → X be the q-th Zariski tangent linear bundle over X, which is, by definition, the linear bundle associated to the O X -coherent sheaf Ω q X . Then a section ω of the sheaf α q X defines a morphism of linear bundles pr : T q → X × C over the open set X \ S. As a function on T q it is locally bounded along pr -1 (S) thanks to property (B) in the theorem 2.0.1. So it defines a meromorphic locally bounded function on the complex space T q and then satisfies an integral dependence equation of the form

ω m + m j=1 a j .ω m-j = 0
locally in an open neighbourhood of S × {0} in T q . As ω is linear on the fibres, we may assume that the holomorphic function a j is a homogeneous polynomial of degree j along the fibres of pr : T q → X, and this conclude the proof, because the symmetric algebra of the sheaf Ω q X /torsion is the algebra of holomorphic functions on T q which are polynomial on the fibres modulo these which vanish on pr -1 (X \ S).

Normalized Nash transform

Notation. For integers n < N we shall denote Gr(n, N) the grassmannian manifold of sub-vector spaces in C N of dimension n.

Let X be a reduced complex space pure of dimension n and let S its singular locus. Assuming that X is embedded in an open set U in C N we have a holomorphic map θ : X \ S → Gr(n, N) sending each point x ∈ X \ S to the n-dimensional vector sub-space of C N which directs the tangent space at x to X. This map is holomorphic on X \S and meromorphic along S. So the closure of its graph in X × Gr(n, N) is a proper modification of X. We shall note N : X → X the projection on X of the normalization of this graph. We shall call the (local) normalized Nash transform of X this modification. Let π : U → Gr(n, N) the universal n-vector bundle of Gr(n, N) and let L q be the sheaf of section of the dual vector bundle to Λ q (U). Let pr : X → Gr(n, N) be the projection.

6 Some examples 6.1 Computation of ω • X for hypersurfaces

We shall need the following elementary lemma.

Γ(X, ω q X ) ⊂ ⊕ k-1 j=0 z j P ′ (z) .Γ(U, Ω q U )
with equality for q = n.

Proof. First will shall prove the following formula, where (j, h) ∈ [0, k] 2 :

det j,h T race X/U ( z j+h P ′ (z) ) = (-1) k.(k-1)/2 .
Assume, without loss of generality, that D is centered at the origin with radius R.

Then for r > R we have, thanks to Cauchy' formula

T race X/U ( z m P ′ (z) ) = 1 2iπ . |z|=r z m .dz P (z) .
Then for m ≤ k -2 put z = r.e i.θ we obtain (m+1-k) .dθ 1 + O(1/r) and letting r → +∞ gives 0. For m = k -1 the same computation gives

T race X/U ( z m P ′ (z) ) = 1 2π . 2π 0 r m+1-k .e i.
T race X/U ( z k-1 P ′ (z) ) = 1 2π . 2π 0 dθ 1 + Q((1/r).e -i.θ )
Where Q is a polynomial without constant term in (1/r).e -i.θ . So we obtain that T race X/U ( z k-1 P ′ (z) ) = 1. This is enough to obtain the formula above.

To prove the inclusion Γ(X, ω q X ) ⊂ 1 P ′ (z) .Γ(U, Ω q U ) take α ∈ ω q X and write

α = |H|=q g H .dt H
where g H are degree ≤ k -1 polynomials in z with meromorphic functions on U as coefficients. As we have P ′ (z).dz = -n h=1 ∂P ∂t h .dt h on X, this is possible. Now for any f ∈ O(X) we have T race X/U [f.α] ∈ Ω q (U) and this implies that for any

H ⊂ [1, n], T race X/U [f.g H ] is in O(U).
Let g be a meromorphic function on X and assume that we write

g = k-1 j=0 a j . z j P ′ (z)
where a j , j ∈ [0, k -1] is a meromorphic function on U. This is always possible for the g H as we can see in what follows. Let m p := T race X/U [z p .g] for p ∈ [0, k -1]. Then we have the linear system in the (a j ), j ∈ [0, k -1]:

k-1 j=0 a j .T race X/U [ z p+j P ′ (z) ] = m p ∀p ∈ [0, k -1].
But the determinant of this linear system is (-1) k.(k-1)/2 , so this implies, if we assume that the functions m p are holomorphic on U, that the functions a j for j ∈ [0, k -1], are holomorphic in U and so that g is in 1 P ′ (z) .O(X). Then our inclusion is proved, as O(X) = k-1 j=0 O(U).z j . Note that in the situation above, the condition in order that α = 1 P ′ (z) .Ω q (U) will be in ω q (X) is that for any j ∈ [0, k -1] the (q + 1)-forms

T race X/U [z j .dz ∧ α]
are holomorphic for all j ∈ [0, k -1]. This is consequence of the fact that for any β ∈ Ω p (X) the (p + 1)-form T race X/U [α ∧ β] must be holomorphic (see [B. 78] for this characterization of the sheaf ω • X ). For q = n this extra condition is empty, so the equality occurs.

Remark. For a general reduced multiform graph X ⊂ U × B where B is now a polydisc in C p , for any linear form l in C p which separates generically the fibers of the projection π : X → U, the map id U × l : U × B → U × C is proper and generically injective on X. If we define Y l := (id U × l)(X), we are in the situation of the lemma above, and, as the direct image by π induces an injective sheaf map π

* : ω • X → π * (ω • Y l ), we obtain the inclusion ω • X ⊂ ⊕ k-1 j=0 l(x) j P ′ l .Ω • U
for any such l, where P l is the canonical equation for Y l (see [B-M 1] chapter II). Note that the canonical equation P l is obtained from the canonical equation of the reduced multiform graph X by the evaluation at l (with z = l(x)); see loc. cit..

Note that, if X is a reduced complex space of pure dimension n, a section α ∈ ω n X is in L n X iff α ∧ ᾱ is locally integrable on X. The analogous characterization for p < n, involves local integrability of α ∧ ᾱ on all p-dimensional irreducible analytic subset Y ⊂ X not contained in the singular set of X; so it may be useful as a necessary condition but very difficult to check as a sufficient condition. which is holomorphic but not in τ * * (Ω 2 S 2 ) ≃ O X .z.dz ∧ da. The assertion is proved for k = 2.

As the assertion is proved for k = 2, we may assume that, for k ≥ 3 the equality is proved for S k-2 . Then let X → X := S k be the blow-up of S k at the singular point x = y = z = 0. In the chart {γ = 0} of X we have the relations da ∧ db z k-4 and, as a consequence of the fact that z k-q-2 is not in the ideal a, b .O S k-2 for q ≥ 1, for no q ≥ 1 the 2-form dx ∧ dy z q is a section of the sheaf τ * * (Ω 2 S k torsion) near the origin a = b = z = 0 in this chart. So the sheaf α 2 S k is equal to Ω 2 S k torsion.

Lemma 6.2.3 For all k ≥ 0 the vector space L 1 S k α 1 S k has dimension p = [(k -1)/2] the integral part of (k -1)/2. A basis is given by the 1-forms x.dy/z q for q in [[k/2] + 1, k -1], for k ≥ 3.

Proof. The lemma is clear for k = 0, 1. We have seen that after blowing-up the singular point in S k for any k ≥ 2 we find only one singular point of the type S k-2 in the chart {γ = 0} and that the form x.dy/z q is given by the following computation in this chart {γ = 0} :

x = z.a, y = z.b, a.b = z k-2 x.dy/z q = a.db/z q-2 + z k-q-1 .dz.

So assuming that k = 2p + 1 ≥ 3 and that we know that the forms a.db/z q-2 for q -2 in {p, . . . , 2p -2} is a basis of the vector space L 1 S 2p-1 α 1 S 2p-1 , this implies that the forms x.dy/z q for q = p + 2, . . . , 2p is a basis of a vector space of dimension p -1 in L 1 S 2p+1 α 1 S 2p+1 because we have a linear map L 1 S 2p+1 /α 1 S 2p-1 /α 1 S 2p-1 given by the pull-back. It is then enough to prove that for q = p + 1 the form x.dy/z p+1 is not in α 1 S 2p+1 and that it completes in a basis the previous free system in L 1 S 2p+1 α 1 S 2p+1 to conclude. In the last chart {γ = 0} in the desingularization process of S 2p+1 by blowing up the unique singular point at each step, we reach the following relations:

x = u p .v p+1 , y = u p+1 .v p , z = u.v x.dy z p+1 = (p + 1).u p-1 .v p .du + p.u p .v modulo (f /t 2 ).Ω 2 C 4 which allows to conclude as t is not a zero divisor in X. Consider now the following sections of ω 1 X :

u := t.ω 1 v := t.ω 2 w := t.ω 3

where ω 2 and ω 3 are deduced from ω 1 respectively by x → y, y → z, z → x and x → z, y → x, z → y. Then we have in the symmetric algebra of Ω 1 X u+v+w = 3t.dt u.v+v.w+w.u = t.(z.dx.dy+x.dy.dz+z.dx.dy) u.v.w = t 3 .dx.dy.dz.

This shows that u, v, w satisfy the following integral dependence relation over the symmetric algebra of Ω 1 X Θ 3 -3t.dt.Θ 2 + t.(z.dx.dy + x.dy.dz + z.dx.dy).Θt 3 .dx.dy.dz = 0.

Note that, because the coefficient of Θ does not belong to (t 2 ), we do not obtain an integral dependence relation over the symmetric algebra of Ω 1 X for Θ/t so for the forms ω i , i = 1, 2, 3 ! In fact they are not sections of the sheaf α 1 X (for instance the restriction of ω 1 to the surface S 3 ≃ {z = 1} ∩ X is not in α 1 S 3 (see sub-section 6.2).

Let us now verify that t.u is not a section of Ω 1 X /torsion. Let assume that we can write y.z.dx = t. λ.dx + µ.dy + ν.dz + θ.dt) modulo f.Ω 1 X + O X .df then, by homogeneity, we may assume that λ, µ, ν are homogeneous of degree 2 and y.z.dx = t. λ.dx + µ.dy + ν.dz + θ.dt) + σ.df where σ is a constant. This implies y.z.(1σ)t.λ = 0, t.µ + σ.x.z = 0 which is already enough to obtain a contradiction, as these equations imply σ = 1 and σ = 0 respectively.

Remark. Using the map ((x, y, z) → (x + y, x + j.y, x + j 2 .y, -z) which sends the previous Y := {x 3 + y 3 + z 3 = 0} to X = {x.y.z = t 3 } allows to find an integral equation over the symmetric algebra of Ω 1 Y of the section (x 2 + y 2x.y).d(x + y) z of α 1 Y .

References.

• [B.78] Barlet, D. Le faisceau ω • X on a reduced complex space, in Sem. F. Norguet III, Lecture Notes, vol. 670, Springer Verlag (1978), p. 187-204. • [B-M.17] Barlet, D. et Magnusson, J. Nearly-smooth complex spaces to appear soon in math-arXiv.

  dependence equations . . . . . . . . . . . . . . . . . . . . . . 22 5.2 Normalized Nash transform . . . . . . . . . . . . . . . . . . . . . . . 23 6 Some examples 25 6.1 Computation of ω • X for hypersurfaces . . . . . . . . . . . . . . . . . . 25 6.2

  holds : if y is a point in X we can write in an open neighbourghood W of y α = |I|=p g I .τ * * (dx I ) and β == |I|=p h I .τ * * (dx I )

  Lemma 6.1.1 Let U be an open polydisc in C n and D an open disc in C. Let X ⊂ U × D be a reduced multiform graph of degree k in U × D with canonical equation P ∈ O(U)[z], which is a monic degree k polynomial in z. Then we have the inclusion

x

  = a.z, y = b.z a.b = z k-2and we find a copy of S k-2 . For k ≥ 3 we havedx ∧ dy = k k -2 .z 2 .da ∧ db = k. So in this chart τ * * (Ω 2 S k torsion) = O S k-2 . a, b .

•

  [B-M 1] Barlet, D. et Magnusson, J. Cycles analytiques complexes I : théorèmes de préparation des cycles, Cours Spécialisés 22, Société Mathématiques de France, Paris 2014. • [B.15] Barlet, D. Strongly quasi-proper maps and the f-flattning theorem, arXiv:1504.01579 (math.CV)

  • X when α is a section of the sheaf α • Y . So let α be a section on Y of the sheaf α p Y . Let τ : Ỹ → Y be a locally projective desingularization of Y . Consider the following commutative diagram Xπ

  .(x 1 .dx 2 ∧ dx 3 + x 2 .dx 3 ∧ dx 1 + x 3 .dx 1 ∧ dx 2 ) ).dx 2 ∧ dx 3 + (x 2 .dx 3x 3 .dx 2 ) ∧ x 1 .dx 1 ) ).dx 2 ∧ dx 3 + (x 2 .dx 3x 3 .dx 2 ) ∧ (-x 2 .dx 2x 3 .dx 3x 4 .dx 4 )) = -x 4 .dx 2 ∧ dx 3x 2 .dx 3 ∧ dx 4 + x 3 .dx 2 ∧ dx 4proving our claim.6.5 The case X := {(x, y, z, t) ∈ C 4 / x.y.z = t 3 } Remark first that the form ω 1 := y.z.dx/t 2 is in ω 1 X because we have, with the notation f := x.y.zt 3 :ω 1 ∧ df = z.t 3 .dx ∧ dy + y.t 3 .dx ∧ dz + 3t 2 .y.z.dx ∧ dt t 2 ∈ Ω 2

	for instance
	x 1 .η x 4	==	x 1 x 4
	= 4 = 1 x 4 .(-(x 2 2 + x 2 3 + x 2 1 x 4 .(-(x 2 2 + x 2 3 + x 2 4 C 4
			p-1 .dv

This means that if f : U → Sym k (X ′ ) is the holomorphic map classifying the fibers of π, the cycle Z w is the cycle-graph of the analytic family of k-tuples in X ′ defined by the restriction of f to V × {w}.

We shall make this precise in the theorem

3.0.2 below.

The ∂-closed (p, 0)-current π * (β ∧ w) on X is independent of the choices of the forms w j , assuming that they are d-closed and satisfy the condition (@).

The section π * (β ∧ w) on X of the sheaf ω p X is a section of the sub-sheaf α p X .

If there exists a section α of the sheaf Ω p X /torsion such that β = π * * (α) on Z, then α = π * (β ∧ w) as a section on X of the sheaf ω p X .Remarks.1. It is enough to prove assertion 1) and 3) of the proposition above for each map π j , j ∈ J because the sheaf ω p X is the direct sum of the sheaves ω p X i , i ∈ I and the restriction of β to Z j is a section of the sheaf π * * j (Ω p X θ(j) ) for each j ∈ J. This is not the case for the assertion 2) of the proposition : the sheaf α p X is a sub-sheaf of the direct sum of the sheaves α p X i , i ∈ I but, in general, strictly smaller than this direct sum. Note also that the condition on β to be a section

See the simple lemma 3.0.5 below.

This means that the restriction of f to |Z| is proper; see [B-M 1] chapter IV.

See, for instance, the lemma 2.1.8 in[B.15].

Proposition 5.2.1 For each integer q there is a canonical isomorphism c q : N * (α q X )/torsion → pr * (L q ).

Proof. Remark first that the pull-back fo holomorphic forms

factorizes via the natural sheaf morphisms

This is a consequence of the fact that L • is the quotient N * (Ω • X /torsion)/torsion. Consider now a desingularization τ : X → X. Then τ := τ •N is a desingularization of X and we have τ * * (Ω

Moreover, if σ is a section of N * (L • ), then τ * (σ) is a section of τ * * (Ω • X /torsion) thanks to the lemma 1.0.4 and then σ is a section of the sheaf α • X .

As a consequence of this proposition we obtain that for a normal complex space we have α q X ≃ N * (L q ) for any integer q ≥ 0.

Lemma 5.2.2 Let X be a reduced complex space and let τ : X → X be any (proper) modification. Then we have a natural inclusion α

Proof. Consider a desingularization θ : X → X and remark that π := θ • τ is a desingularization of X. Then we have

concluding the proof.

Remark. This shows that when we consider a sequence of successive modifications over a reduced complex space X, the sequence of coherent sub-sheaves (τ ν ) * (α • Xν ) is locally stationary on X. For instance, this is the case for iterated normalized Nash transforms over a given X.

The case

Notation. After blow-up (x, y, z) in C 3 the homogeneous coordinates in P 2 will be (α, β, γ). The symetry between x and y allows to consider only the chart {α = 0} on which we put b := β/α, c := γ/α and the chart {γ = 0} on which we put a := α/γ, b := β/γ.

Our first example will be the normal complex spaces, where k ∈ N, k ≥ 2

Note that S 0 and S 1 are smooth complex surfaces. Lemma 6.2.1 For any k ≥ 2 the normal complex space S k is nearly smooth 6 . So we have L

Proof. Let ζ be a k-th primitive root of 1. Then S k is somorphic to the quotient of C 2 by the action of the automorphism θ(u, v)

A rather easy computation shows that the quotient ω 1 X Ω 1 X is generated on O X by the image of x.dy z k-1 = -y.dx/z k-1 + k.dz which is annihilated in this quotient by x, y and z k-1 . Lemma 6.2.2 For any k ≥ 2 the sheaf α 2 S k coincides with Ω 2 S k torsion.

Proof. Remark that for k = 0, 1 the lemma is obvious as S k is smooth. We shall prove the lemma by induction on k ≥ 2. We have to consider the case k = 2 first because it appears that the computation is special in this case (see the denominator k -2 in the computation for k ≥ 3).

For k = 2 after blowing-up the origin we have a smooth manifold:

where (u, v) ∈ C 2 is a local coordinate system. Now assume that there exists holomorphic functions λ, µ, ν near the origin on C 2 and a polynomial Q(z) := 1 + p j=1 q j .z j such that we have

which is a contradiction. Note that the same computation gives

x.dy/z p = (p + 1).u p .v p+1 .du + p.u p+1 .v p .dv = (p + 1).x.du + p.y.dv and as xu = yv = z p+1 = (p + 1).(dz p+1u.dx) + p.(dz p+1v.dy)

Now assume that k = 2p with p ≥ 2 then in the last chart {γ = 0} we shall have, with coordinates (z, u) with u = 0

x = z p .u, y = z p /u so x.dy/z p+1 = p.z p-2 .dzz p-1 .du/u.

Then assume that, again with Q(z)

which gives a contradiction for z = 0. Note that it easy to see that z p .du = dxp.u.z p-1 .dz ∈ τ * * (Ω 1 S 2p torsion).

6.3

The case

where x, y are the coordinates on C 2 . An easy computation shows that the form α := (x.dyy.dx)/z 2 generates ω 1 X Ω 1 X .

Let τ : X → X the blowing-up at the origin of X is a desingularization. In the chart {α = 0} we have

Then we can choose (x, u) or (x, v) as local coordinates and α = du/v 2 = -dv/u 2 . This shows that ω 1 X = L 1 X . But α does not vanish on the exceptional divisor, so α is not a section of α 1 X . But, in the first chart,

X ) and in the second chart

x.α = x.du/v 2 = dy/v 2u.dx/v 2 = -dz/u 2 + v.dx/u 2 also belong to τ * * (Ω 1 X ). Then x.α, y.α and z.α are sections of α 1 X and the quotient L 1 X α 1 X is a vector space of dimension 1 with basis α.

Note that x.y.z.α is not a section of Ω 1 X /torsion because if this is not the case, we can write

x.y.(x.dyy.dx) = z. λ.dx + µ.dy + ν.dz + ρ.df + σ.f in C 3 , where λ, µ, ν where homogeneous of degree 2, ρ is a complex number and where σ := u.dx + v.dy + w.dz with u, v, w complex numbers. This gives, for instance -x.y 2 = z.λ + 3z.ρ.x 2 + u.f which is impossible. So the vector space α 1 X /Ω 1 X has dimension at least 2. The complete determination of the quotient α 1 X /Ω 1 X is a non trivial exercise left to the reader...

Proof. After blowing-up (x, y, z) in C 3 we consider the chart {γ = 0} as above.

We have

. Then x.ω, y.ω, zω are holomorphic in this chart, as we have x = z.a and y = z.b and this chart is enough as dx ∧ dy/z 2 = dy ∧ dz x 2 = dz ∧ dx y 2 so x.ω, y.ω, z.ω belongs to L 2 X . But this is not the case for ω. So dim ω 2 /L 2 X = 1. The sheaf τ * * (Ω 2 X torsion) in this chart is generted by

Then it is equal to z.Ω 2 X in this chart. So a section in

. This intersection is generated by x.y.ω, y.z.ω, z.x.ω as a O X -module. The vector space L 2 X /α 2 X is generated by x.ω, y.ω, z.ω because x 2 .ω, y 2 .ω, z 2 .ω are in Ω 2 X ⊂ α 2 X . We let to the reader the proof that they give a basis of L 2 X /α 2 X .

Let us prove that x.y.z.ω is not in Ω 2 X /torsion. Assume that x.y.z.ω ∈ Ω 2 X /torsion. Then we can write on C 3 :

where we can assume that λ, µ, ν are linear forms on C 3 and a, b, c are complex number, using the homogeneity of the situation. The coefficient of dx ∧ dy in this identity is equal to x.yz.λa.y 2 + b.x 2 which cannot be identically zero. Contradiction.

As it is easy to see that x.y.ω = y.z.ω = z.x.ω and x.y.z.ω are linearly independent over C (different homogeneities) we conclude that dim α 2 X /Ω 2 X = 2.

Remark. We have on X

because on X we have x 2 .dx ∧ dy = -z 2 .dz ∧ dy and y 2 .dx ∧ dy = -z 2 .dx ∧ dz. This gives an integral dependence relation for x.y.ω on Ω 2 X /torsion.

6.4

The case X := {(x, y, u, v) ∈ C 4 / x.y = u.v} Lemma 6.4.1 The sheaf L 3 X is equal to ω 3 X and is given by O X .ω where we define ω := dy ∧ du ∧ dv y .

Moreover, ω does not belong to α 3 X .

proof. On X we have x.dy + y.dx = u.dv

To see that ω

where f := x.yu.v. This is clear.

Using the symetries between the coordinates, it is enough to see that τ * (ω) is holomorphic in the first chart of the strict transform X of X by the blow-up at the origin in

where x, µ, ν are the coordinates for X in this chart (and λ = µ.ν). So ω ∈ L 3 X . To see that ω is not in α 3 X it is enough to see that ω does not belongs to τ * * (Ω 3 X ) in the first chart above. An easy computation show that τ * * (Ω 3 X ) is generated by π * * (dx ∧ du ∧ dv) = x 2 .dx ∧ dµ ∧ dν and so ω = -x.dx ∧ dµ ∧ dν does not belong to τ * * (Ω 3 X ).

Lemma 6.4.2 The meromorphic form w := u.dv ∧ dx/x is a section of α 2 X but it is not a section of Ω 2 X /torsion and its differential is not a section of α 3 X .

proof. As u.dv ∧ dx/x + v.du ∧ dx/x = dx ∧ dy is holomorphic on X, u and v play the same role for this form modulo holomorphic forms. Also u.dv ∧ (dx/x + dy/y) = dv ∧ du so x and y play also the same role modulo holomorphic forms on X. So it is enough to see that in the first chart of the strict transform X of X by the blow-up at the origin in C 4 the form τ * * (w) is a section of τ * * (Ω 2 X ) to prove that w is a section of α 2 X . Using the same coordinates as above we obtain τ * * (w) = µ.x.d(ν.x) ∧ dx/x = µ.x.dν ∧ dx = µ.dv ∧ dx which is a section of τ * * (Ω 2 X ). To see that w is not a section of Ω 2 X /torsion assume the contrary. Then, by symetry 7 w ′ := v.du ∧ dx/x is also a section of Ω 2 X /torsion and the differential of ww ′ must be a section of Ω 3 X /torsion. But we have already seen that 2.ω = -d(ww ′ ) is not a section of α 3 X . Contradiction.

Note that an integral dependence relation on the symmetric algebra of the sheaf Ω 2 X /torsion for w is given in the second example of the begining of the section 5.1.

Then thanks to the lemma 6.1.1 we have ω

X and to show that it is a section of Ω 1 X /torsion. But then T race π (v∧dx 4 ) must be a holomorphic form on C 3 , where π : X → C 3 is the projection which makes X a branched covering of degree 2. This condition implies (x 1 .dx 1 + x 2 .dx 2 + x 3 .dx 3 ) ∧ (a.dx 1 + b.dx 2 + c.dx 3 ) = 0 and then (a.dx 1 + b.dx 2 + c.dx 3 ) = u.x 4 .dx 4 on X, where u is holomorphic on C 3 . So v is a section of Ω 1 X /torsion. Lemma 6.4.4 We have ω 2 X = Ω 2 X /torsion ⊕ C .η where η := x 1 .dx 2 ∧ dx 3 + x 2 .dx 3 ∧ dx 1 + x 3 .dx 1 ∧ dx 2 x 4 .

Proof. Write ω := (a.dx 1 ∧ dx 2 + b.dx 2 ∧ dx 3 + c.dx 3 ∧ dx 1 )/x 4 where a, b, c are holomorphic on C 3 . Then ω is in ω 2 X if and only if T race π (dx 4 ∧ ω) is a section of Ω 3

C 3 . This is satisfyed if and only if a.x 3 + b.x 1 + c.x 2 is a multiple of ξ := x 2 1 + x 2 2 + x 2 3 in O C 3 . This gives the relation (ag.x 3 ).x 3 + (bg.x 1 ).x 1 + (cg.x 2 ).x 2 = 0. And, as x 1 , x 2 , x 3 is a regular sequence, this implies a = g.x 3 + λ.x 1 + µ.x 2 , b = g.x 1 + λ ′ .x 2λ.x 3 , c = g.x 2λ ′ .x 1µ.x 3 where λ, λ ′ , µ are in O C 3 . This shows that ω 2 X is generated as a O X -module by Ω 2 X and η. Note that we already know that η is not a section of Ω 2 X /torsion as we have shown that ω 2 X is not equal to Ω 2 X /torsion and that x i .η is in Ω 2 X /torsion for i = 1, 2, 3, 4 : 7 or using (u.dv + v.du) ∧ dx/x = dy ∧ dx.


Aymeric Vié
email: aymeric.vie@stanford.edu

Benedetta Franzelli
email: franzelb@stanford.edu

Yang Gao
email: yang.gao@uconn.edu

Tianfeng Lu

Hai Wang
email: hai.wang@stanford.edu

Matthias Ihme
email: mihme@stanford.edu

Analysis of segregation and bifurcation in turbulent spray flames: A 3D counterflow configuration

Keywords: spray flame, 3D counterflow, turbulence, bifurcation, hysteresis

come   L'archive ouverte pluridisciplinaire

Introduction

Because of high energy density of liquid fuel, spray combustion is one of the most common mechanisms to convert liquid fuels into thermal energy in industrial applications. In aeronautical combustors, the spray is usually injected into the combustion chamber at the exit of a swirl stage. The main function of this swirl stage is to enhance the mixing as well as to shorten the flame by limiting its axial expansion. In such systems, the interaction between the spray, the flame and the flow field is highly intricate, and the overall behavior and stabilization processes that take place are the consequence of these complex mechanisms. Consequently, fundamental investigations are needed to understand the underlying physical processes that are associated with the coupling between turbulence, evaporation, mixing, and combustion [1,[START_REF] Sirignano | Fluid dynamics and transport of droplets and sprays[END_REF][START_REF] Jenny | [END_REF].

At laminar conditions, spray, mixing and combustion are coupled by mutual interactions: the spray evaporation generates the gaseous fuel which mixes with air into a flammable mixture. This gaseous reactive mixture feeds the flame which in turn heats the droplets and thus sustains fuel supply. Several studies have been devoted to the investigation of these interactions, see [4,5] and references therein for exhaustive parametric studies. In turbulent flows, the physics becomes even more intricate: the turbulence may act on the spray, by inducing preferential concentration, which modifies the local mixture fraction field. In addition, the interaction of the turbulence with the flame results in wrinkling and stretching of the flame.

Several studies have investigated 2D unsteady configurations [6,7,8], highlighting preferential concentration effects and the influence of local strain rate. A 3D swirled spray flame has also been investigated in [9], showing the coexistence of premixed and diffusion regime, and the role of rich premixed zones on flame stabilization. The goal of the present work is to investigate the structure of turbulent spray flames, by considering a 3D statistically stationary counterflow system. Compared to previous studies, in which global one-step reaction mechanisms have been used to describe the chemistry, a 24-species reduced mechanism for n-dodecane is used, which provides an accurate representation of the chemical phenomena.

The remainder of this paper is organized as follows. In Sec. 2, the modeling approach and the numerics are briefly summarized. The 3D turbulent counterflow configuration, and all relevant non-dimensional parameters that characterize the configuration are summarized in Sec. 3. Results for monodispersed spray injection and different droplet diameters are presented in Sec. 4. The focus of the analysis is on characterizing the bimodal flame structure and on describing the hysteresis and the spray-flame bifurcations. The paper finishes with conclusions.

Modeling and numerics

Gas-phase equations

The gas phase is described by the conservation equations for mass, momentum, species, and energy:

D t ρ = Ṡ m , (1)
D t (ρu i ) = Ṡ u i -∂ i p + ∂ j (σ i j ) , (2)
D t (ρY k ) = ∂ j ρD k W k W ∂ j X k + ωk + Ṡ m δ kF , (3)
D t (ρT ) = ∂ j λ c p ∂ j T + λ c 2 p ∂ j T ∂ j c p - N s k=1 h k ωk c p + Ṡ T , (4)
where

D t φ = ∂ t φ + ∂ i u i φ, ∂ i ≡ ∂ x i
, ρ is the gas density, u i the gas velocity, Ṡ m , Ṡ u i , and Ṡ T the source terms due to droplet evaporation, drag force, and heat transfer. The pressure is denoted by p, and

σ i j = µ ∂ i u j + ∂ j u i - 2 3 ∂ k u k δ i j is the viscous stress tensor.
The mass fraction and mole fraction of species k are denoted by Y k and X k , respectively. The molecular weight of species k is denoted by W k and W is the mixtureaveraged molecular weight. The diffusivity of species k is denoted by D k , ωk is the reaction source term of species k, and δ kF is the Kronecker function that is unity for fuel and zero for all other species. The temperature is denoted by T , λ is the thermal conductivity, c p is the heat capacity, and h k is the enthalpy of species k.

Dispersed-phase equations

For the dispersed phase, a Lagrangian point-particle approach is used (see [10] for details). The equations describing each droplet are written as:

d t x d,i = u d,i , (5)
d t u d,i = f d,i = f 1 τ d u i (x d ) -u d,i , (6)
d t T d = Ṫd = Nu 3Pr c p c l f 2 τ d [T (x d ) -T d ] + ṁd l v m d c l , (7)
d t m d = ṁd = - Sh 3Sc m d τ d ln(1 + B M ) , (8)
where x d is the position of the droplet, u d its velocity, T d its temperature and m d its mass. Nu is the Nusselt number, Pr the Prandtl number, Sh the Sherwood number, Sc the Schmidt number, and B M the Spalding number.

The relaxation time of the droplet is τ d = ρ l d 2 /18µ, ρ l its density, d its diameter, c l its heat capacity and l v the latent heat of vaporization. The drag factor is f 1 , accounting for high Reynolds number effects and f 2 is a correction factor to account for effects of heat-exchange on the evaporation [10].

The coupling terms with the gas phase are obtained by integrating the contributions from all droplets contained in the control volume ∆V:

Ṡ m = -{d t m d } , (9)
Ṡ u i = -d t m d u d,i , (10)
Ṡ T = - 1 c p c l m d d t T d + (c p T d + l v )d t m d , (11)
where

{•} = 1 ∆V d∈∆V .

Reaction chemistry

In the present study, a 24-species mechanism for ndodecane is used, which is based on the JetSurF 1.0-l1 mechanism [START_REF] Sirjean | presented at the 6th US National Combustion Meeting[END_REF], originally consisting of 123 species and 977 reactions. JetSurF 1.0-l1 is a simplified version of JetSurF 1.0. It features a lumped model for n-alkane cracking and the detailed USC Mech II [START_REF] You | [END_REF] for the pyrolysis and oxidation of C1-C4 hydrocarbons. The pyrolysis of n-dodecane is described by a lumped model of three species for cracking to form C1-C4 fragments [START_REF] Wang | USC Mech Version II[END_REF]. The lumping presents a considerable simplification over the detailed chemistry in JetSurF, but it retains the accuracy of the full model [START_REF] Sirjean | presented at the 6th US National Combustion Meeting[END_REF] over a wide range of conditions. In the present work, the USC Mech II part of the mechanism was systematically reduced with directed relation graph (DRG), DRG-aided sensitivity analysis, and linearized quasi steady state (QSS) approximations [START_REF] Lu | [END_REF]. The reduced mechanism is validated in auto-ignition and perfectly stirred reactors (PSR), and results for stochiometric conditions are shown in Fig. 1. Similar agreement was also achieved for equivalence ratios of 0.5 and 1.5. Detailed thermodynamic and transport properties are considered. The species diffusivities D k are calculated assuming constant but not equal species Lewis number Le k , i.e. D k = λ (ρc p Le k ), provided as a supplementary material together with the 24-species mechanism.

Numerics

The governing equations are solved in the low Mach number limit using the structured 3DA code [START_REF] Shashank | High fidelity simulation of reactive liquid fuel jets[END_REF]. A QUICK scheme is used for the discretization of the scalar advection operators, and a second-order central scheme is used for solving the momentum and pressure equations, in combination with the HYPRE library for solving the Poisson equation. A staggered representation is used where the velocity is defined at the cell face, while the scalars are located at the cell center. Time-integration is performed using a second-order Crank-Nicholson scheme. The chemical source terms are evaluated using the DVODE library,which uses an adaptive time stepping to advance the system of ODEs.

Three-dimensional turbulent spray counterflow configuration

Setup of the configuration

We consider a 3D counterflow configuration which consists of two opposed square slots. The separation distance of L = 0.02 m is selected to be comparable to experiments, see [START_REF] Li | [END_REF]. The direction x 1 = x is the injection direction, the direction y = x 2 the outflow, and the direction z = x 3 is periodic. The mesh consists in 256 × 384 × 256 cells. Grid-convergence studies were performed to confirm that this resolution is adequate for resolving the flame structure and all turbulence scales.

At the fuel side, pure air is injected with a fuel spray composed of n-dodecane at ambient conditions (T F d = T F = 300 K). Here and in the following the superscripts "F" and "O" refer to the fuel side and the oxidizer side, respectively. The equivalence ratio is φ = s ṁF ṁO = 2.0 (where s is the stoichiometric mass ratio), and the corresponding liquid mass flow rate is 6.18 × 10 -4 kg/s. The droplet positions at injection are randomly drawn over the entire slot, resulting in a statistically homogeneous distribution. This operating condition ensures that the liquid phase is not fully evaporated before reaching the flame. The injection velocity of the liquid phase is identical to that of the gas phase, u F = u F d . At the fuel side, a turbulence velocity profile is prescribed to assess the effects of turbulence on the evaporation, mixing, and subsequent combustion. The injection of the turbulent velocity field is accomplished by superimposing a synthetic turbulence field over the mean injection velocity, following the Taylor hypothesis Details on the parameters are given in the next section.

The oxidizer side is composed of hot air at T O = 1500 K. The elevated temperature is representative of the recirculation region in gas-turbine combustors to ensure flame stabilization. The injection velocity is prescribed from the constraint of equal gaseous mass-flow rates at both injection streams:

u O = -T O T F u F d . 3

Non-dimensional parameters

The injection velocity on the fuel side is set to u F = 1.0 m/s, resulting in a global strain rate of 600 s -1 , which is comparable to experiments at laminar conditions [START_REF] Li | [END_REF]. The turbulent Reynolds number on the fuel side of the turbulence field that is injected at the fuel side is Re t = 50, and this turbulence field is generated by separately performing a simulation of a homogeneous isotropic turbulence satisfying a Passot-Pouquet spectrum [17]. With the present conditions for temperature and global equivalence ratio the turbulent Damköhler and Karlovitz numbers are, respectively, Da t = τ L /τ c = 4.35 and Ka t = τ c /τ K = 2.43 where τ c is the chemical time , τ K is the Kolmogorov time scale and τ L is the integral time scale. This combination of parameters indicates potential scale interactions between the small scales of the turbulence and the reaction zones. For the droplet phase, three droplet diameters with d = 20, 40 and 80 µm are investigated, referred to as D20, D40 and D80. These conditions are representative for aeronautical combustor applications [START_REF] Lecourt | Proceedings of ASME Turbo Expo 2011: Power for Land[END_REF][START_REF] Providakis | [END_REF], resulting in the Stokes numbers with respect to the Kolomogorov time scale τ K of St K = τ d /τ k = 1.0, 4.0 and 16.0, respectively. The corresponding Stokes numbers with respect to the global strain rate a are St a = aτ d = 0.49, 1.98, and 7.80. Thus, preferential concentration effects are expected and droplets of d = 40 and 80 µm are expected to cross the stagnation plane if they do not evaporate. The initial droplet spacing is 0.25, 0.5 and 1.10 mm respectively, ensuring sufficient scale separation.

Following the scale analysis, the Kolmogorov length scale is of the order of magnitude of the droplet interspacing. Therefore, we can expect interactions between the evaporation-drag processes and the turbulence [START_REF] Sirignano | Fluid dynamics and transport of droplets and sprays[END_REF]. However, these effects are not taken into account in the closure models used here. This question is outside the scope of the present investigation, and will require further analysis. Consequently, the present simulations are referred to as mesoscale/flow-field resolved simulations [20]: given mesoscale closures (evaporation, drag force, combustion), our simulations are fully resolved.

In the following, the three cases are identified by the droplet diameter. In additional, a gas-phase DNS at identical injection conditions and global composition is performed for comparison, and referred to as GAS.

Results

Computations where performed over 50 ms, with an initialization phase of 20 ms, and a statistical analysis phase of 30 ms. In Figure 2, comparisons of instantaneous temperature fields at the center plane y = 0 are presented for the gaseous and the liquid injection cases. For the case GAS, the fuel is already injected in gaseous form, so that the mixture fraction Z, which is here defined following Bilger's formulation [21], decreases from the fuel side to the oxidizer side. 1 The flame front is wrinkled by the turbulence and the flame thickness varies depending on the local instantaneous stretch. For the case D20, the mixture fraction increases first due to the liquid fuel evaporation, is first generated by the evaporation of the liquid fuel and then decreases because of mixing between fuel and oxidizer streams. The flame is located after the evaporation zone, and the structure is similar to that of the gaseous flame. For D40, the flame structure is more complex: while the overall structure is similar to that of GAS and D20, isolated reaction zones can be found upstream of the coherent flame front, and will be further analyzed. For the case D80, the flame structure is considerably different from the previous cases. Specifically, the distinct formation of two coherent reaction zones is evident. The first one is located after the evaporation line, and the second one is found on the oxidizer side. Between both zones is a high-temperature region without significant chemical reaction. This region is occupied by large droplets that did not evaporate in the first reaction zone, and will eventually evaporate in the second reaction zone.

Since the cases D20 and D40 are close to the critical Stokes number of order unity, segregation effects are preferential concentration is expected [22]. To analyze this effect, we investigate the droplet interspace distribution, by means of a Delaunay triangulation on the particle position. The generated tetrahedrons allow to evaluate the interspace volume between droplets, and thus the interspace distance. In Fig. 3, the droplet interspacing is plotted against the axial position, showing a spreading as the droplets are going in to the flame, identified by high temperature. It is also exhibited in Fig. 4, where the distribution close to the injection (x > 0.9 mm) and in the flame (T g > 470 K) are presented. These results show the effect of segregation for both cases, the distribution being larger in the flame, indicating that the droplet distribution is approximately log-normal.

Spray flame structure bifurcation and hysteresis

To further analyze the flame structure, results of the laminar spray flame are used in the following to interpret the turbulent flame behavior. The flame structure for the three considered droplet diameters are pre-sented in Fig. 5. The probability in mixture fractiontemperature space is evaluated for the turbulent flames. Results from this analysis at the mid-plane y = 0 are plotted in Fig. 6, and these data are collected over 30 uncorrelated DNS snapshots. Results for laminar spray flame calculations have also been added. For the purely gaseous case, the turbulence has only a marginal effect, exhibiting only a marginal scattering around the laminar results. Following classical flamelet theory [23], the turbulent GAS solution can be considered as a collection of laminar one-dimensional flames, connecting the oxidizer side at high temperature with the fuel side at low temperature. The flame structure of the gaseous flame is composed of a primary rich premixed reaction region, for which the mixture fraction is approximately constant while the temperature increases, and a secondary recombination zone between the unburnt species and the products of the premixed flame with the oxidizer stream.

For the spray flame cases, the scenario is more complex. At the fuel injection side(point F), the mixture fraction Z is zero. Then, Z increases through evaporation and subsequently decreases by mixing with the air coming from the oxidizer side (point O). Secondly, compared to the gaseous case, the turbulence strongly affects the flame. For all results, mixture fractions higher than the one reached at the corresponding laminar case are found. This is due to the preferential concentration that has been identified previously: by accumulating droplets, the achievable mixture fraction is higher, leading to a broader scatter of the points in the mixture fraction-temperature space.

Finally, differences in the flame structure between the three spray cases are observed from the scatterplot in Fig. 6. One preferential pathway, i.e., one characteristic flame structure, is obtained for cases D20 and D80. In contrast, two distinct flame-structure profiles can be identified for the case D40. Moreover, D20 and D80 present very different flame structures.

In analogy with the laminar results presented in Fig. 5 for flame structure in physical space, the flame D20 is composed of an evaporation zone, in which the mixture fraction increases, and a single reaction zone comprising both the fuel oxidation and the recombination of the products with the oxidizer (identified by a peak of OH mass fraction for the laminar fsolution in Fig. 5). In contrast, the case D80 exhibits two local maxima of temperature, which is characteristic of a double-reaction spray flame structure, identified by two peaks of OH in the laminar solution in Fig. 5. Indeed, a first reaction zone is found on the fuel side, which is connected to the oxidizer side through a second reaction zone. These different flame structures have already been observed for lam-inar counterflow flames [4,24,25,26,27]. The dependence on the droplet diameter observed here is in agreement with the laminar results in [26]. In both cases, the turbulent spray flame can be represented as a collection of laminar flames, even if the space spanned by the turbulent solution for the D80 flame is significantly reduced compared to the D20 case. This is due to the larger inertia of the droplets inducing weaker segregation effects, as discussed in Fig. 3.

In the D40 case, two possible flame-structure pathways are likely to occur. As such, the overall structure presents both aspects of D20 and D80 flames, that is single-reaction and double-reaction structures. These two flame structures are identified in the instantaneous snapshots in Fig. 2 and could be retrieved from the laminar case, see Fig. 5. Indeed, for this operating conditions (d = 40 µm, a = 600 s -1 ) multiple spray flame structures have been observed in the laminar case. The existence of multiple solutions for counterflow spray flames has been previously postulated by Continillo and Sirignano [24] on the basis of the high non-linearity of the spray equations and this has been numerically confirmed by Gutheil [28] for laminar spray flames.

In an attempt to understand why this bimodal structure is particular to the case D40, additional laminar simulations are performed by considering a parametric study of droplet diameter and strain rate values. First, the flame structure dependence on the diameter is studied, at the same global strain rate as the turbulent cases. In Fig. 7a., the mean temperature along the axial position normalized by the value of the gaseous flame, is represented. For the case that multiple solutions are found, the normalized mean temperature is higher for the double-reaction flame (where the two reaction zones are separated by a high temperature region) than for the single-reaction flame, allowing to study the transition between these two possible flame structures. Starting from the single-reaction solution obtained for d = 20 µm, the droplet diameter is successively increased. A single-reaction flame is detected for D < 60 µm, after which a bifurcation occurs [START_REF] Hirsh | Differential Equations, Dynamical Systems, and an Introduction to Chaos[END_REF]: the flame structure switches to a double-reaction flame for all diameters up to the studied maximum value of d = 80 µm. Then, to detect an upper branch of the hysteresis curve, we begin with d = 80 µm and successively decrease the droplet diameter. Another bifurcation occurs at d = 20 µm, for which the double-reaction flame collapses again onto a single-reaction zone. Consequently, the laminar spray flame presents a hysteresis [START_REF] Brokate | Hysteresis and Phase Transitions[END_REF] with in droplet diameter, i.e. the solution depends on the history and not only on the boundary conditions, as shown in Fig. 7a.

The dependence of the flame structure on the strain rate is also investigated by fixing the droplet diameter to d = 40 µm. Results for the mean temperature normalized by the corresponding gaseous flame value at the same strain rate are shown in Fig. 7b. In this case, two branches are detected: a lower single-reaction branch is discerned for a ∈ [500, 1100] s -1 , and an upper branch that contains single-reaction for small strain rate a ∈ [300, 400] s -1 and double-reaction for higher strain rate. The lower branch is generated by starting from the single-reaction solution at a = 600 s -1 , and the second branch is obtained by starting either from the lower (a = 300 s -1 ) or the higher (a = 1350 s -1 ) strain rate. Compared to the diameter variations, there is no hysteresis loop for strain-rate variations. Starting from the lower branch, the solution can bifurcate to the upper branch for low (a < 500 s -1 ) or high (a > 1100 s -1 ) strain rates. However when a solution is on the upper upper branch, there is no way to go back to the lower one by varying the strain rate.

A complete diagram of the laminar counterflow spray flame structure is presented in Fig. 8 Being able to integrate this bimodal character into turbulent combustion models appears to be challenging, since the spray flame structure is highly sensitive to conditions of the spray and the flow, i.e. strain rate and droplet diameter as well as their history, which could be strongly affected by turbulent combustion models. For instance, flamelet tabulation methods [START_REF] Pitsch | [END_REF][START_REF] Gicquel | Proc. Combust. Inst[END_REF][START_REF] Hollmann | [END_REF] require recognizing all possible states for each flamelet, which necessitates introducing additional parameters that identify each possible pathway.

Conclusions

In the present work, a canonical DNS configuration was introduced to study turbulent spray flames at statistically stationary conditions over a wide range of parameters and operating conditions, where the reaction was described using a 24-species reduced mechanism for dodecane.

A parametric study was conducted to systematically examine effects of the turbulence interaction with the liquid spray phase and the flame. Segregation effects due to turbulent mixing were characterized using a Delaunay triangulation to evaluate the droplet interspacing. It was also shown that preferential concentration affects the local mixture-fraction field. The flame structure was analyzed, showing that this preferential concentration allows to span a larger mixture-fraction space. We have also observed a regime for which the turbulent flame can be mapped by a collection of laminar flamelets, indicating that turbulent spray flames can be represented this way. Moreover, the sensitivity of the flame structure with respect to droplet diameter as well as local strain rate was identified. From this study, a local bifurcationmode was identified which arises from the sensitivity to variations in strain rate and diameter variations. The hysteresis of the spray flame structure due to droplet diameter variations has also been observed for laminar cases. Potential issues of the presence of these bifurcations for industrial applications and associated challenges for the modeling of turbulent spray flames were discussed.

  as a function of the injection droplet diameter and the global strain rate. Single-reaction and double-reaction flames are represented by open and closed symbols, respectively. The bimodal shaded region identifies the hysteresis for which multiple solutions are observed. It explains the bimodal structure of D40 and the unimodal results for D20 and D80, since turbulence induces local variations of the strain rate and droplet diameters at the flame front. Compared to the cases D20 and D80, the D40 flame is more sensitive to such oscillations, showing a bimodal character in analogy to the laminar behavior.

Figure 1 :

 1 Figure 1: Ignition delay as a function of initial temperature in constant-pressure auto-ignition (bottom-left) and temperature as a function of residence time in PSR with inlet temperature of 300 K (top-right), for stoichiometric n-dodecane/air at atmospheric pressure, calculated with the detailed and reduced mechanisms, respectively.

Figure 2 :

 2 Figure 2: Instantaneous results at the cut plane at y = 0: Temperature for GAS, D20, D40 and D80. Isoline of mixture fraction (Z = 0.001,white) and of heat release (H R = 2.10 7 J/kg, black). The points represent the droplet positions.

Figure 3 :

 3 Figure 3: (a) Droplet interspacing based on Delaunay Triangulation versus axial position, colored by gaseous temperature at droplet location, for D20 (left), D40 (center) and D80 (right). The black symbols corresponds to droplet positions for which the gas temperature is less than their boiling temperature, thus delimiting the flame position. The white lines correspond to the mean isoline of mixture fraction (Z=0.001) and the blue squares identify mean reaction zones (H R > 2.10 7 J/kg).

Figure 4 :

 4 Figure 4: Probability of droplet interspacing close to injection (black line) and in the flame (red lines) for D20 (left), D40 (center) and D80 (right). The dashed line corresponds to the log-normal fitting.

Figure 5 :

 5 Figure 5: Flame structure for the laminar spray cases: gas temperature (black left axis) and OH mass fraction (blue right axis) for case D20 (left), D40 (center) and D80 (right). For D40, the single-reaction solution is in full line, the double-reaction in dashed lines.

Figure 6 :

 6 Figure 6: Instantaneous temperature in mixture fraction-temperature space colored by the probability density. Results at the center plane: GAS (upper left), D20 (upper right), D40 (lower left), D80 (lower right). Black lines correspond to the laminar spray-flame solutions of Fig. 5.

Figure 7 :

 7 Figure 7: Normalized mean temperature over the axial position at y = 0 for laminar conditions. In (a), increasing (square with blue line) and decreasing (circles with red line) injection droplet diameter are considered. In (b), strain rate variations are considered, that exhibit a single-reaction branch (blue line) that can bifurcates (black line) into a stable branch (red line) with either single-reaction or double-reaction flame structure. Open and closed symbols denote single-reaction and double-reaction flames, respectively. The vertical lines identify the reference boundary conditions.

Figure 8 :

 8 Figure 8: Structure diagram for laminar counterflow spray flames as function of the injection droplet diameter and global strain rate. Open and closed symbols are for single-reaction and double-reaction flames, respectively. The shaded area identifies the bi-modal region.

Since non-equal species diffusivities are considered, Bilger's mixture fraction is not monotonic.

Acknowledgements

The authors thanks H. Pitsch for permission to use the 3DA code, and to greatly acknowledge Shashank and J. Urzay for help on the code. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ASC130004, and resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. The work at the University of Connecticut was supported by the Air Force Office of Scientific Research under Grant FA9550-13-1-0057.