Jean-Pierre Demailly

Extension of holomorphic functions defined on non reduced analytic subvarieties

Keywords: Holomorphic function, plurisubharmonic function, multiplier ideal sheaf, L 2 extension theorem, Ohsawa-Takegoshi theorem, log canonical singularities, non reduced subvariety, K ahler metric. MSC Classification 2010. 14C30, 14F05, 32C35

The goal of this contribution is to investigate L 2 extension properties for holomorphic sections of vector bundles satisfying weak semi-positivity properties. Using techniques borrowed from recent proofs of the Ohsawa-Takegoshi extension theorem, we obtain several new surjectivity results for the restriction morphism to a non necessarily reduced subvariety, provided the latter is defined as the zero variety of a multiplier ideal sheaf. These extension results come with precise L 2 estimates and (probably) optimal curvature conditions.

Introduction

Bernhard Riemann is considered to be the founder of many branches of geometry, especially by providing the foundations of modern differential geometry [START_REF] Riemann | Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, Habilitationsschrift, 1854[END_REF] and by introducing a deep geometric approach in the theory of functions of one complex variable [START_REF] Riemann | Grundlagen f ur eine allgemeine Theorie der Functionen einer ver anderlichen complexen Gr osse, Inauguraldissertation[END_REF][START_REF] Riemann | Theorie der Abel'schen Functionen[END_REF]. The study of Riemannian manifolds has since become a central theme of mathematics. We are concerned here with the study of holomorphic functions of several variables, and in this context, Hermitian geometry is the relevant special case of Riemannian manifolds -as a matter of fact, Charles Hermite, born in 1822, was a contemporary of Bernhard Riemann.

The more specific problem we are considering is the question whether a holomorphic function f defined on a subvariety Y of a complex manifold X can be extended to the whole of X. More generally, given a holomorphic vector bundle E on X, we are interested in the existence of global extensions F ∈ H 0 (X, E) of a section f ∈ H 0 (Y, E ↾Y ), assuming suitable convexity properties of X and Y , suitable growth conditions for f , and appropriate curvature positivity hypotheses for the bundle E. This can also be seen as an interpolation problem when Y is not connected -possibly a discrete set.

In his general study of singular points and other questions of function theory, Riemann obtained what is now known as the Riemann extension theorem: a singularity of a holomorphic function at a point is "removable" as soon as it is bounded -or more generally if it is L 2 in the sense of Lebesgue; the same is true for functions of several variables defined on the complement of an analytic set. Using the maximum principle, a consequence is that a holomorphic function defined in the complement of an analytic set of codimension 2 automatically extends to the ambient manifold.

The extension-interpolation problem we are considering is strongly related to Riemann's extension result because we allow here Y to have singularities, and f must already have a local extension near any singular point x 0 ∈ Y ; for this, some local growth condition is needed in general, especially if x 0 is a non normal point. A major advance in the general problem is the Ohsawa-Takegoshi L 2 extension theorem [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF] (see also the subsequent series of papers II-VI by T. Ohsawa). It is remarkable that Bernhard Riemann already anticipated in [START_REF] Riemann | Grundlagen f ur eine allgemeine Theorie der Functionen einer ver anderlichen complexen Gr osse, Inauguraldissertation[END_REF] the use of L 2 estimates and the idea of minimizing energy, even though his terminology was very different from the one currently in use.

The goal of the present contribution is to generalize the Ohsawa-Takegoshi L 2 extension theorem to the case where the sections are extended from a non necessarily reduced subvariety, associated with an arbitrary multiplier ideal sheaf. A similar idea had already been considered in D. Popovici's work [START_REF] Popovici | L 2 extension for jets of holomorphic sections of a Hermitian line bundle[END_REF], but the present approach is substantially more general. Since the extension theorems do not require any strict positivity assumptions, we hope that they will be useful to investigate further properties of linear systems and pluricanonical systems on varieties that are not of general type. The exposition is organized as follows: section 2 presents the main definitions and results, section 3 (which is more standard and which the expert reader way wish to skip) is devoted to recalling the required Khler identities and inequalities, section 4 elaborates on the concept of jumping numbers for multiplier ideal sheaves, and section 5 explains the technical details of the L 2 estimates and their proofs. We refer to [START_REF] Berndtsson | The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman[END_REF], [START_REF] Berndtsson | A proof of the Ohsawa-Takegoshi theorem with sharp estimates[END_REF], [START_REF] B␣locki | Suita conjecture and the Ohsawa-Takegoshi extension theorem[END_REF], [START_REF] Chen | A simple proof of the Ohsawa-Takegoshi extension theorem[END_REF], [START_REF] Demailly | Extension theorems, Non-vanishing and the existence of good minimal models[END_REF], [START_REF] Guan | Strong openness conjecture for plurisubharmonic functions[END_REF]15], [START_REF] Manivel | Un th« eor" eme de prolongement L 2 de sections holomorphes d'un fibr« e vectoriel[END_REF], [START_REF] Mcneal | Analytic inversion of adjunction: L 2 extension theorems with gain[END_REF], [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, II[END_REF]94,95,01,03,05], [START_REF] Popovici | L 2 extension for jets of holomorphic sections of a Hermitian line bundle[END_REF], [START_REF] Varolin | Three variations on a theme in complex analytic geometry, Analytic and Algebraic Geometry[END_REF] for related work on L 2 extension theorems.

The author adresses warm thanks to Mihai Pȃun and Xiangyu Zhou for several stimulating discussions around these questions.

Statement of the main extension results

The Ohsawa-Takegoshi theorem addresses the following extension problem: let Y be a complex analytic submanifold of a complex manifold X ; given a holomorphic function f on Y satisfying suitable L 2 conditions on Y , find a holomorphic extension F of f to X, together with a good L 2 estimate for F on X. For this, suitable pseudoconvexity and curvature conditions are needed. We start with a few basic definitions in these directions.

All manifolds and complex spaces are assumed to be paracompact (and even countable at infinity).

(2.1) Definition. A compact complex manifold X will be said to be weakly pseudoconvex if it possesses a smooth plurisubharmonic exhaustion function ρ (say ρ : X → R + ).

(Note: in [START_REF] Nakano | Vanishing theorems for weakly 1-complete manifolds, Number Theory, Algebraic Geometry and Commutative Algebra[END_REF] and later work by Ohsawa, the terminology "weakly 1-complete" is used instead of "weakly pseudoconvex").

(2.2) Definition. A function ψ : X → [-∞, +∞[ on a complex manifold X is said to be quasi-plurisubharmonic (quasi-psh) if ϕ is locally the sum of a psh function and of a smooth function (or equivalently, if i∂∂ψ is locally bounded from below). In addition, we say that ψ has neat analytic singularities if every point x 0 ∈ X possesses an open neighborhood U on which ψ can be written

ψ(z) = c log 1≤j≤N |g k (z)| 2 + w(z)
where c ≥ 0, g k ∈ O X (U ) and w ∈ C ∞ (U ).

(2.3) Definition. If ψ is a quasi-psh function on a complex manifold X, the multiplier ideal sheaf I(ψ) is the coherent analytic subsheaf of O X defined by In case ψ has log canonical singularities, it is easy to see that I(ψ) is a reduced ideal, i.e. that Y = V (I(ψ)) is a reduced analytic subvariety of X. In fact if f p ∈ I(ψ) x for some integer p ≥ 2, then |f | 2p e -ψ is locally integrable, hence, by openness, we have also |f | 2p e -(1+ε)ψ for ε > 0 small enough (see [START_REF] Guan | Strong openness conjecture for plurisubharmonic functions[END_REF] for a general result on openness that does not assume anything on ψ -the present special case follows in fact directly from Hironaka's theorm on resolution of singularities, in case ψ has analytic singularities). Using the fact that e -(1-ε)ψ is integrable for every ε ∈ ]0, 1[, the Hlder inequality for the conjugate exponents 1/p + 1/q = 1 implies that

I(ψ) x = f ∈ O X,x ; ∃U ∋ x ,
|f | 2 e -ψ = |f | 2p e -(1+ε)ψ 1/p e -ψ(1/q-ε/p)
is locally integrable, hence f ∈ I(ψ) x , as was to be shown. If ω is a Khler metric on X, we let dV X,ω = 1 n! ω n be the corresponding Kähler volume element, n = dim X. In case ψ has log canonical singularities along Y = V (I(ψ)), one can also associate in a natural way a measure dV Y

• ,ω [ψ] on the set Y • = Y reg of regular points of Y as follows. If g ∈ C c (Y •
) is a compactly supported continuous function on Y • and g a compactly supported extension of g to X, we set (2.4)

Y • g dV Y • ,ω [ψ] = lim sup t→-∞ {x∈X , t<ψ(x)<t+1} ge -ψ dV X,ω .
By Hironaka, it is easy to see that the limit does not depend on the continuous extension g, and that one gets in this way a measure with smooth positive density with respect to the Lebesgue measure, at least on an (analytic) Zariski open set in Y • (cf. Proposition 4.5, in the special case f = 1, p = 1). In case Y is a codimension r subvariety of X defined by an equation σ(x) = 0 associated with a section σ ∈ H 0 (X, S) of some hermitian vector bundle (S, h S ) on X, and assuming that σ is generically transverse to zero along Y , it is natural to take (2.5)

ψ(z) = r log |σ(z)| 2 h S .
One can then easily check that dV

Y • ,ω [ψ] is the measure supported on Y • = Y reg such that (2.6) dV Y • ,ω [ψ] = 2 r+1 π r (r -1)! 1 |Λ r (dσ)| 2 ω,h S dV Y,ω where dV Y,ω = 1 (n -r)! ω n-r ↾Y • .
For a quasi-psh function with log canonical singularities, dV Y • ,ω [ψ] should thus be seen as some sort of (inverse of) Jacobian determinant associated with the logarithmic singularities of ψ. Finally, the following positive real function will make an appearance in several of our final L 2 estimates :

(2.7)

γ(x) =    exp(-x/2) if x ≥ 0, 1 1 + x 2 if x ≤ 0.
The first generalized extension theorem we are interested in is a variation of Theorem 4 in [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, V : Effects of generalization[END_REF]. The first difference is that we do not require any specific behavior of the quasi-psh function defining the subvariety: any quasi-psh function with log canonical singularities will do; secondly, we do not want to make any assumption that there exist negligible sets in the ambient manifold whose complements are Stein, because such an hypothesis need not be true on a general compact Kähler manifold -one of the targets of our study. Recall that a hermitian tensor

Θ = i 1≤j,k≤n, 1≤λ,µ≤r c jkλµ dz j ∧ dz k ⊗ e * λ ⊗ e µ ∈ C ∞ (X, Λ 1,1 T * X ⊗ Hom(E, E))
is said to be Nakano semi-positive (resp. positive) if the associated hermitian quadratic form

H Θ (τ ) = 1≤j,k≤n, 1≤λ,µ≤r c jkλµ τ j,λ τ k,µ is semi-positive (resp. positive) on non zero tensors τ = τ j,λ ∂ ∂z j ⊗ e λ ∈ T X ⊗ E. It is said to be Griffiths semi-positive (resp. positive) if H Θ (τ ) ≥ 0 (resp. H Θ (τ ) > 0) for all non zero decomposable tensors τ = ζ ⊗ v ∈ T X ⊗ E.
(2.8) Theorem (general L 2 extension result for reduced subvarieties). Let X be a weakly pseudoconvex Khler manifold, and ω a Khler metric on X. Let (E, h) be a holomorphic vector bundle equipped with a smooth hermitian metric h on X, and let ψ : X → [-∞, +∞[ be a quasi-psh function on X with neat analytic singularities. Let Y be the analytic subvariety of X defined by Y = V (I(ψ)) and assume that ψ has log canonical singularities along Y , so that Y is reduced. Finally, assume that the Chern curvature tensor of (E, h) is such that the sum iΘ E,h + α i∂∂ψ ⊗ Id E is Nakano semipositive for all α ∈ [1, 1 + δ] and some δ > 0. Then for every section

f ∈ H 0 (Y • , (K X ⊗ E) ↾Y • ) on Y • = Y reg such that Y • |f | 2 ω,h dV Y • ,ω [ψ] < +∞, there exists an extension F ∈ H 0 (X, K X ⊗ E) whose restriction to Y • is equal to f , such that X γ(δψ) |F | 2 ω,h e -ψ dV X,ω ≤ 34 δ Y • |f | 2 ω,h dV Y • ,ω [ψ].
(2.9) Remarks. (a) Although |F | 2 ω,h and dV X,ω both depend on ω, it is easy to see that the product |F | 2 ω,h dV X,ω actually does not depend on ω when F is a (n, 0)-form. The same observation applies to the product |f | 2 ω,h dV Y • ,ω [ψ], hence the final L 2 estimate is in fact independent of ω. Nevertheless, the existence of a Kähler metric (and even of a complete Kähler metric) is crucial in the proof, thanks to the techniques developped in [START_REF] Andreotti | Carleman estimates for the Laplace-Beltrami equation in complex manifolds[END_REF] and [START_REF] Demailly | Estimations L 2 pour l'op« erateur ∂ d'un fibr« e vectoriel holomorphe semi-positif au dessus d'une vari« et« e k ahl« erienne compl" ete[END_REF].

(b) By approximating non smooth plurisubharmonic weights with smooth ones, one can see that the above result still holds when E is a line bundle equipped with a singular hermitian metric h = e -ϕ . The curvature condition then reads

iΘ E,h + α i∂∂ψ = i ∂∂(ϕ + tψ) ≥ 0, α ∈ [1, 1 + δ],
and should be understood in the sense of currents.

(c) The constant 34 δ given in the above L 2 inequality is not optimal. By exercising more care in the bounds, an optimal estimate could probably be found by following the techniques of Blocki [START_REF] B␣locki | Suita conjecture and the Ohsawa-Takegoshi extension theorem[END_REF] and Guan-Zhou [START_REF] Guan | A solution of an L 2 extension problem with an optimal estimate and applications[END_REF], at the expense of replacing γ(δψ) with a more complicated and less explicit function of δ and ψ. Notice also that in the L 2 estimate, δ can be replaced by any δ ′ ∈ ]0, δ].

We now turn ourselves to the case where non reduced multiplier ideal sheaves and non reduced subvarieties are considered. This situation has already been considered by D. Popovici [START_REF] Popovici | L 2 extension for jets of holomorphic sections of a Hermitian line bundle[END_REF] in the case of powers of a reduced ideal, but we aim here at a much wider generality, which also yields more natural assumptions. For m ∈ R + , we consider the multiplier ideal sheaf I(mψ) and the associated non necessarily reduced subvariety Y (m) = V (I(mψ)), together with the structure sheaf O Y (m) = O X /I(mψ), the real number m being viewed as some sort of multiplicity -the support |Y (m) | may increase with m, but certainly stabilizes to set of poles P = ψ -1 (-∞) for m large enough. We assume the existence of a discrete sequence of positive numbers 0 = m 0 < m 1 < m 2 < . . . < m p < . . . such that I(mψ) = I(m p ψ) for m ∈ [m p , m p+1 [ (with of course I(m 0 ψ) = O X ); they are called the jumping numbers of ψ. The existence of a discrete sequence of jumping numbers is automatic if X is compact. In general, it still holds on every relatively compact open subset

X c := {x ∈ X , ρ(x) < c} ⋐ X,
but requires some some of uniform behaviour of singularities at infinity in the noncompact case. We are interested in extending a holomorphic section

f ∈ H 0 (Y (m p ) , O Y (m p ) (K X ⊗ E ↾Y (m p ) ) := H 0 (Y (m p ) , O X (K X ⊗ C E) ⊗ O X O X /I(m p ψ)).
[Later on, we usually omit to specify the rings over which tensor products are taken, as they are implicit from the nature of objects under consideration]. The results are easier to state in case one takes a nilpotent section of the form

f ∈ H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(m p-1 ψ)/I(m p ψ)).
Then I(m p-1 ψ)/I(m p ψ)) is actually a coherent sheaf, and its support is a reduced subvariety Z p of Y (m p ) (see Lemma 4.2). Therefore I(m p-1 ψ)/I(m p ψ)) can be seen as a vector bundle over a Zariski open set Z • p ⊂ Z p . We can mimic formula (2.4) and define some sort of infinitesimal "m p -jet"

L 2 norm |J m p f | 2 ω,h dV Z • p ,ω [ψ]
(a purely formal notation), as the measure on Z • p defined by (2.10)

Z • p g |J m p f | 2 ω,h dV Z • p ,ω [ψ] = lim sup t→-∞ {x∈X , t<ψ(x)<t+1} g | f | 2 ω,h e -m p ψ dV X,ω
for any g ∈ C c (Z • p ), where g ∈ C c (X) is a continuous extension of g and f a smooth extension of f on X such that ff ∈ I(m p ψ) ⊗ O X C ∞ (this measure again has a smooth positive density on a Zariski open set in Z • p , and does not depend on the choices of f and g, see Prop. 4.5). We extend the measure as being 0 on Y

(m p ) red Z p , since I(m p-1 ψ)/I(m p ψ)) has support in Z • p ⊂ Z p .
In this context, we introduce the following natural definition.

(2.11) Definition. We define the restricted multiplied ideal sheaf

I ′ (m p-1 ψ) ⊂ I(m p-1 ψ)
to be the set of germs F ∈ I(m p-1 ψ) x ⊂ O X,x such that there exists a neighborhood U of x satisfying

Y (m p ) ∩U |J m p F | 2 ω,h dV Y (m p ) ,ω [ψ] < +∞.
This is a coherent ideal sheaf that contains I(m p ψ). Both of the inclusions

I(m p ψ) ⊂ I ′ (m p-1 ψ) ⊂ I(m p-1 ψ)
can be strict (even for p = 1).

The proof is given in Section 4 (Proposition 4.5). One of the geometric consequences is the following "quantitative" surjectivity statement, which is the analogue of Theorem 2.8 for the case when the first non trivial jumping number m 1 is replaced by a higher jumping number m p .

(2.12) Theorem. With the above notation and in the general setting of Theorem 2.8 (but without the hypothesis that the quasi-psh function ψ has log canonical singularities), let 0 = m 0 < m 1 < m 2 < . . . < m p < . . . be the jumping numbers of ψ. Assume that

iΘ E,h + α i∂∂ψ ⊗ Id E ≥ Nak 0 is Nakano semipositive for all α ∈ [m p , m p + δ], for some δ > 0. (a) Let f ∈ H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I ′ (m p-1 ψ)/I(m p ψ))
be a section such that

Y (m p ) |J m p f | 2 ω,h dV Y (m p ) ,ω [ψ] < +∞.
Then there exists a global section

F ∈ H 0 (X, O X (K X ⊗ E) ⊗ I ′ (m p-1 ψ))
which maps to f under the morphism

I ′ (m p-1 ψ) → I(m p-1 ψ)/I(m p ψ), such that X γ(δψ) |F | 2 ω,h e -m p ψ dV X,ω [ψ] ≤ 34 δ Y (m p ) |J m p f | 2 ω,h dV Y (m p ) ,ω [ψ].
(b) The restriction morphism

H 0 (X, O X (K X ⊗ E) ⊗ I ′ (m p-1 ψ)) → H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I ′ (m p-1 ψ)/I(m p ψ)) is surjective.
If E is a line bundle and h a singular hermitian metric on E, a similar result can be obtained by approximating h. However, the L 2 estimates require to take into account the multiplier ideal sheaf of h, and we get the following result.

(2.13) Theorem. Let (X, ω) be a weakly pseudoconvex Kähler manifold, ψ a quasi-psh function with neat analytic singularities and E a holomorphic line bundle equipped with a singular hermitian metric h. Let 0 = m 0 < m 1 < m 2 < . . . < m p < . . . be the jumping numbers for the family of multiplier ideal sheaves I(he -mψ ), m ∈ R + . Assume that iΘ E,h + α i∂∂ψ ≥ 0 in the sense of currents, for all α ∈ [m p , m p + δ] and δ > 0 small enough. Define I ′ (he -m ℓ ψ ) by the same L 2 convergence property as in the case when h is non singular. Then the restriction morphism

H 0 (X, O X (K X ⊗ E) ⊗ I ′ (he -m p-1 ψ )) → H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I ′ (he -m p-1 ψ )/I(he -m p ψ ))
is surjective. Moreover, the L 2 estimate (2.12) (b) still holds in this situation.

One of the strengths of the above theorems lies in the fact that no strict curvature hypothesis on iΘ E,h is ever needed. On the other hand, if we assume a strictly positive lower bound iΘ E,h + m p i∂∂ψ ≥ εω > 0, then the Nadel vanishing theorem [START_REF] Nadel | Multiplier ideal sheaves and K ahler-Einstein metrics of positive scalar curvature[END_REF] implies H 1 (X, O X (K X ⊗E) ⊗ I(he -m p ψ )) = 0, and we immediately get stronger surjectivity statements by considering the relevant cohomology exact sequence, e.g. that

H 0 (X, O X (K X ⊗ E) ⊗ I(he -m ℓ ψ )) → H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(he -m ℓ ψ )/I(he -m p ψ ))
is surjective for all ℓ < p. In case X is compact, it turns out that this qualitative surjectivity property still holds true with a semi-positivity assumption only. In view of [START_REF] Demailly | Extension theorems, Non-vanishing and the existence of good minimal models[END_REF], this probably has interesting applications to algebraic geometry which we intend to discuss in a future work:

(2.14) Theorem. Let (X, ω) be a compact Kähler manifold, ψ a quasi-psh function with neat analytic singularities and E a holomorphic vector bundle equipped with a hermitian metric h. Let 0 = m 0 < m 1 < m 2 < . . . < m p < . . . be the jumping numbers for the family of multiplier ideal sheaves I(he -mψ ), m ∈ R + .

(a) The rank of E being arbitrary, assume that h is smooth and that for some index ℓ = 0, 1, . . . , p -1 and all α ∈ [m ℓ+1 , m p + δ], we have

iΘ E,h + α i∂∂ψ ⊗ Id E ≥ 0 in the sense of Nakano,
for δ > 0 small enough. Then the restriction morphism

H 0 (X, O X (K X ⊗ E) ⊗ I(e -m ℓ ψ )) → H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(e -m ℓ ψ )/I(e -m p ψ ))
is surjective.

(b) Assume that E is a line bundle equipped with a singular metric h and that for some ℓ = 0, 1, . . . , p -1 and all α ∈ [m ℓ+1 , m p + δ], we have iΘ E,h + α i∂∂ψ ≥ 0 in the sense of currents, for δ > 0 small enough. Then the restriction morphism

H 0 (X, O X (K X ⊗ E) ⊗ I(he -m ℓ ψ )) → H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(he -m ℓ ψ )/I(he -m p ψ ))
is surjective.

(2.15) Question. It would be interesting to know whether Theorem 2.14 can be strengthened by suitable quantitative L 2 estimates. The main difficulty is already to define the norm of jets when there is more than one jump number involved. Some sort of "Cauchy inequality" for jets would be needed in order to derive the successive jet norms from a known global L 2 estimate for a holomorphic section defined on the whole of X.

We do not know how to proceed further at this point.

Fundamental estimates of K ahler geometry

3.A. Basic set-up

We refer to [START_REF] Griffiths | The extension problem in complex analysis II; embeddings with positive normal bundle[END_REF]69] and [Dem-X] for general background results on the geometry of Kähler manifolds and hermitian bundles. Let X be a complex n-dimensional manifold equipped with a smooth Hermitian metric

ω = 1≤j,k≤n ω jk (z) dz j ⊗ dz k .
As usual, it is convenient to view ω rather as a real (1, 1)-form ω = i ω jk (z) dz j ∧ dz k . The metric ω is said to be Kähler if dω = 0, i.e. ω also defines a symplectic structure. It can be easily shown that ω is Kähler if and only if there are holomorphic coordinates (z 1 , . . . , z n ) centered at any point x 0 ∈ X such that the matrix of coefficients (ω jk ) is tangent to identity at order 2, i.e.

(3.1)

ω jk (z) = δ jk + O(|z| 2 ) at x 0 . Now, let (E, h) is a Hermitian vector bundle over X. Given a smooth (p, q)-form u on X with values in E, that is, a section of C ∞ (X, Λ p,q T * X ⊗ E), we consider the global L 2 norm (3.2) u 2 = M |u(x)| 2 dV X,ω (x)
where |u(x)| = |u(x)| ω,h is the pointwise Hermitian norm of u(x) in the tensor product Λ p,q T * X ⊗ E and dV X,ω = ω n n! the Hermitian volume form on X ; for simplicity, we will usually omit the dependence on the metrics in the notation of |u(x)| ω,h . We denote by •, • the inner product of the Hilbert space L 2 (X, Λ p,q T * X ⊗ E) of L 2 sections for the norm • . Let D = D E,h be the Chern connection of (E, h), that is, the unique connection D = D 1,0 + D 0,1 = D ′ + D ′′ that makes h parallel, for which the (0, 1) component D ′′ coincides with ∂. It follows in particular that D ′2 = 0, D ′′2 = 0 and

D 2 = D ′ D ′′ + D ′′ D ′ . The Chern curvature tensor of (E, h) is the (1, 1)-form (3.3) Θ E,h ∈ C ∞ (X, Λ 1,1 T * X ⊗ Hom(E, E)) such that D 2 u = (D ′ D ′′ + D ′′ D ′ )u = Θ E,h ∧ u.
Next, the complex Laplace-Beltrami operators are defined by

(3.4) ∆ ′ = D ′ D ′ * + D ′ * D ′ , ∆ ′′ = D ′′ D ′′ * + D ′′ * D ′′
where P * denotes the formal adjoint of a differential operator

P : C ∞ (X, Λ p,q T * X ⊗ E) → C ∞ (X, Λ p+r,q+s T * X ⊗ E), deg P = r + s
with respect to the corresponding inner products •, • . If u is a compactly supported section, we get in particular

∆ ′ u, u = D ′ u 2 + D ′ * u 2 ≥ 0, ∆ ′′ u, u = D ′′ u 2 + D ′′ * u 2 ≥ 0.

3.B. Bochner-Kodaira-Nakano identity

Great simplifications occur in the operators of Hermitian geometry when the metric ω is Kähler. In fact, if we use normal coordinates at a point x 0 (cf. (3.1)), and a local holomorphic frame (e λ ) 1≤λ≤r of E such that De λ (x 0 ) = 0, it is not difficult to see that all order 1 operators D ′ , D ′′ and their adjoints D ′ * , D ′′ * admit at x 0 the same expansion as the analogous operators obtained when all Hermitian metrics on X or E are constant. From this, the basic commutation relations of Kähler geometry can be checked. If A, B are differential operators acting on the algebra C ∞ (X, Λ •,• T * X ⊗ E), their graded commutator (or graded Lie bracket) is [A, B] = AB -(-1) ab BA where a, b are the degrees of A and B respectively. If C is another endomorphism of degree c, the following purely formal Jacobi identity holds:

(-1) ca A, [B, C] + (-1) ab B, [C, A] + (-1) bc C, [A, B] = 0.
(3.5) Fundamental Kähler identities. Let (X, ω) be a Kähler manifold and let L be the Lefschetz operator defined by Lu = ω ∧ u and Λ = L * its adjoint, acting on E-valued forms. The following identities hold for the Chern connection D = D ′ + D ′′ on E and the associated complex Laplace operators ∆ ′ and ∆ ′′ .

(a) Basic commutation relations

[D ′′ * , L] = iD ′ , [Λ, D ′′ ] = -iD ′ * , [D ′ * , L] = -iD ′′ , [Λ, D ′ ] = iD ′′ * . (b) Bochner-Kodaira-Nakano identity ([Boc48], [Kod53a,b], [AN54], [Nak55]) ∆ ′′ = ∆ ′ + [iΘ E,h , Λ].

Idea of proof. (a)

The first step is to check the identity [d ′′ * , L] = id ′ for constant metrics on X = C n and the trivial bundle E = X × C, by a brute force calculation. All three other identities follow by taking conjugates or adjoints. The case of variable metrics follows by looking at Taylor expansions up to order 1.

(b) The last equality in (a) yields

D ′′ * = -i[Λ, D ′ ], hence ∆ ′′ = [D ′′ , D ′′ * ] = -i[D ′′ , Λ, D ′ ] .
By the Jacobi identity we get

D ′′ , [Λ, D ′ ] = Λ, [D ′ , D ′′ ]] + D ′ , [D ′′ , Λ] = [Λ, Θ E,h ] + i[D ′ , D ′ * ], taking into account that [D ′ , D ′′ ] = D 2 = Θ E,h . The formula follows.
One important (well known) fact is that the curvature term [iΘ E,h , Λ] operates as a hermitian (semi-)positive operator on L 2 (X, Λ n,q T * X ⊗ E as soon as i Θ E,h is Nakano (semi-)positive.

3.C. The twisted a priori inequality of Ohsawa and Takegoshi

The main a priori inequality that we are going to use is a simplified (and slightly extended) version of the original Ohsawa-Takegoshi a priori inequality [START_REF] Ohsawa | On the extension of L 2 holomorphic functions[END_REF][START_REF] Ohsawa | On the extension of L 2 holomorphic functions, II[END_REF], along the lines proposed by Manivel [START_REF] Manivel | Un th« eor" eme de prolongement L 2 de sections holomorphes d'un fibr« e vectoriel[END_REF] and Ohsawa [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, V : Effects of generalization[END_REF]. Such inequalities were originally introduced in the work of Donnelly-Fefferman [START_REF] Donnelly | L 2 -cohomology and index theorem for the Bergman metric[END_REF] and Donnelly-Xavier [START_REF] Donnelly | On the differential form spectrum of negatively curved Riemann manifolds[END_REF]. The main idea is to introduce a modified Bochner-Kodaira-Nakano inequality. Although it has become classical in this context, we reproduce here briefly the calculations for completeness, and also for the sake of fixing the notation.

(3.6) Lemma (Ohsawa [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, V : Effects of generalization[END_REF]). Let E be a Hermitian vector bundle on a complex manifold X equipped with a Kähler metric ω. Let η, λ > 0 be smooth functions on X. Then for every form u ∈ C ∞ c (X, Λ p,q T * X ⊗ E) with compact support we have

(η + λ) 1 2 D ′′ * u 2 + η 1 2 D ′′ u 2 + λ 1 2 D ′ u 2 + 2 λ -1 2 d ′ η ∧ u 2 ≥ [η iΘ E -i d ′ d ′′ η -iλ -1 d ′ η ∧ d ′′ η, Λ]u, u .
Proof. We consider the "twisted" Laplace-Beltrami operators

D ′ ηD ′ * + D ′ * ηD ′ = η[D ′ , D ′ * ] + [D ′ , η]D ′ * + [D ′ * , η]D ′ = η∆ ′ + (d ′ η)D ′ * -(d ′ η) * D ′ , D ′′ ηD ′′ * + D ′′ * ηD ′′ = η[D ′′ , D ′′ * ] + [D ′′ , η]D ′′ * + [D ′′ * , η]D ′′ = η∆ ′′ + (d ′′ η)D ′′ * -(d ′′ η) * D ′′ ,
D ′′ ηD ′′ * + D ′′ * ηD ′′ -D ′ ηD ′ * -D ′ * ηD ′ = η[iΘ E , Λ] + (d ′′ η)D ′′ * -(d ′′ η) * D ′′ + (d ′ η) * D ′ -(d ′ η)D ′ * . (3.7)
Moreover, the Jacobi identity yields

[D ′′ , [d ′ η, Λ]] -[d ′ η, [Λ, D ′′ ]] + [Λ, [D ′′ , d ′ η]] = 0, whilst [Λ, D ′′ ] = -iD ′ * by the basic commutation relations 3.5 (a). A straightforward computation shows that [D ′′ , d ′ η] = -(d ′ d ′′ η) and [d ′ η, Λ] = i(d ′′ η) * . Therefore we get i[D ′′ , (d ′′ η) * ] + i[d ′ η, D ′ * ] -[Λ, (d ′ d ′′ η)] = 0, that is, [i d ′ d ′′ η, Λ] = [D ′′ , (d ′′ η) * ] + [D ′ * , d ′ η] = D ′′ (d ′′ η) * + (d ′′ η) * D ′′ + D ′ * (d ′ η) + (d ′ η)D ′ * .
After adding this to (3.7), we find

D ′′ ηD ′′ * + D ′′ * ηD ′′ -D ′ ηD ′ * -D ′ * ηD ′ + [i d ′ d ′′ η, Λ] = η[iΘ E , Λ] + (d ′′ η)D ′′ * + D ′′ (d ′′ η) * + (d ′ η) * D ′ + D ′ * (d ′ η).
We apply this identity to a form u ∈ C ∞ c (X, Λ p,q T * X ⊗ E) and take the inner bracket with u. Then (

D ′′ ηD ′′ * )u, u = ηD ′′ * u, D ′′ * u = η 1 2 D ′′ * u 2 ,
and likewise for the other similar terms. The above equalities imply

η 1 2 D ′′ * u 2 + η 1 2 D ′′ u 2 -η 1 2 D ′ u 2 -η 1 2 D ′ * u 2 = [η iΘ E -i d ′ d ′′ η, Λ]u, u + 2 Re D ′′ * u, (d ′′ η) * u + 2 Re D ′ u, d ′ η ∧ u .
By neglecting the negative termsη

1 2 D ′ u 2 -η 1 2 D ′ * u 2 and adding the squares λ 1 2 D ′′ * u 2 + 2 Re D ′′ * u, (d ′′ η) * u + λ -1 2 (d ′′ η) * u 2 ≥ 0, λ 1 2 D ′ u 2 + 2 Re D ′ u, d ′ η ∧ u + λ -1 2 d ′ η ∧ u 2 ≥ 0 we get η 1 2 D ′′ * u 2 + λ 1 2 D ′′ u 2 + λ 1 2 D ′ u 2 + λ -1 2 d ′ η ∧ u 2 + λ -1 2 (d ′′ η) * u 2 ≥ [η iΘ E -i d ′ d ′′ η, Λ]u, u . (3.8)
Finally, we use the identity a * = i[a, Λ] for any (1, 0)-form a to get

(d ′ η) * (d ′ η) -(d ′′ η)(d ′′ η) * = i[d ′′ η, Λ](d ′ η) + i(d ′′ η)[d ′ η, Λ] = [id ′′ η ∧ d ′ η, Λ], which implies (3.9) λ -1 2 d ′ η ∧ u 2 -λ -1 2 (d ′′ η) * u 2 = -[iλ -1 d ′ η ∧ d ′′ η, Λ]u, u .
The inequality asserted in Lemma 3.6 follows by adding (3.8) and (3.9).

In the special case of (n, q)-forms, the forms D ′ u and d ′ η ∧ u are of bidegree (n + 1, q), hence the estimate takes the simpler form

(3.10) (η + λ) 1 2 D ′′ * u 2 + η 1 2 D ′′ u 2 ≥ [η iΘ E -i d ′ d ′′ η -iλ -1 d ′ η ∧ d ′′ η, Λ]u, u .

3.D. Abstract L 2 existence theorem for solutions of ∂-equations

Using standard arguments from functional analysis -actually just basic properties of Hilbert spaces -the a priori inequality (3.10) implies a powerful L 2 existence theorem for solutions of ∂-equations.

(3.11) Proposition. Let X be a complete Kähler manifold equipped with a (non necessarily complete) Kähler metric ω, and let (E, h) be a Hermitian vector bundle over X.

Assume that there are smooth and bounded functions η, λ > 0 on X such that the (Hermitian) curvature operator

B = B n,q E,h,ω,η,λ = [η iΘ E,h -i d ′ d ′′ η -iλ -1 d ′ η ∧ d ′′ η, Λ ω ]
is positive definite everywhere on Λ n,q T * X ⊗ E, for some q ≥ 1. Then for every form g ∈ L 2 (X, Λ n,q T * X ⊗ E) such that D ′′ g = 0 and X B -1 g, g dV X,ω < +∞, there exists f ∈ L 2 (X, Λ n,q-1 T * X ⊗ E) such that D ′′ f = g and

X (η + λ) -1 |f | 2 dV X,ω ≤ X B -1 g, g dV X,ω .
Proof. Assume first that ω is complete Kähler metric. Let v ∈ L 2 (X, Λ n,q T * X ⊗ E), and v = v 1 + v 2 ∈ (Ker D ′′ ) ⊕ (Ker D ′′ ) ⊥ the decomposition of v with respect to the closed subspace Ker D ′′ and its orthogonal. Since g ∈ Ker D ′′ , The Cauchy-Schwarz inequality yields

| g, v | 2 = | g, v 1 | 2 = | B -1 2 g, B 1 2 v 1 | 2 ≤ X B -1 g, g dV X,ω X Bv 1 , v 1 dV X,ω ,
and provided that v ∈ Dom D ′′ * , we find v 2 ∈ (Ker D ′′ ) ⊥ ⊂ (Im D ′′ ) ⊥ = Ker D ′′ * , and so

D ′′ v 1 = 0, D ′′ * v 2 = 0, whence X Bv 1 , v 1 dV X,ω ≤ (η + λ) 1 2 D ′′ * v 1 2 + η 1 2 D ′′ v 1 2 = (η + λ) 1 2 D ′′ * v 2 .
Combining both inequalities, we obtain

| g, v | 2 ≤ X B -1 g, g dV X,ω (η + λ) 1 2 D ′′ * v 2 .
The Hahn-Banch theorem applied to the linear form (η + λ)

1 2 D ′′ * v → v, g implies the existence of an element w ∈ L 2 (X, Λ n,q T * X ⊗ E) such that w 2 ≤ X B -1 g, g dV X,ω and
v, g = (η + λ) 1 2 D ′′ * v, w ∀g ∈ Dom D ′′ ∩ Dom D ′′ * .
It follows that f = (η + λ)

1 2 w satisfies D ′′ f = g as well as the desired L 2 estimate. If ω is not complete, we set ω ε = ω + ε ω with some complete Kähler metric ω. The final conclusion is then obtained by passing to the limit and using a monotonicity argument (the integrals are easily shown to be monotonic with respect to ε, see [START_REF] Demailly | Estimations L 2 pour l'op« erateur ∂ d'un fibr« e vectoriel holomorphe semi-positif au dessus d'une vari« et« e k ahl« erienne compl" ete[END_REF]).

We need also a variant of the L 2 -estimate, so as to obtain approximate solutions with weaker requirements on the data :

(3.12) Proposition. With the notation of 3.11, assume that B + εI > 0 for some ε > 0 (so that B can be just semi-positive or even slightly negative; here I is the identity endomorphism). Given a section g ∈ L 2 (X, Λ n,q T * X ⊗ E) such that D ′′ g = 0 and

M (ε) := X (B + εI) -1 g, g dV X,ω < +∞,
there exists an approximate solution f ε ∈ L 2 (X, Λ n,q-1 T * X ⊗ E) and a correcting term

g ε ∈ L 2 (X, Λ n,q T * X ⊗ E) such that D ′′ f ε = g -g ε and X (η + λ) -1 |f ε | 2 dV X,ω + 1 ε X |g ε | 2 dV X,ω ≤ M (ε).
If g is smooth, then f ε and g ε can be taken smooth.

Proof. The arguments are almost unchanged, we rely instead on the estimates

| g, v 1 | 2 ≤ X (B + εI) -1 g, g dV X,ω X (B + εI)v 1 , v 1 dV X,ω ,
and

X (B + εI)v 1 , v 1 dV X,ω ≤ (η + λ) 1 2 D ′′ * v 2 + ε v 2 .
This gives a pair (w ε , w

′ ε ) such that w ε 2 + w ′ ε 2 ≤ M (ε) and (3.13) v, g = (η + λ) 1 2 D ′′ * v, w ε + ε 1/2 v, w ′ ε for all v ∈ Dom D ′′ * , hence f ε = (η + λ)
1 2 w ε is the expected approximate solution with error term g ε = ε 1/2 w ′ ε . By (3.13), we do get D ′′ f ε + g ε = g, and the expected L 2 estimates hold as well. In fact one can take

f ε = (η + λ)D ′′ * -1 ε g and g ε = ε -1 ε g where ε = D ′′ (η + λ)D ′′ * + D ′′ * (η + λ)D ′′ +
εI is an invertible self-adjoint elliptic operator. Then f ε and g ε are smooth.

Openness of multiplier ideal sheaves and jumping numbers

Let X be complex manifold and ϕ, ψ quasi-psh functions on X. To every m ∈ R + we associate the multiplier ideal sheaf I(ϕ+mψ) ⊂ O X . By Nadel [START_REF] Nadel | Multiplier ideal sheaves and K ahler-Einstein metrics of positive scalar curvature[END_REF], this is a coherent ideal sheaf, and I(ϕ + m ′ ψ) ⊂ I(ϕ + mψ) for m ′ ≥ m. The recent result of Guan-Zhou [START_REF] Guan | Strong openness conjecture for plurisubharmonic functions[END_REF] implies that one has in fact I(ϕ + (m + α)ψ) = I(ϕ + mψ) for every m ∈ R + and α ∈ [0, α 0 (m)[ sufficiently small. From this we conclude without any restriction that there exists a discrete sequence of numbers (4.2) Lemma. For every p > 0, the ideal J p O X of germs of holomorphic functions h such that h I(ϕ + m p-1 ψ) ⊂ I(ϕ + m p ψ) is reduced, i.e. J p = J p . Moreover J p contains I(m p ψ).

Proof. Let h k ∈ J p,x for some exponent k ≥ 2. Pick f ∈ I(ϕ + m p-1 ψ) x . By definition of the jumping numbers, |f | 2 e -ϕ-(m p -ε)ψ is integrable near x for every ε > 0. Since h k f ∈ I(ϕ + m p ψ), the openness property shows that |h k f | 2 e -ϕ-(m p +δ)ψ is integrable for some δ > 0. For a suitable neighborhood U of x and ε > 0 smaller than δ/k, the Hölder inequality applied with the measure dµ = |f | 2 e -ϕ-m p ψ dλ and the functions v = |h| 2 e -εψ , w = e εψ for the conjugate exponents 1/k + 1/ℓ = 1 implies

U |hf | 2 e -ϕ-m p ψ dλ = U vwdµ ≤ U |h| 2k |f | 2 e -ϕ-(m p +kε)ψ dλ 1/k U |f | 2 e -ϕ-(m p -ℓε)ψ dλ 1/ℓ < +∞,
hence hf ∈ I(ϕ + m p ψ). Since this is true for every f ∈ I(ϕ + m p-1 ψ) we conclude that h ∈ J p,x , thus J p = J p . The last assertion is equivalent to J p ⊃ I(m p ψ) and follows similarly from the inequality

U |hf | 2 e -ϕ-m p ψ dλ ≤ U |h| 2k e -(m p +kε/ℓ)ψ dλ 1/k U |f | 2ℓ e -ℓϕ-(m p -ε)ψ dλ 1/ℓ < +∞ ;
we fix here ε > 0 so small that I(m p ψ) = I((m p +ε)ψ) and, by openness, ℓ sufficiently close to 1 to make the last integral convergent whenever f ∈ I(ϕ+m p-1 ψ) = I(ϕ+(m p -ε)ψ).

A consequence of Lemma 4.2 is that the zero variety Z p = V (J p ) is a reduced subvariety of Y (m p ) = V (I(m p ψ)) and that the quotient sheaf I(ϕ + m p-1 ψ)/I(ϕ + m p ψ) is a coherent sheaf over O Z p = O X /J p . Therefore I(ϕ + m p-1 ψ)/I(ϕ + m p ψ) can be seen as a vector bundle on some Zariski open set Z

• p ⊂ Z p ⊂ Y (m p )
red . In the sequel, a case of special interest is when ψ has analytic singularities, that is, every point x 0 ∈ X possesses an open neighborhood V ⊂ X on which ψ can be written

ψ(z) = c log 1≤j≤N |g k (z)| 2 + u(z)
where c ≥ 0, g k ∈ O X (V ) and u ∈ C ∞ (V ). The integrability of |f | 2 e -mψ on a coordinate neighborhood V then means that

(4.3) V |f (z)| 2 |g(z)| 2m dλ(z) < +∞.
By Hironaka [START_REF] Hironaka | Resolution of singularities of an algebraic variety over a field of characteristic zero[END_REF], there exists a principalization of the ideal

J = (g k ) ⊂ O X , that is, a modification µ : X → X such that µ * J = (g k • µ) = O X (-∆) where ∆ =
a k ∆ k is a simple normal crossing divisor on X. We can also assume that the Jacobian Jac(µ) has a zero divisor B = b k ∆ k contained in the exceptional divisor. After a change of variables z = µ(w) and a use of local coordinates where ∆ k = {w k = 0} and µ * J is the principal ideal generated by the monomial w a = w a k k , we see that (4.3) is equivalent to the convergence of

µ -1 (V ) |f (µ(w))| 2 | Jac(µ)| 2 |g(µ(w)| 2mc dλ(w),
which can be expressed locally as the convergence of

µ -1 (V ) |f (µ(w))| 2 |w b | 2 |w a | 2mc dλ(w).
For this, the condition is that f • µ(w) be divisible by w s with s k = ⌊mca kb k ⌋ + . In other words, the multipler ideal sheaves I(mψ) are given by the direct image formula (4.4)

I(mψ) = µ * O X - k ⌊mca k -b k ⌋ + ∆ k .
The jumps can only occur when m is equal to one of the values b k +N ca k , N ∈ N, which form a discrete subset of R + .

(4.5) Proposition. Let 0 = m 0 < m 1 < . . . < m p be the jumping numbers of the quasi-psh function ψ, which is assumed to have neat analytic singularities. Let ℓ a local holomorphic generator of K X ⊗ E at a point x 0 ∈ X, E being equipped with a smooth hermitian metric h, let Z p = Supp(I(m p-1 ψ)/I(m p ψ)), and take a germ f ∈ I(m p-1 ψ) x 0 .

Y (m p ) ∩U |J m p (F ℓ)| 2 ω,h dV Y (m p ) ,ω [ψ] < +∞ is a coherent ideal sheaf such that I(m p ψ) ⊂ I ′ (m p-1 ψ) ⊂ I(m p-1 ψ).
Both of the inclusions can be strict (even for p = 1).

(c) The function (1 + |ψ|) -(n+1) |f ℓ| 2 ω,h e -m p ψ is locally integrable at x 0 .

Proof. (a) As above, we use a principalization of the singularities of ψ and apply formula (4.4). Over a generic point x 0 ∈ Z p , the component of Z p containing x 0 is dominated by exactly one of the divisors ∆ k , and the jump number m p is such that m p ca kb k = N for some integer N . The previous jump number m p-1 is then given by m p-1 ca kb k = N -1. On a suitable coordinate chart of the blow-up µ : X → X, let us write ψ • µ(w) = c log |w a | 2 + u(w) where u is smooth, and let w b = 0 be the zero divisor of Jac(µ). By definition, the measure

|J m p f | 2 ω,h dV Z • p ,ω [ψ]
is the direct image of measures defined upstairs as the limit

g ∈ C c (Z p , R) → lim sup t→-∞ t<c log |w a | 2 +u(w)<t+1 e -m p u µ * g β(w) | v(w)| 2 e -ϕ•µ |w k | 2 dλ(w).
Here ϕ is the weight of the metric h on E, v(w) is the holomorphic function representing the section µ * (f ℓ)/(w m p ca-b /w k ) (the denominator divisor cancels with the numerator by construction), and β(w) is a smooth positive weight arising from the change of variable formula, given by µ * dV X,ω /|w b | 2 [one would still have to take into account a partition of unity on the various coordinate charts covering the fibers of µ, but we will avoid this technicality for the simplicity of notation]. Let us denote w = (w ′ , w k ) ∈ C n-1 × C and dλ(w) = dλ(w ′ ) λ(w k ) the Lebesgue measure on C n . At a generic point w where w j = 0 for j = k, the domain of integration is of the form t(w ′ ) < ca k log |w k | 2 < t ′ (w ′ ) + 1. It is easy to check that the limsup measure is a limit, equal to

g ∈ C c (Z p , R) → πca k e -m p u µ * g | v(w)| 2 β(w)e -ϕ•µ ↾w k =0 dλ(w ′ ).
We then have to integrate the right hand side measure over the fibers of µ to get the density of this measure along Z p . Since µ can be taken to be a composition of blow-ups with smooth centers, the fibration has (upstairs) a locally trivial product structure over a Zariski open set Z • p ⊂ Z p . Smooth local vector fields on Z • p can be lifted to smooth vector fields in the corresponding chart of X, and we conclude by differentiating under the integral sign that the density downstairs on Z p is generically smooth.

(b) One typical example is given by ψ

(w) = log |w 1 | 2 |w 2 | 2 . Let us put ∆ k = {w k = 0}, k = 1, 2.
Then the jumping numbers are m p = p ∈ N, and we get

I(m p ψ) = O X (-p(∆ 1 + ∆ 2 )).
For any r 0 > 0 fixed, we have

|w 1 |<r 0 , |w 2 |<r 0 , e t <|w 1 | 2 |w 2 | 2 <e t+1 1 |w 1 | 2 |w 2 | 2 dλ(w 1 ) dλ(w 2 ) ≥ π e t+1 /r 2 0 <|w 2 | 2 <r 2 0 1 |w 2 | 2 dλ(w 2 )
and the limit as t → -∞ is easily seen to be infinite. Therefore I ′ (m 0 ψ) = I ∆ 1 ∩∆ 2 , and we do have

O X (-(∆ 1 + ∆ 2 )) = I(m 0 ψ) I ′ (m 0 ψ) I(m 0 ψ) = O X
in this case. In general, with our notation, it is easy to see that I ′ (m p-1 ψ) is given as the direct image (4.6)

I ′ (m p-1 ψ) = µ * O X - k ⌊m p-1 ca k -b k ⌋ + ∆ k ⊗ I R where R = ∆ ℓ ∩ ∆ ℓ ′ is the union of pairwise intersections of divisors ∆ ℓ for which m p ca ℓ -b ℓ = m p ca k -b k ( = m p-1 ca k -b k + 1
), k being one of the indices achieving the values of m p , m p-1 at the given point x ∈ Z p . This is a coherent ideal sheaf by the direct image theorem.

(c) Near a point where ψ • µ(w) has singularities along normal crossing divisors w j = 0, 1 ≤ j ≤ k, it is sufficient to show that integrals of the form

I k = |w 1 |<1/2,...,|w k |<1/2 -log |w 1 | 2 . . . |w k | 2 -(k+1) |w 1 | 2 . . . |w k | 2 dλ(w 1 ) . . . dλ(w k )
are convergent. This is easily done by induction on k, by using a change of variable w ′ k = w 1 . . . w k , and by applying the Fubini formula together with an integration in w ′ k . We then get

I k ≤ 4π k I k-1 (notice that |w ′ k | < |w 1 | . . . |w k-1 |)
, and I 1 = π/(2 log 2), thus all I k are finite.

Proof of the L 2 extension theorems

Unless otherwise specified, X denotes a weakly pseudoconvex complex n-dimensional manifold equipped with a (non necessarily complete) Kähler metric ω, ρ : X → [0, +∞[ a smooth psh exhaustion on X, (E, h) a smooth hermitian holomorphic vector bundle, and ψ : X → [-∞, +∞[ a quasi-psh function with neat analytic singularities. Before given the technical details of the proofs, we start with a rather simple observation.

(5.1) Observation. Let µ : X → X be a proper modification. Assume that X is equipped with a Kähler metric ω.

(a) For every m ≥ 0, there is an isomorphism

µ * : H 0 (X, O(K X ⊗ E) ⊗ I(mψ)) → H 0 (X, O(K X ⊗ µ * E) ⊗ I(mψ • µ))
whose inverse is the direct image morphism µ * .

(b) For any holomorphic section

F of E, the L 2 norm X |F • µ| 2 e -mψ•µ dV X, ω coincides with X |F | 2 e -mψ dV X,ω .
(c) On the regular part of the subvariety Y (m p ) = V (I(m p ψ)), for any

f ∈ H 0 (Y, O X (K X ⊗ E) ⊗ I(m p-1 ψ)/I(m p ψ)), the area measure |f | 2 ω,h dV Y (m p )
,ω (which is independent of ω) is the direct image of its counterpart defined by ψ = ψ • µ and f = µ * f on the strict transform of Y (m p ) , i.e. the union of components of Y (m p ) = µ -1 (Y (m p ) ) that have a dominant projection to a component of Y (m p ) .

The proof of (c) is immediate by a change of variable z = µ(w) in the integrals {t<ψ<t+1} ... and by passing to the limits. It follows from the observation and the discussion of section 4 that after blowing up the proof of our theorems can be reduced to the case where ψ has divisorial singularities along a normal crossing divisor.

Proof of Theorem 2.8. This will be only a mild generalization of the techniques used in [START_REF] Ohsawa | On the extension of L 2 holomorphic functions, V : Effects of generalization[END_REF], with a technical complication due to the fact that we do not assume any Steinness of complements of negligible sets. With the notation of Theorem 2.8, let f ∈ H 0 (Y • , (K X ⊗ E) ↾Y • ). We view f as an E-valued (n, 0)-form defined over Y • and apply Proposition 3.11 after replacing the metric h of E by the singular metric

h ψ = h -ψ , whose curvature is i Θ E,h ψ = i Θ E,h + i d ′ d ′′ ψ.
In order to avoid the singularities, we shrink X to a relatively compact weakly pseudoconvex domain X c = {ρ < c}, and work on X c Y instead of X. In fact, we know by [Dem82, Theorem 1.5] that X c Y is complete Kähler. Let us first assume that Y is non singular, i.e. Y • = Y , and that ψ ≤ 0 ; of course, when X is compact, one can always subtract a constant to ψ to achieve ψ ≤ 0, but there could exist non compact situations when it is interesting to take ψ unbounded from above. In any case, we claim that there exists a smooth section f

∈ C ∞ (X, Λ n,0 T * X ⊗ E) such that (5.2) f coincides with f on Y , (5.3) D ′′ f = 0 at every point of Y , (5.4) |D ′′ f | 2
ω,h e -ψ is locally integrable near Y . For this, consider a locally finite covering of Y by coordinates patches U j ⊂ X biholomorphic to polydiscs, on which E ↾U j is trivial, and such that either

Y ∩ U j = ∅ or Y ∩ U j = {z ∈ U j ; z 1 = • • • = z r = 0}.
We can find holomorphic sections

f j ∈ C ∞ (X, Λ n,q T * X ⊗ E)
which extend f ↾Y ∩U j [such sections can be obtained simply by viewing functions of the form g(z r+1 , . . . , z n ) as independent of z 1 , . . . , z r on each polydisc]. For some partition of unity (χ j ) subordinate to (U j ), we then set f := j χ j f j . Clearly (5.2) f ↾Y = f holds. Since we have

D ′′ f = D ′′ ( f -f k ) = D ′′ j χ j ( f j -f k ) = j d ′′ χ j ∧ ( f j -f k ),
properties (5.3) and (5.4) also hold, as f jf k = 0 on Y and I(ψ) = I Y is reduced by our assumption that ψ has log canonical singularities. The main idea is to apply Proposition 3.11 to solve the equation (5.5)

D ′′ u t = v t := D ′′ (θ(ψ -t) f ), t ∈ ] -∞, -1],
where

θ : [-∞, +∞[ → [0, 1] is a smooth non increasing function such that θ(τ ) = 1 for τ ∈ ] -∞, ε/3], θ(τ ) = 0 for τ ∈ [ε/3, +∞[ and |θ ′ | ≤ 1 + ε, for any positive ε ≪ 1.
First assume for simplicity that D ′′ f = 0 without any error (i.e. that f is globally holomorphic). Then

(5.6)

v t = D ′′ (θ(ψ -t) f ) = θ ′ (ψ -t) d ′′ ψ ∧ f
has support in the tubular domain W t = {t < ψ < t + 1}. At the same time, we adjust the functions η = η t and λ = λ t used in Prop. 3.11 to create enough convexity on W t . For this, we take (5.7)

η t = 1 -δχ t (ψ)
where χ t : ] -∞, 0] → R is a negative smooth convex increasing function with the following properties:

(5.8 a) χ t (0) = 0 and inf

τ ≤0 χ t (τ ) = -M t > -∞, (5.8 b) 0 ≤ χ ′ t (τ ) ≤ 1 2 for τ ≤ 0, (5.8 c) χ ′ t (τ ) = 0 for τ ∈ ] -∞, t -1], χ ′ t (τ ) > 0 for τ ∈ ]t -1, 0], (5.8 d) χ ′′ t (τ ) ≥ 1 -ε 4 for τ ∈ [t, t + 1].
(5.8 e)

χ ′′ t (τ ) χ ′ t (τ ) 2 ≥ 2δ π(1 + δ 2 τ 2 ) for τ ∈ ]t -1, 0].
The function χ t can be easily constructed by taking (5.9)

χ ′′ t (τ ) = δ 2π(1 + δ 2 τ 2 ) β(τ -t) + 1 -ε 4 ξ(τ -t)
with support in [t -1, 0], where β : R → [0, 1] is a smooth non decreasing function such that β(τ ) = 0 for τ < -1, β(τ ) = 1 for τ ≥ 0 and ξ(τ

) = β(τ )β(1 -τ ). We then have Supp(β) = [-1, +∞[, Supp(ξ) = [-1, 2], and ξ(τ ) = β(τ ) on [-1, 0]. On [-1
, 0[, we can take β so small that 0 -1 β(τ ) dτ < ε/2. By symmetry we find 2 -1 ξ(τ ) dτ < 1+ε. Clearly (5.8 a,c,d) hold if we adjust the integration constant so that χ ′ t (t -1) = 0. Now, the right hand side of (5.8) has a total integral on ]-∞, 0] that is less than (1-ε) 4 (1 + ε) + 1 2π π 2 < 1 2 , thus (5.8 b) is satisfied. This implies that (5.8 e) holds at least on the interval [t, 0]. An integration by parts yields

χ ′ t (τ ) ≤ δ 2π + 1 -ε 4 β(τ -t) on [t -1, t],
where β ≥ 0 is a primitive of β vanishing at τ = -1. On [t -1, t] we find (5.10)

χ ′′ t (τ ) χ ′ t (τ ) 2 ≥ 1 -ε 4 δ 2π + 1 -ε 4 -2 β(τ -t) β(τ -t) 2 . On [-1, 0] we have β(τ ) ≤ (1 + τ )β(τ ) ≤ β(τ ) and β(τ ) ≤ 0 -1 β(τ ) dτ < ε/2
, and we see that the right hand side of (5.10) can be taken arbitrary large when ε is small. Therefore (5.8 e) can also be achieved on [t -1, t].

We now come back to the choice of η t and λ t . Since we have assumed ψ ≤ 0 at this step, we have by definition η t ≥ 1. Moreover,

d ′ η t = -δ χ ′ t (ψ) d ′ ψ and (5.11) i d ′ d ′′ χ t (ψ) = i χ ′ t (ψ) d ′ d ′′ ψ + i χ ′′ t (ψ) d ′ ψ ∧ d ′′ ψ,
hence we see that

R t := η t i Θ E,h + i d ′ d ′′ ψ -i d ′ d ′′ η t -λ -1 t i d ′ η t ∧ d ′′ η t = η t i Θ E,h + (1 + δη -1 t χ ′ t (ψ)) i d ′ d ′′ ψ + δχ ′′ t (ψ) -λ -1 t δ 2 χ ′ t (ψ) 2 i d ′ ψ ∧ d ′′ ψ. The coefficient (1 + δη -1 t χ ′ t (ψ)) lies in [1, 1 + δ 2 ]
, and our curvature assumption implies that the first term in the right hand side is non negative. We take (5.12)

λ t = π(1 + δ 2 ψ 2 ).
By (5.8 e), this ensures that λ -1 t δ 2 χ ′ t (ψ) 2 ≤ 1 2 δχ ′′ t (ψ), hence we get the crucial lower bounds

R t ≥ 1 2 δχ ′′ t (ψ) i d ′ ψ ∧ d ′′ ψ ≥ 0 on X c , (5.13) R t ≥ (1 -ε)δ 8 i d ′ ψ ∧ d ′′ ψ on W t = {t < ψ < t + 1}. (5.14) A standard calculation gives the formula [Λ ω , i d ′ ψ ∧ d ′′ ψ]v, v ω,h = |(d ′′ ψ) * v| 2
ω,h for any (n, 1)-form v. Therefore, for every (n, 0)-form u we have

| d ′′ ψ ∧ u, v | 2 = | u, (d ′′ ψ) * v | 2 ≤ |u| 2 |(d ′′ ψ) * v| 2 = |u| 2 [Λ ω , i d ′ ψ ∧ d ′′ ψ]v, v .
From this and (5.14) we infer that the curvature operator

B t = [Λ ω , R t ] satisfies B -1 t (d ′′ ψ ∧ u), d ′′ ψ ∧ u = |B -1/2 t (d ′′ ψ ∧ u)| 2 ≤ 8 (1 -ε)δ |u| 2 on W t .
In particular, since

|θ ′ | ≤ 1 + ε, we see that the form v t = θ ′ (ψ -t) d ′′ ψ ∧ f defined in (5.6) satisfies (5.15) B -1 t v t , v t ≤ 8(1 + ε) 2 (1 -ε)δ | f | 2 . Now, (5.8 b) implies |χ t (τ )| ≤ 1 2 |τ |, thus η t = 1 -δχ t (ψ) ≤ 1 -1 2 δψ and
(5.16)

η t + λ t ≤ 1 -1 2 δψ + π(1 + δ 2 ψ 2 ) ≤ 4.21 (1 + δ 2 ψ 2 )
(the optimal constant is

√ 5+2 4 + π < 4.21). As 4.21 × 8 × (1 + ε) 2 /(1 -ε) < 34 for ε ≪ 1, Proposition 3.11 produces a solution u t such that D ′′ u t = v t on X c Y and X c Y (1 + δ 2 ψ 2 ) -1 |u t | 2 ω,h e -ψ dV X,ω ≤ 34 δ {t<ψ<t+1} | f | 2 ω,h e -ψ dV X,ω .
The function F t = θ(ψt) fu t is essentially the extension we are looking for. For any α > 0 we have

|F t | 2 ≤ (1 + α) |u t | 2 + (1 + α -1 ) |θ(ψ -t)| 2 | f | 2 , hence X c Y (1 + δ 2 ψ 2 ) -1 (1 + α 2 ψ 2 ) -(n-1)/2 |F t | 2 ω,h e -ψ dV X,ω ≤ 34(1 + α) δ X c ∩{t<ψ<t+1} | f | 2 ω,h e -ψ dV X,ω (5.17 1 ) + (1 + α -1 ) X c ∩{ψ<t+1} (1 + δ 2 ψ 2 ) -1 (1 + α 2 ψ 2 ) -(n-1)/2 | f | 2 ω,h e -ψ dV X,ω . (5.17 2 )
The local integrability of (1 + |ψ|) -(n+1) | f | 2 e -ψ asserted by Proposition 4.5 (c) and the Lebesgue dominated convergence theorem imply that the last integral (5.17 2 ) converges to 0 as t → -∞. Therefore we obtain lim sup

α→0 + lim t→-∞
(5.17 1 ) + (5.17

2 ) ≤ 34 δ X c ∩Y |f | 2 ω,h dV Y,ω [ψ].
by definition of the measure in the right hand side (cf. (2.4)). Since u t is in L 2 with respect to the singular weight e -ψ , it is also locally L 2 with respect to a smooth weight. A standard lemma (cf. [Dem82, Lemme 6.9]) shows that the equation D ′′ u t = v t extends to X c , and the hypoellipticity of D ′′ implies that u t is smooth on X c . As e -ψ is non integrable along Y , we conclude that u t must vanish on X c ∩ Y . Therefore F t is indeed a holomorphic extension of f on X c . By letting t tend to -∞ and then c to +∞, the uniformity of our L 2 inequalities implies that one can extract a weakly convergent sequence F t ν → F , such that F ∈ H 0 (X, K X ⊗ E) is an extension of f and

X (1 + δ 2 ψ 2 ) -1 |F | 2 ω,h e -ψ dV X,ω ≤ 34 δ Y |f | 2 ω,h dV Y,ω [ψ].
The proof also works when Y is singular, because the equations can be considered on X Y sing , and X c Y sing is again complete Kähler for every c > 0.

In case ψ is no longer negative, we put ψ + A = 1 A log(1 + e Aψ ) and replace ψ with

ψ A = ψ -ψ + A < 0
which converges to ψψ + = -ψ -as A → +∞. We then solve D ′′ u t = v t with v t = D ′′ (θ(ψ At) f ), and use the functions η t = 1δχ t (ψ A ) and λ t = π(1 + δ 2 ψ 2 A ) in the application of proposition 3.11. The expression of the curvature term R t becomes

R t,A = η t i Θ E,h + i d ′ d ′′ ψ + δη -1 t χ ′ t (ψ A )) i d ′ d ′′ ψ A + δχ ′′ t (ψ A ) -λ -1 t δ 2 χ ′ t (ψ A ) 2 i d ′ ψ A ∧ d ′′ ψ A .
All bounds are then essentially the same, except that we have an additional negative term (...

)i d ′ ψ ∧ d ′′ ψ in i d ′ d ′′ ψ A , i.e. i d ′ d ′′ ψ A = 1 1 + e Aψ i d ′ d ′′ ψ - Ae Aψ (1 + e Aψ ) 2 i d ′ ψ ∧ d ′′ ψ.
Because of this term, the first term η t (...) in R t,A is a priori no longer ≥ 0. However, this can be compensated by adding an extra weight δ 2 ψ + A to the metric of (E, h), so that the total weight is now ψ + δ 2 ψ + A . The contribution of the new weight to the term η t (...) in the modified R t,A reads

i Θ E,h + 1 + δη -1 t χ ′ t (ψ A ) 1 1 + e Aψ + δ 2 e Aψ 1 + e Aψ i d ′ d ′′ ψ + δ Ae Aψ (1 + e Aψ ) 2 1 2 -η -1 t χ ′ t (ψ A ) i d ′ ψ ∧ d ′′ ψ,
and as the coefficient of i d ′ d ′′ ψ still lies in [1, 1 + δ] (remember that η t ≥ 1 and χ ′ t ≤ 1 2 ), we conclude that we have again

R t,A ≥ 1 2 δχ ′′ t (ψ A ) i d ′ ψ A ∧ d ′′ ψ A ≥ 0 on X c , (5.18) R t,A ≥ (1 -ε)δ 8 i d ′ ψ A ∧ d ′′ ψ A on W t,A = {t < ψ A < t + 1}. (5.19) Since ψ -ψ A → 0 along Y = ψ -1 (-∞)
, it is easy to see that the measures dV Y,ω [ψ] and dV Y,ω [ψ A ] coincide. After making those corrections, we get an extension F A such that

X (1 + δ 2 ψ 2 A ) -1 |F A | 2 ω,h e -ψ-δ 2 ψ + A dV X,ω ≤ 34 δ Y |f | 2 ω,h dV Y,ω [ψ].
By letting A → +∞ and extracting a limit F A → F , we get

X (1 + δ 2 ψ 2 -) -1 |F | 2 ω,h e -ψ-δ 2 ψ + dV X,ω ≤ 34 δ Y |f | 2 ω,h dV Y,ω [ψ],
which is equivalent to the final bound given in Theorem 2.8.

The next point we have to justify is that unfortunately we cannot expect D ′′ f ≡ 0 as we assumed a priori (unless we already know for some reason that a global holomorphic extension exists). In fact, we have to solve an equation

D ′′ u = v t := D ′′ (θ(ψ A -t) f )
with an extra term in the right hand side, namely (5.20)

D ′′ u = v t = v (1) t + v (2) t , v (1) t = θ ′ (ψ A -t)d ′′ ψ A ∧ f , v (2) t = θ(ψ A -t)D ′′ f .

The first term v

(1) t of (5.20) has been already estimated, but we have to show that the second term v

(2) t becomes "negligible" when we take limits as t → -∞. For this we solve (5.20) by means of Prop. 3.12 instead of Prop. 3.11. We get an approximate L 2 solution

D ′′ u t,ε = v t -w t,ε , whence D ′′ (θ(ψ A -t) f -u t,ε ) = w t,ε . Moreover, this solution satisfies the L 2 estimate (η t + λ t ) -1/2 u t,ε 2 + 1 ε w t,ε 2 ≤ X c Y (B t,A + εI) -1 v t , v t e -ψ-δ 2 ψ + A dV X,ω ≤ (1 + α) X c Y B -1 t,A v (1) t , v (1) t e -ψ-δ 2 ψ + A dV X,ω (5.21 1 ) + (1 + α -1 )ε -1 X c Y v (2) t , v (2) t e -ψ-δ 2 ψ + A dV X,ω (5.21 2 )
for α > 0 arbitrary. The integral (5.21 1 ) is bounded by means of (5.17 i ) -or its analogue for the modified weight ψ + δ 2 ψ + A -and the integral (5.21 2 ) is equal to

(5.22) (1 + α -1 )ε -1 X c ∩{t<ψ<t+1} |θ(ψ A -t)| 2 |D ′′ f | 2 e -ψ-δ 2 ψ + A dV X,ω .
We now put all estimates together for the section F t,ε = θ(ψ At) fu t,ε , which satisfies

|F t,ε | 2 ≤ (1 + α) |u t,ε | 2 + (1 + α -1 ) |θ(ψ A -t)| 2 | f | 2 .
We get in this way from (5.16), (5.17 i ), (5.21 i ) and (5.22)

X c Y (1 + δ 2 ψ 2 A ) -1 (1 + α 2 ψ 2 A ) -(n-1)/2 |F t,ε | 2 ω,h e -ψ-δ 2 ψ + A dV X,ω + 1 ε X c Y |w t,ε | 2 ω,h e -ψ-δ 2 ψ + A dV X,ω ≤ 34(1 + α) 2 δ X c ∩{t<ψ A <t+1} | f | 2 e -ψ-δ 2 ψ + A dV X,ω (5.23 1 ) + (1 + α -1 ) X c ∩{ψ A <t+1} (1 + δ 2 ψ 2 A ) -1 (1 + α 2 ψ 2 A ) -(n-1)/2 | f | 2 e -ψ-δ 2 ψ + A dV X,ω (5.23 2 ) + (1 + α)(1 + α -1 ) ε X c ∩{ψ A <t+1} |D ′′ f | 2 e -ψ-δ 2 ψ +
A dV X,ω . (5.23 3 ) By Proposition 4.5 (c), the integral (5.23 2 ) converges to 0 as t → -∞. Since |D ′′ f | 2 e -ψ is locally integrable on X, the last integral (5.23 3 ) also converges to 0 as t → -∞. We let ε and α converge to 0 afterwards, and extract a limit F = lim ε→0 lim t→-∞ F t,ε as already explained, to recover the expected L 2 estimate.

(5.24) Final regularity argument. One remaining non trivial point is to check that we get smooth solutions and that the resulting limit F of F t,ε = θ(ψ -t) f -u t,ε is actually an extension of f , one particular issue being that F t,ε is not exactly holomorphic (for the simplicity of notation we assume here that ψ ≤ 0 since the difficulty is purely local near Y , and thus skip the ψ A approximation process in what follows). For this, we apply observation 5.1 and use a composition of blow-ups µ : X → X such that the singularities of ψ = ψ•µ are divisorial, given by some normal crossing divisor ∆ = c j ∆ j in X whose support contains the exceptional divisor of ψ. If g is a germ of section in O

X,x (K X ⊗ E), the section g = µ * g takes values in µ * (K X ⊗ E) = K X ⊗ (O X (-∆ ′ ) ⊗ µ * E), where ∆ ′ =
c ′ j ∆ j has support in |∆| and the c ′ j are non negative integers. As ψ has log canonical singularities on X, we see by taking g invertible that ∆ -∆ ′ has coefficients c jc ′ j ≤ 1 on X. Let us set ∆ -∆ ′ = ∆ (1) + ∆ ′′ where ∆ (1) consists of the sum of components of multiplicity c jc ′ j = 1, and ∆ ′′ is the sum of all other ones with c jc ′ j < 1. The metric involved in our

L 2 estimates is µ * h ψ = µ * he -ψ . When viewed as a metric on G = K X ⊗ µ * E ⊗ O(-∆ ′ -∆ (1) ), it possesses a weight ψ G := ψ -log |σ ∆ ′ | 2 -log |σ ∆ (1) | 2 = log |σ ∆ ′′ | 2 mod C ∞ ,
hence ψ G is non singular at the generic point of any component ∆ j of ∆ (1) . Solving a ∂-equation in µ * (K X ⊗ E) with respect to the singular weight e -ψ•µ = e -ψ amounts to solving the same ∂-equation with values in G with respect to the weight ψ G . The standard ellipticity results imply that the solutions u t,ε on X given by 3.12 are smooth as sections of G. This argument shows that we can assume right away that ψ has divisorial singularities, and consider only the case µ = Id X and ∆ = ∆ (1) + ∆ ′′ , in which case Y = |∆ (1) | and O X (-∆ (1) ) = O X (-Y ). We thus simplify the notation by removing all hats, and set

G = K X ⊗ E ⊗ O X (-Y ). Notice that F t,ε = τ (ψ -t) f -u t,ε
is not exactly holomorphic. We have instead D ′′ F t,ε = w t,ε where w t,ε is a section of Λ

0,1 T * X ⊗ G that is smooth on the Zariski open set Y • = Y ∆ k ⊂Y ∆ k , and its L 2 norm with respect to ψ G = ψ -log |σ Y | 2 satisfies w t,ε = O(ε 1/2 ). If F 0 is a fixed local extension of f near a generic point x 0 ∈ ∆ j , then F t -F 0 is a local section of G and D ′′ (F t,ε -F 0 ) = w t,ε .
We can find, say by the standard Hörmander L 2 estimates on a coordinate ball B(x 0 , r) ([Hör65,66], see also [START_REF] Kohn | Harmonic integrals on strongly pseudo-convex manifolds I[END_REF]64]), a smooth L 2 section s t,ε of G such that s t,ε = O(ε 1/2 ) and D ′′ s t,ε = w t,ε on B(x 0 , r). Then F t,ε -F 0s t,ε is a holomorphic section of G on B(x 0 , r). Its limit F -F 0 is a limit in L 2-α for every α > 0, thanks to the Hölder inequality and the fact that 1 + δ 2 ψ 2 is in L p for every p > 1. Thus F -F 0 is a holomorphic section of G on B(x 0 , r), and so

F ↾Y = f on Y • .
Proof of Remark 2.9 (b). By the technique of proof of the regularization theorem [Dem92, Theorem 1.1], on any relatively compact subset X c ⋐ X, there are quasi-psh approximations ϕ ν ↓ ϕ of ϕ with neat analytic singularities, such that the curvature estimate suffers only a small error ≤ 2 -ν ω, namely, for

h ν = e -ϕ ν ≤ h, we have i Θ E,h ν + α i d ′ d ′′ ψ ν ≥ -2 -ν ω on X c , uniformly for α ∈ [1, 1 + δ].
As a consequence, the operator

B t,ν = η t (i Θ E,h ν + α i d ′ d ′′ ψ ν -i d ′ d ′′ η t -λ -1 t i d ′ η t ∧ d ′′ η t , Λ ω ]
has a slightly negative lower bound -M t 2 -ν by (5.8 a). This can be absorbed by means of an additional positive term εI in B t,ν + εI, with ε = O(2 -ν ). Moreover the set of poles of ϕ ν is an analytic set P ν and we can work on the complete Kähler manifold X c (Y ∪ P ν ) to avoid any singularities. Then Prop. 3.12 provides an approximate solution

D ′′ u t,ν ≈ v t with error O(ε 1/2 ) = O(2 -ν/2 ), satisfying the estimate X c Y exp(-δ 2 1 A log(1 + e Aψ )) 1 + δ 2 ψ 2 A |u t,ν | 2 ω,h ν e -ψ dV X,ω ≤ 34 δ X c ∩{t<ψ A <t+1} | f | 2 ω,h ν e -ψ A dV X,ω .
The right hand side is uniformly bounded by a similar norm where h ν is replaced by h = e -ϕ . The conclusion follows by letting ν converge to +∞ (before doing anything else), and by extracting limits.

Proof of Theorem 2.12. The proof is essentially identical to the proof of Theorem 2.8, we simply make a "rescaling": we replace ψ by m p ψ, t by m p t, δ by δ/m p , θ by θ(τ ) = θ(τ /m p ) and use Lemma 4.5 in its full generality. Property (5.4) [namely the integrability of |D ′′ ( f )| 2 e -m p ψ ] still holds here since f has local extensions f j such that the differences f jf k lie by construction in I(m p ψ). In the final regularity argument 5.24, the vanishing of f prescribed by I(m p-1 ψ) subtracts a further divisor to ∆ -∆ ′ -∆ m p-1 where µ * I(m p-1 ψ) = O(-∆ m p-1 ). The definition of jumps leads to the fact that we only have to considers components of multiplicity 1 in that difference, and the rest of the argument is the same.

Proof of Theorem 2.13. The argument is identical to the proof of Remark 2.9 (b), when we make the same rescaling, especially by replacing ψ with m p ψ.

Proof of Theorem 2.14. Since X is compact, all coherent cohomology groups involved are finite dimensional and Hausdorff for their natural topology. We can assume ψ < 0 by subtracting a constant (and thus avoid the ψ A approximation process), and X = X c . First assume ℓ = p -1. Theorem 2.12 provides an extension F of f in case I(m p-1 ψ) is replaced by I ′ (m p-1 ψ), but otherwise the limiting L 2 integral of f computed on Y (m p ) may diverge. The main idea, however, is that the integrals are still convergent when considered on each tube {t < ψ < t + 1}, and this can be used to check at least the qualitative part of the surjectivity theorem. In fact, we apply the arguments used in the proof of Theorems 2.8-2.12, especially estimates (5.23 i ), taken with α = 1, and ψ = ψ A , t, θ(τ ), δ replaced respectively with m p ψ, m p t, θ(τ /m p ), δ/m p ; we also assume δ ≤ 1 here. This produces a C ∞ extension F t,ε = θ(ψt) fu t,ε such that D ′′ F t,ε = w t,ε and This is true for all t < 0 and ε > 0, and the idea is to adjust the choice of ε as a function of t. The integral (5.25 2 ) is uniformly bounded by Proposition 4.5 (c) and can be disregarded. By construction the coefficients D ′′ f can be expressed as a combination of functions f jf j in I(m p ϕ), thus by openness, there exists α > 0 such that X |D ′′ f | 2 ω,h e -(m p +α)ψ dV X,ω < +∞.

X (1 + m 2 p ψ 2 ) -(n+1)/2 |F t,ε | 2 ω,h e -m p ψ dV X,ω + 1 ε X |w t,ε | 2 ω,
We therefore get an upper bound (5.26)

{ψ<t+1}
|D ′′ f | 2 ω,h e -m p ψ dV X,ω ≤ C e αt as t → -∞.

On the other hand, since X | f | 2 ω,h e -(m p -β)ψ dV X,ω < +∞ for every β > 0, we conclude that (5.27)

{t<ψ<t+1} | f | 2
ω,h e -m p ψ dV X,ω ≤ C ′ (β) e -βt as t → -∞.

A possible choice of ε is to take ε = e (α+β)t for some β > 0. Then (5.25 i ) and (5.26-5.27) imply X |w t,ε | 2 ω,h e -m p ψ dV X,ω ≤ C ′′ (δ, β)e αt .

We conclude that the error term w t,ε converges uniformly to 0 in L 2 norm as t → -∞ and ε = e (α+β)t → 0. The main term F t,ε , however, is not under control, but we can cope with this situation. By taking a principalization of the ideal defining the singularities of ψ, we can assume that these singularities are divisorial, and thus that our solutions are smooth at the generic point of Z p = Supp(I(m p-1 ψ)/I(m p ψ)). Observe that w t,ε is a coboundary for the Dolbeault complex associated with the cohomology group H 0 (X, O X (K X ⊗ E) ⊗ I(m p-1 ψ)). Now, X is compact and Čech cohomology can be calculated on finite Stein coverings by spaces of Čech cocycles equipped with the topology given by L 2 norms with respect to the weight e -m p-1 ψ . We conclude via an isomorphism between Čech cohomology and Dolbeault cohomology that there is a smooth global section s t,ε of C ∞ (K X ⊗ E) ⊗ I(m p-1 ψ) such that D ′′ s t,ε = w t,ε and X |s t,ε | 2 ω,h e -m p-1 ψ dV X,ω = O(e tα ).

On the other hand, on any coordinate ball B = B(x 0 , r) ⊂ X, we can apply the standard L 2 estimates of Hörmander for bounded pseudoconvex domains, and find another local solution s B,t,ε such that D ′′ s B,t,ε = w t,ε and

(5.28)

B(x 0 ,r)

| s B,t,ε | 2 ω,h e -m p ψ dV X,ω = O(e tα ).

The difference s t,εs B,t,ε is a holomorphic section of O X (K X ⊗E)⊗I(m p-1 ψ) on B(x 0 , r) which converges to 0 in L 2 norm, thus uniformly on any smaller ball B(x 0 , r ′ ). Notice that s B,t,ε has a vanishing order prescribed by I(m p ψ) by (5.28). Therefore F t,εs B,t,ε is a local holomorphic section of O X (K X ⊗ E) ⊗ I(m p-1 ψ) that maps exactly to f . By what we have seen

F t,ε -s t,ε ∈ H 0 (X, O X (K X ⊗ E) ⊗ I(m p-1 ψ))
is a global holomorphic section whose restriction converges uniformly to f . Since X is compact, we are dealing with finite dimensional spaces of sections and thus the restriction morphism must be surjective. The proof of the case ℓ = p -1 is complete. Now, assume that the result has been proved for pℓ = d and take pℓ = d + 1. By reducing f mod I(m p-1 ψ), the induction hypothesis provides an extension F ′ in H 0 (X, O X (K X ⊗ E) ⊗ I(m ℓ ψ)) of f ′ := f mod I(m p-1 ψ) in H 0 (Y (m p-1 ) , O X (K X ⊗ E) ⊗ I(m ℓ ψ)/I(m p-1 ψ)).

Thus f p := f -(F ′ mod I(m p ψ)) defines a section f p ∈ H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(m p-1 ψ)/I(m p ψ)).

The case d = 1 provides an extension

F p ∈ H 0 (X, O X (K X ⊗ E) ⊗ I(m p-1 ψ))
and F = F ′ + F p is the extension we are looking for.

(b) Assume finally that E is a line bundle and that h is a singular hermitian metric satisfying the curvature estimate in the sense of currents only. We can reduce ourselves to the case when ψ has divisorial singularities. The regularization techniques of [START_REF] Demailly | On the cohomology of pseudoeffective line bundles[END_REF] (see Remark 3 before section 3, based on the solution of the openness conjecture) produces a singular metric h ε withe analytic singularities, such that the multiplier ideal sheaves I(he -m ℓ ψ ) involved are unchanged when h is replaced by h ε , and such that there is an arbitrary small loss in the curvature. We absorb this new adverse negative term by considering again B t + εI, and by applying the same tricks as above.

U

  |f | 2 e -ψ dλ < +∞ where U is an open coordinate neighborhood of x, and dλ the standard Lebesgue measure in the corresponding open chart of C n . We say that the singularities of ψ are log canonical along the zero variety Y = V (I(ψ)) if I((1ε)ψ) ↾Y = O X↾Y for every ε > 0.

  where η, (d ′ η), (d ′′ η) are abbreviated notations for the multiplication operators η•, (d ′ η) ∧ •, (d ′′ η) ∧ •. By subtracting the above equalities and taking into account the Bochner-Kodaira-Nakano identity ∆ ′′ -∆ ′ = [iΘ E , Λ], we get

  (4.1) 0 = m 0 < m 1 < m 2 < . . . < m p < . . . such that I(ϕ + mψ) = I(ϕ + m p ψ) for m ∈ [m p , m p+1 [ andI(ϕ) = I(ϕ + m 0 ψ) I(ϕ + m 1 ψ) . . . I(ϕ + m l ψ) . . . .If ψ is smooth, we have of course m 1 = +∞ already, and the sequence stops there; in the sequel we assume that ψ has non empty logarithmic poles to avoid this trivial situation -the sequence m p is then infinite since the multiplicities of germs of functions in I(ϕ + mψ) x tend to infinity at every point x where the Lelong number ν(ψ, x) is positive.

( a )

 a The measure |J m p (f ℓ)| 2 ω,h dV Z • p ,ω [ψ] defined by (2.10) has a smooth positive density with respect to the Lebesgue measure on a Zariski open set Z • p of Z p . (b) The sheaf I ′ (m p-1 ψ) of germs F ∈ I(m p-1 ψ) x ⊂ O X,x such that there exists a neighborhood U of x satisfying

( a )

 a We proceed by induction on pℓ. Letf ∈ H 0 (Y (m p ) , O X (K X ⊗ E) ⊗ I(m ℓ ψ)/I(m p ψ)).

  h e -m p ψ dV X,ω≤ 136 δ {t<ψ<t+1} | f | 2 e -m p ψ dV X,ω (5.25 1 ) + 2 {ψ<t+1} (1 + m p ψ 2 ) -(n+1)/2 | f | 2 e -m p ψ dV X,ω (5.25 2 ) + 4 ε {ψ<t+1} |D ′′ f | 2 e -m p ψ dV X,ω . (5.25 3 )

* This work is supported by the ERC grant ALKAGE


Filip Křikava
email: filip.krikava@fit.cvut.cz

Solving the TTC'15 Train Benchmark Case Study with SIGMA

This paper describes a solution for the Transformation Tool Contest 2015 (TTC'15) Train Benchmark case study using SIGMA, a family of Scala internal Domain-Specific Languages (DSLs) that provides an expressive and efficient API for model consistency checking and model transformations.

Introduction

The purpose of the TTC'15 Train Benchmark case study [START_REF] Szárnyas | The TTC 2015 Train Benchmark Case for Incremental Model Validation[END_REF] is to systematically assess the scalability of consistency checking and repair of large scale models. It presents a scenario from the railway domain for which the solution requires to implement 5 constraints and repair operations of increasing complexity. An associated framework is then used to evaluate the correctness and performance of the solutions over large model instances.

In this paper we present our solution using SIGMA [START_REF] Krikava | SIGMA: Scala Internal Domain-Specific Languages for Model Manipulations[END_REF], a family of Scala1 internal DSLs for model manipulation tasks such as model validation, model to model (M2M), and model to text (M2T) transformations. Scala is a statically typed production-ready General-Purpose Language (GPL) that supports both object-oriented and functional styles of programming. It uses type inference to combine static type safety with a "look and feel" close to dynamically typed languages. Furthermore, it is supported by the major integrated development environments bringing EMF modeling to other IDEs than traditionally Eclipse (e.g. IntelliJ IDEA was used for this solution).

SIGMA DSLs are embedded in Scala as a library allowing one to manipulate models using high-level constructs similar to the ones found in the external model manipulation DSLs. The intent is to provide an approach that developers can use to implement many of the practical model manipulations within a familiar environment, with a reduced learning overhead as well as improved usability and performance.

The solution is based on the Eclipse Modeling Framework (EMF) [START_REF]EMF: Eclipse Modeling Framework (2nd Edition)[END_REF], which is a popular metamodeling framework widely used in both academia and industry, and which is directly supported by SIGMA. The complete source code is available on Github2 in the fork of the original case study repository.

Solution Description

The solution for this transformation case study consist of a set of queries that check for violations of a number of model constrains and repair transformations that in turn fixes them. SIGMA provides a dedicated model consistency checking DSL with the ability to provide quick fixes repairing invariant validations. However, given the benchmark framework used in the case study, we decided to provide a more dedicated support for the given query/repair tasks in a form of an internal DSL. The reason is that (1) it allows for an easy comparison between the reference implementations in Java and EMF-IncQuery and (2) it shows the expressiveness of the language allowing one in few lines of code to bridge the gap between the problem-level abstractions (query, repair transformation) and the implementation-level concepts (e.g., classes, higher-order functions). We therefore only rely on the SIGMA operations for model navigation (i.e. projecting information from models) and modification (i.e. changing model properties or elements). Essentially, these operations bridge the model classes (Ecore classes in this case) to be compatible with Scala allowing for example one to use the powerful Scala collection library.

On the top of SIGMA, we have created a small internal DSL that allows us to solve the given benchmark cases in an expressive and compact way. Following the case study description, the top-level domain concept is a constraint. A constraint is composed of a model query that finds all model instances violating a certain model restriction and a repair transformation correcting the failed instances. Concretely, a query is a function that given a model element-i.e. a context of the constraint in the classical model consistency checking-returns a set of matches. A match can either be a single instance or a tuple of instances of model elements that are related to the violations.

The following description of the solution is split in two parts: (1) the core part that describes the queries and repair transformations, (2) the integration part gives an overview how it has been integrated in the case study source code.

Queries and Repair Transformations DSL

A typical way of creating an internal DSL in Scala is by designing a library that allows one to write fragments of code with domain-specific syntax. These fragments are woven within Scala's own syntax so that it appears different.

One way to represent the above concepts is using a Scala case class:

case class Constraint[A <: EObject, B <: AnyRef]( query: (A) => Iterable[B], repair: (B) => Unit )
This defines a case class with a field for both query and repair. A case class in Scala is like a regular class with some additional properties out which, in our case, the important one is that it can be instantiated without the new keyword and thus limiting the language noise. The two type parameters A, B specify the model context for the query and the types of matches the query produces. The input type is further constrained to be a subtype of an EObject. The query and repair are defined as functions A → Iterable[B] and B → Unit where Unit is like void in Java.

In some cases the match returned by the query is of the same type as the query context. The query can be therefore simplified to a boolean expression selecting instances on which it evaluates to true. For these types of queries we provide a dedicated construct called BooleanConstraint: For example, the first query, PosLength, which finds all the segments with negative length can be written as:

BooleanConstraint[Segment]( query = segment => segment.length < 0, repair = segment => segment.length += -segment.length + 1 )
We do not have to specify the types of the parameter nor the result as they will be inferred by the Scala compiler.

Another example using more complex expression is the SwitchSet constraint: This is a more complex constraint that matches a tuple of model elements. It is using a for comprehension, a lightweight notation for expressing sequence comprehensions 3 . Scala for comprehensions have the form for (enumerators) yield e, where enumerators refers to a list of enumerators. An enumerator is either a generator which introduces new variables, or it is a filter. A comprehension evaluates the body e for each binding generated by the enumerators and returns a sequence of these values.

In this concrete example, the generator is the optional value of the Route.entry reference. It either generates a single value in the case the actual instance contains one or it does not produce anything. There is a small inconsistency in the model, the Route.entry should have the cardinality set to 0..1 instead of 1, and that is why we need to explicitly convert the reference to an Option.

Finally, the repair function is defined using a pattern matching construct allowing us to concisely assign variables from the matching tuple.

Operationalization and Integration

The integration consists in making our solution work within the provided benchmark framework. First, next to constraint syntax, we also need to define its semantics. For that we define a validator which operationalizes the DSL executing the checks and consequent repairs of the incorrect model instances. It is defined as an abstract class with two methods that correspond to the two operations: The implementation is straight forward. For all elements contained in a container, we first collect all instances of the required context type and then query them using the query function provided by the given constraint. The repair simply executes the constraint repair function on the matching element. Finally, we instantiate all the constraints, plugs them into the validator and connects the result to the provided framework. The integration schema is shown in Appendix B. We also create a SigmaBenchmarkComparator that is used to compare the matches as required by the case study. It is a general comparator that either compares single instances (results from boolean constraints violations) or tuples (regular constraints violations).

Evaluation

In this section we provide an evaluation of our solution following the categories specified in the case study.

Correctness and Completeness of Model Queries and Transformations. We developed a solution for all of the tasks required by the case study and the solution passes the provided tests.

Conciseness. The solution itself consists of 52 lines of Scala code the internal DSL developed for this case study. The DSL itself has been implemented using 20 lines of Scala code using SIGMA. The integration part consists of three files with the total of 65 lines. All measures are source lines only excluding comments and new lines. Given these measures, we believe that the code is rather concise.

Readability. Next to being concise, the solution is also quite expressive. This means that the given problem (queries and repair transformations) naturally maps into the implementation. The higher-level abstraction provided by both SIGMA and the internal DSLs helps to facilitate it making a significant improvement over the Java reference implementation. The code is also type-safe as Scala is statically typed language. A notable consequence is that it is very easy to use the DSL with an IDE like Eclipse or IntelliJ that provides a robust code completing functionalities, outline views and other features increasing one's productivity.

In summary, while readability is a subjective matter and largely depends on the background and experience of users, we believe that SIGMA scores well. Thanks to the syntax of Scala which is close to one of Java/C++ and hence shall be familiar to many developers. The expressiveness of the first-order logic collection operation should be familiar to anyone knowing OCL or any other function language.

Performance on Large Models. The tests have been performed on an 2.3 GHz Intel Core i7 machine with 16 GB of RAM being dedicated to the JVM process. We ran our solution together with the reference implementation in Java. We used the model instances from size 1 to 8192 and set 8GB memory to be dedicated to the JVM. The performance is similar to the Java reference implementation which has been expected due to the fact that Scala compiles directly to Java bytecode and we use the same underlying libraries for accessing EMF models. This shows that we can leverage from concise and expressive queries without sacrificing performance. The overhead of using SIGMA is mostly on the compile time where the implicit conversions are inlined by the Scala compiler.

It is important to note that we do not developed any extra functionality for these benchmarksi.e. no caching or incremental validations. On the other hand, functional approach we have selected makes it perfect for further parallelization.

Conclusion

This paper presents a solution for the Train Benchmark case study of the 2015 Transformation Tool Contest. It demonstrates some of the features of the SIGMA internal DSLs for model manipulation as well as the extensibility, expressiveness and scalability of the Scala host language. The solution is realized as a tiny internal DSL in Scala that mixes in SIGMA common infrastructure for EMF model querying and manipulation. There is a significant improvement in the readability and conciseness of the solution, yet the performance is similar to the reference Java version.

Again based on the for comprehension. Additionally, we provide a shortcut using the route1.exit != null so immediately skip the route instances that do not have an exit semaphore set.

B Integration with the Train Benchmark Framework

C Performance Comparison

The performance comparison charts (cf. Figures 2,3, 4 and 5) have been generated by the case study benchmark. They present a performance comparison between SIGMA and the reference implementation in Java on the model instances from size 1 to 8192 using 8GB memory dedicated to the JVM. The corresponding results are shown in the figures 2 and 3. We compare them to the Java solution which is shown in the figures 4 and 5.

q q q q q q q q q q q q q q 221.09 Sigma, fixed, Function: read+check (Y: Log2) (X: Log2)

Figure 2: SIGMA fixed validation batch q q q q q q q q q q q q q q 8.38 Sigma, fixed, Function: repair+recheck (Y: Log2) (X: Log2)

Figure 3: SIGMA fixed revalidation q q q q q q q q q q q q q q 205.83 Java, fixed, Function: read+check (Y: Log2) (X: Log2)

Figure 4: Java fixed validation batch q q q q q q q q q q q q q q 5.14

  case class BooleanConstraint[A <: EObject : ClassTag]( query: (A) => Boolean, repair: (A) => Unit )

  Constraint[SwitchPosition, (Semaphore, Route, SwitchPosition, Switch)]( query = swP => { for { semaphore <-Option(swP.route.entry) if semaphore.signal == Signal.GO sw = swP.switch if sw.currentPosition != swP.position } yield (semaphore, swP.route, swP, sw) }, repair = { case (_, _, swP, sw) => sw.currentPosition = swP.position } )

  abstract class Validator[A <: EObject, B <: AnyRef] { def check(container: EObject): Iterator[B] def repair(matches: Iterator[B]): Unit }

  case class ConstraintValidator[A <: EObject, B <: AnyRef](constraint: Constraint[A, B]) extends Validator[A, B] { override def check(container: EObject) = container.eAllContents collect { case x: A => x } flatMap constraint.query override def repair(matches: Iterator[B]) = matches foreach constraint.repair }

Figure 1

 1 Figure1shows the various layers of integration of the solution into the train benchmark framework.

Figure 1 :

 1 Figure 1: Integration schema

Figure 5 :

 5 Figure 5: Java fixed revalidation

http://scala-lang.org

https://github.com/fikovnik/trainbenchmark-ttc in the hu.bme.mit.trainbenchmark. ttc.benchmark.sigma module

http://docs.scala-lang.org/tutorials/tour/sequence-comprehensions.html

http://www.datalyse.fr

Acknowledgments. This work is partially supported by the Datalyse project 4 .

A Constraints

In the following we describe the individual constraints that were part of the case study (case study tasks) except the PosLength and SwitchSet which have already been shown above (cf. Section 2).

-SwitchSensor

The isEmpty is a method that is defined on an Option type (coming from the standard Scala library) representing a type which may or may not have a value. Since in the model, the sensor reference of the Switch class is defined as optional (with cardinality 0..1), in SIGMA we represent the reference using the Option class. Not only makes this the cardinality expressed in the type definition, but it also prevents some of the NullPointerExceptions caused by traversing unset references. Technically, this is realized by implicit conversions (cf. Krikava et al. [START_REF] Krikava | SIGMA: Scala Internal Domain-Specific Languages for Model Manipulations[END_REF]).

-RouteSensor