Takegoshi [OT] in the case when Y is a hyperplane of a bounded pseudoconvex domain X in Cn, L0 is the trivial bundle and q = 0

VINCENT KOZIARZ

Abstract. We prove an extension theorem of “Ohsawa-Takegoshi type” for Dolbeaultq- classes of cohomology (q≥1) on smooth compact hypersurfaces in a weakly pseudoconvex Kähler manifold.

1. Introduction

Let Y be a complex submanifold of a Kähler manifold X and let L⁰ be a Hermitian line bundle onX. First consider the following

Problem. Let f be a smooth D⁰⁰-closed section of Λ^0,qT_X^? ⊗L⁰ over Y satisfying a suitable L² condition. Can we find a smoothD⁰⁰-closed extensionF of f toX together with a goodL² estimate for F on X?

The first result of this kind was obtained by T. Ohasawa and K. Takegoshi [OT] in the case when Y is a hyperplane of a bounded pseudoconvex domain X in Cⁿ, L⁰ is the trivial bundle and q = 0. It was further generalized by L. Manivel [Ma] (with a simplified proof by J.-P. Demailly [De3]) in the following setting: X is a weakly pseudoconvex manifold,Y is the zero set of a holomorphic section of a rank r Hermitian bundle over X,L⁰ =K_X ⊗L where L is a Hermitian line bundle whose curvature satisfies appropriate positivity properties, KX

is the canonical bundle of X, and q = 0. Whenq ≥1, the method leads to a new technical difficulty occurring in the regularity argument for(0, q)forms. In [De3], Demailly suggests an approach to overcome this difficulty but, to our knowledge, the complete arguments did not appear anywhere. In this paper, we rather consider the

Modified problem. Let q ≥1 and f be a smooth D⁰⁰-closed section of Λ^0,qT_X^? ⊗L⁰ over Y satisfying a suitable L² condition. Can we find a smooth D⁰⁰-closed extension F of f toX as a cohomology class (i.e.[F_|Y] = [f]∈H^q(Y, L⁰)) together with a goodL² estimate forF on X?

Observe that ifY is a Stein submanifold (this happens e.g. whenXis a Stein manifold), the modified problem is not relevant whenq ≥1since the Dolbeault groupH^q(Y, L⁰)vanishes. In contrast, we will focus here on the case whenY is a smooth compact hypersurface of a weakly pseudoconvex Kähler manifoldX. This situation naturally happens, for example, whenXis a compact Kähler manifold, or whenXis a holomorphic family of projective algebraic manifolds fibered over the unit disc.

Theorem 1.1. Let(X, ω)be a weakly pseudonconvexn-dimensional Kähler manifold, and let Y ⊂ X be the zero set of a holomorphic section s ∈ H⁰(X, E) of a Hermitian line bundle (E, h_E); the subvariety Y is assumed to be compact and nonsingular. Let L be a line bundle endowed with a smooth Hermitian metric hL such that

√−1Θ(L) +√

−1d⁰d⁰⁰log|s|²≥0, (1.1)

√−1Θ(L) +√

−1d⁰d⁰⁰log|s|²≥α⁻¹√

−1Θ(E) for someα ≥1, (1.2)

|s|²≤e^−α (1.3)

Date: May 25, 2010.

1

on X. Let0< κ≤1 and letΩ⊂X be a relatively compact open subset containingY. Then, for any q≥0 and every smooth D⁰⁰-closed (0, q)-form f with values inKX ⊗L over Y, there exists a smooth extensionF off toΩas a cohomology class (i.e. [F_|Y] = [f]∈H^q(Y, K_X⊗L)) such that

Z

Ω

|F|²

|s|^2(1−κ)dVω ≤ C κ

Z

Y

|f|²

|ds|²dVY,ω

whereC is a numerical constant depending only on Ω,E, L and q.

The norm of the forms with values in bundles will always be computed with respect to the one induced byω, h_E and h_L. Also, Θ(E) (resp. Θ(L)) will always denote the curvature of the Hermitian line bundle(E, h_E) (resp. (L, h_L)). When the metrics will be twisted by some positive functions, the weights will appear explicitely in the formulae.

Our proof follows many of the ideas outlined in [De3]. First, using the weight bumping technique (and the adapted Bochner-Kodaira-Nakano inequality) initiated by Ohsawa and Takegoshi, for allε >0, we build extensions off of classC¹whoseL² norm is controlled, and which are “approximately” D⁰⁰-closed (in the sense that the L² norm of their D⁰⁰-derivative is bounded by a constant times ε). Philosophically, passing to the limit as ε → 0 should provide the desired extension but the limiting elliptic differential system is singular along Y and this forbids the direct use of elliptic regularity arguments. Then, at this point, our strategy differs from Demailly’s. Instead, we construct “approximate” q-cocycles ζ_ε in Čech cohomology corresponding to the previous extensions via an effective Leray’s isomorphism, in a similar fashion as Y.-T. Siu in [Si]. During the process, we solve local D⁰⁰-equations by standard techniques of L. Hörmander. Then, we can take the limit as ε → 0 and use the ellipticity of the Laplacian in bidigree(0,0)to ensure the smoothness of the extending cocycle ζ. Finally, reversing the process, we get a smooth extension F of f as a cohomology class.

Notice that the constant C in Theorem 1.1 is mainly related to a finite covering of Ω ⊃ Y by Stein open subsets (which is used to apply Leray’s isomorphism) and the norm of the derivatives of a partition of unity subordinate to this finite covering. This explains in part why we needY to be compact.

A consequence of Theorem 1.1 is a qualitative surjectivity theorem for restriction morphisms in Dolbeault cohomology:

Corollary 1.2. LetX,Y,E andLbe as in Theorem 1.1 i.e. satisfying(1.1),(1.2)and(1.3).

Then the restriction morphism

H^q(X, KX ⊗L)−→H^q(Y,(KX ⊗L)|Y) is surjective for any q≥0.

Applying Theorem 1.1 toE =Cand to any semi-positive line bundleL(for instanceL=C), we also easily get the following corollary which contains a special case of the invariance of the Hodge numbers for a family of compact Kähler manifolds (a result due to K. Kodaira and D. Spencer):

Corollary 1.3. Letπ :X→∆be a proper holomorphic submersion over the unit disc andLa semi-positive line bundle onX. Assume thatXis a Kähler manifold of dimensionn+1. Then, for any q≥0,h^n,q(X_t, L) := dimH^n,q(X_t, L) is independent of t∈∆(where X_t=π⁻¹(t)).

Acknowledgments. I would like to thank Mihai Păun for many valuable discussions. I would also like to thank Jean-Pierre Demailly for explaining me details on his article [De3], as well as Benoît Claudon and Dror Varolin for useful comments on an earlier version of this paper.

2. Preliminary material

From now on, we assume thatX,Y,L andE satisfy the hypotheses of Theorem 1.1.

Let c ∈ R such that Ω ⊂ Xc := {x ∈ X , ψ(x) < c}, where ψ is the plurisubharmonic exhaustion of X. Let U = {U_j}_j∈J be a finite covering of the closure of Ω by coordinate chartsφj :B −→Uj whereB is the unit ball inCⁿ, and such thatUj ⊂Xcfor allj. Denoting by ν the standard Hermitian norm onCⁿ, we assume that the functionsϕ_j :=ν◦φ⁻¹_j satisfy

(2.1) √

−1Θ(L)−β√

−1Θ(E) +√

−1d⁰d⁰⁰ϕ_j ≥ω

on Uj for any β ∈[0,1](this is always possible if theUj’s are chosen small enough). For any multi-index (j0, . . . , j`), we shall denote byϕj0,...,j` the function P`

i=0ϕji which is defined on the intersection U_{j0,...,j`</sub> :=U<sub>j0</sub>∩ · · · ∩U<sub>j`}.

IfF is a smooth Hermitian vector bundle over XandU ⊂X is an open subset then, for all integer k, we denote by E^k(U, F) the space of sections ofF over U which are of classC^k and by E_c^k(U, F) those with compact support. We also denote by W^k(U, F) the Sobolev space of sections whose derivatives (in the sense of distribution theory) up to order kare inL².

Let us recall three useful results taken from [De1] (Remark 1.6, Lemma 3.3 and Lemma 6.9):

Proposition 2.1. (a) Xc\Y is complete Kähler.

(b) Let ω and ω⁰ be two Hermitian forms on T_X such that ω ≤ ω⁰. Let E be a Hermitian vector bundle on X. Then, for anyq ≥0 and anyu∈Λ^n,qT_X^? ⊗E,|u|²_ω0dV_ω⁰ ≤ |u|²_ωdVω. (c) Let Ω be an open subset of Cⁿ and Y a complex analytic subset of Ω. Assume that v is a (p, q−1)-form with L²_loc coefficients and w a (p, q)-form with L¹_loc coefficients such that d⁰⁰v=w on Ω\Y (in the sense of distribution theory). Then d⁰⁰v=w on Ω.

The following lemma is a consequence of a classical result (see [De2], Corollary 5.3):

Lemma 2.2. Let m and p be positive integers.

(a) Let v∈W^m(U_j0,...,j`,Λ^n,pT_X^? ⊗L⊗E⁻¹) such that D⁰⁰v= 0 and Z

Uj0,...,j`

|v|²e^{−ϕj0,...,j`}dVω <+∞.

Then there exists a (n, p−1) form u ∈ W^m+1(Uj0,...,j_`,Λ^n,p−1T_X^? ⊗L⊗E⁻¹) such that D⁰⁰u=v and

Z

Uj0,...,j`

|u|²e^{−ϕj0,...,j`}dV_ω ≤ 1 p

Z

Uj0,...,j`

|v|²e^{−ϕj0,...,j`} dV_ω.

(b) Let 0< κ≤1 and ε >0. Let v∈W^m(Uj0,...,j_`,Λ^n,pT_X^? ⊗L) such that D⁰⁰v= 0 and Z

Uj0,...,j`

|v|²

(|s|²+ε²)^1−κe^{−ϕj0,...,j`}dV_ω <+∞.

Then there exists a (n, p−1) form u∈W^m+1(U_j0,...,j`,Λ^n,p−1T_X^? ⊗L) such that D⁰⁰u=v and

Z

Uj0,...,j`

|u|²

(|s|²+ε²)^1−κe^{−ϕj0,...,j`} dVω≤ 1 p

Z

Uj0,...,j`

|v|²

(|s|²+ε²)^1−κe^{−ϕj0,...,j`}dVω.

Proof. We only check the hypotheses of Corollary 5.3 in [De2]. The open subsetUj0,...,j` ⊂X is Stein and onUj0,...,j`, the line bundleL⊗E⁻¹, resp. L, endowed with its metric twisted by e^{−ϕj0,...,j`</sup>, resp. (|s|<sup>2</sup>+ε<sup>2</sup>)<sup>−(1−κ)</sup>e<sup>−ϕj0,...,j`}, has curvature

√−1Θ(L)−√

−1Θ(E) +√

−1d⁰d⁰⁰ϕ_j0,...,j`,

resp. √

−1Θ(L) + (1−κ)√

−1d⁰d⁰⁰log(|s|²+ε²) +√

−1d⁰d⁰⁰ϕj0,...,j`, which, by inequality (3.2) and assumption (2.1), is bounded from below by

√−1Θ(L)−√

−1Θ(E) +√

−1d⁰d⁰⁰ϕj0 ≥ω, resp.

√−1Θ(L)−(1−κ)h√

−1Θ(E)s, si

|s|²+ε² +√

−1d⁰d⁰⁰ϕj0 ≥ω.

The fact thatu can be chosen in the Sobolev space W^m+1 comes from the ellipticity of the Laplacian. Let us explain why in the case(a), the case (b) being completely similar. In fact, theD⁰⁰-equation is solved using complete metricsω_ε onU_j0,...,j`, such thatω_ε≥ωandω_ε →ω as ε → 0. For any ε > 0, the corresponding minimal solution u_ε (i.e. the one satisfying D⁰⁰u_ε=v and u_ε∈(KerD⁰⁰)^⊥ωε) is such that

Z

Uj0,...,j`

|u_ε|²_ωεe^{−ϕj0,...,j`}dV_ωε ≤ 1 p

Z

Uj0,...,j`

|v|²_ωεe^{−ϕj0,...,j`}dV_ωε ≤ 1 p

Z

Uj0,...,j`

|v|²e^{−ϕj0,...,j`} dV_ω where the latter inequality comes from Proposition 2.1(b). Then, there exists a sequence (ε_µ) converging to 0 such that u_εµ converges weakly to some u in L²_loc as µ → +∞: u satisfies D⁰⁰u=v,

Z

Uj0,...,j`

|u|²e^{−ϕj0,...,j`} dV_ω≤ 1 p

Z

Uj0,...,j`

|v|²e^{−ϕj0,...,j`}dV_ω

but also u∈(KerD⁰⁰)^⊥ω = Im (D⁰⁰)^?ω (and therefore(D⁰⁰)^?ωu= 0). Indeed, L²(ω) ⊂L²(ω_ε) because |.|²_ω

εdV_ωε ≤ |.|²dV_ω and since ω_ε → ω, we get u ∈ (KerD⁰⁰)^⊥ω by dominated convergence theorem. Finally, u satisfies D⁰⁰u = v and (D⁰⁰)^?u = 0 which is an elliptic differential system and standard arguments giveu∈W^m+1 if v∈W^m.

Notice that we can skip the extraction of a weak limit if we only need a solution with the same estimate on a slightly smaller relatively compact open subset ofUj0,...,j_{`</sub>: all we have to do is take a complete metric onU<sub>j0,...,j`} which coincides with ω on the smaller subset.

Finally, we also select a smooth partition of unity {σ_j}j∈J subordinate to U (i.e. for each j,σj ∈ E_c^∞(Uj,[0,1]) andP

j∈Jσj(x) = 1 for anyx∈Ω).

3. Proof of the theorem

Recall that, by assumption,Y ⊂X is a smooth hypersurface so thatE ' O_X(Y).

3.1. Construction of smooth extensions. In this section, we prove the following Lemma 3.1. For any k≥0, there exists a smooth section

fe∞∈ E^∞(X,Λ^n,qT_X^? ⊗L) such that

(a) fe∞ coincides with f in restriction to Y, (b) |fe∞|=|f|at every point of Y,

(c) D⁰⁰fe∞= 0 at every point of Y,

(d) s⁻¹D⁰⁰fe∞∈ E^k(X,Λ^n,q+1T_X^? ⊗L⊗ O_X(−Y)).

Proof . Let us cover Y by coordinate patches W_j ⊂ X biholomorphic to polydiscs and with the following property: if we denote the corresponding coordinates by (z_j, w_j) ∈ ∆×∆ⁿ⁻¹, wherew_j = (w_j¹, . . . , wⁿ⁻¹_j ), then W_j∩Y ={z_j = 0}. On eachW_j, we fix some holomorphic σj ∈Γ(Wj, K_X ⊗L) which trivialize K_X ⊗L.

As explained in [De3], the restriction map (Λ^0,qT_X^?)_|Y −→ Λ^0,qT_Y^? can be viewed as an orthogonal projection onto aC^∞ subbundle of(Λ^0,qT_X^?)_|Y. One might extend this subbundle fromW_j∩Y toW_j and then extendf onW_j by some smooth formfb_j ∈ E^∞(W_j,Λ^n,qT_X^? ⊗L).

Using a smooth partition of unity θj ∈ E_c^∞(Wj,R), P

jθj = 1 on a neighbourhood of Y, we get a global smooth extensionfb=P

jθjfbj off which fulfills conditions (a) and (b). Since (D⁰⁰f)b_|Y = (D⁰⁰fb_|Y) =D⁰⁰f = 0,

we can write D⁰⁰fb= d¯zj ∧gj on Wj ∩Y for some smooth (0, q)-forms gj which we extend arbitrarily toWj. Then

fe∞:=fb−X

j

θjz¯jgj

coincides with fbon Y and satisfies (c).

We proceed by induction to get(d). Assume that on each W_j, D⁰⁰fe∞=z_jf_j(z_j, w_j) + ¯z_j^kh

d¯z_j ∧ X

|I|=q

a_I(w_j)σ_jdw¯_j^I+ X

|I⁰|=q+1

b_I⁰(w_j)σ_jdw¯^I_j⁰i

+ ¯z_j^k+1h_j(z_j, w_j) for some fj, hj ∈ E^∞(Wj,Λ^n,q+1T_X^? ⊗L),aI, b_I⁰ ∈ E^∞(∆ⁿ⁻¹,C),k≥1, and where the multi- indices I, I⁰ are increasing. We say that fe∞ enjoys property (P_k). Remark that such an equality implies that s⁻¹D⁰⁰fe∞∈ E^k−2(X,Λ^n,q+1T_X^? ⊗L⊗ O_X(−Y))if k≥2. Moreover, the extensionfe∞ we just constructed satisfies property(P1) becauseD⁰⁰fe∞= 0 alongY.

Of course, D⁰⁰(D⁰⁰fe∞) = 0, but also the direct computation gives D⁰⁰(D⁰⁰fe∞) =zjD⁰⁰fj(zj, wj) +k¯z_j^k−1d¯zj∧h X

|I⁰|=q+1

b_I⁰(wj)σjdw¯_j^I0 i

+ ¯z_j^kh⁰_j(zj, wj) for some h⁰_j ∈ E^∞(W_j,Λ^n,q+2T_X^? ⊗L), hence theb_I⁰’s must vanish identically. So if we take

fe_∞⁰ =fe∞−X

j

θj

¯ z_j^k+1 k+ 1

X

|I|=q

aI(wj)σjdw¯_j^I, we have

D⁰⁰fe_∞⁰ = D⁰⁰fe∞−X

j

z¯_j^k+1

k+ 1d⁰⁰θ_j+θ_jz¯_j^kd¯z_j

∧ X

|I|=q

a_I(w_j)σ_jdw¯^I_j

−X

j

θ_j z¯_j^k+1 k+ 1D⁰⁰

X

|I|=q

a_I(w_j)σ_jdw¯^I_j

= X

j

θ_j

D⁰⁰fe∞−z¯^k_jd¯z_j∧ X

|I|=q

a_I(w_j)σ_jdw¯^I_j

+X

j

¯

z^k+1_j h⁰⁰_j(z_j, w_j)

= X

j

zjθjfj(zj, wj) +X

j

¯

z_j^k+1 θjhj(zj, wj) +h⁰⁰_j(zj, wj) for some h⁰⁰_j ∈ E_c^∞(W_j,Λ^n,q+1T_X^? ⊗L). Then, fe_∞⁰ enjoys property(P_k+1).

3.2. Construction of approximate extensions with control. Let θ : R −→ [0,1] be a smooth function with support in (−∞,1), such that θ ≡ 1 on (−∞,1/2] and |θ⁰| ≤ 4, and consider the truncated extension off

feε:=θ(ε⁻²|s|²)fe∞

wherefe∞ is the extension provided by Lemma 3.1, such thats⁻¹D⁰⁰fe∞∈ E^k(X,Λ^n,q+1T_X^? ⊗ L⊗ O_X(−Y)) for some k ≥ 1 which will be determined later. We wish to solve on X the equation

D⁰⁰u_ε=D⁰⁰fe_ε

with estimate, and the additional constraint thatu_ε vanishes along Y. We also expect some regularity on uε in order to justify that feε−uε is a (D⁰⁰-closed) extension of f. In general, we are not able to get this by the method we use here, and we can only produce approximate solutions.

The fundamental tool is the following existence result (see [De3], [Pa]):

Theorem 3.2. Let X be a complete Kähler manifold of dimension n equipped with a (not necessarily complete) Kähler metric ω, and let L be a line bundle endowed with a smooth Hermitian metric. Assume that there exist two smooth bounded functions η, λ > 0 on X satisfying

(3.1) η√

−1Θ(L)−√

−1d⁰d⁰⁰η−√

−1d⁰η∧d⁰⁰η

λ ≥√

−1τ d⁰µ∧d⁰⁰µ

for some positive function τ and some function µ. Let us consider the (densely defined) modified D⁰⁰ operators

T u:=D⁰⁰(p

η+λu) and Su:=√

η(D⁰⁰u)

acting on forms with values inL. Let g=d⁰⁰µ∧g₀+g₂ be aL² form of (n, q+ 1)type (q≥0) with values inL such that

(a) D⁰⁰g= 0,

(b) g₀ ∈L²(X,Λ^n,qT_X^? ⊗L), (c) C(g₀, τ) :=R

X1/τ|g₀|²dV_ω <+∞,

(d) |g₂|² ≤γ C(g₀, τ) almost everywhere for some positive constantγ. Then, for anyu∈DomT^?∩DomS, we have

Z

X

hg, uidV_ω

2 ≤C(g0, τ) kT^?uk²+kSuk²+γkuk² .

In particular, there existv∈L²(X,Λ^n,qT_X^? ⊗L) and w∈L²(X,Λ^n,q+1T_X^? ⊗L) such that T v+γ^1/2w=g

together with the estimate Z

X

|v|²dVω+ Z

X

|w|²dVω ≤C(g0, τ).

As before, let c ∈ R be such that Ω ⊂ Xc. For simplicity, we will assume in the sequel that X = X_c. We are going to apply Theorem 3.2 to D⁰⁰fe_ε on X\Y. By Proposition 2.1 (a),X\Y can be equipped with a complete Kähler metric. As for the bundleL, we endow it with its original metric multiplied with the weight|s|⁻² in order to force the vanishing of the approximate solution alongY.

For any ε > 0, we set σ_ε := −log(ε² +|s|²). Remark that, because of the condition

|s| ≤ e^−α, this function is positive for ε small enough. Let χ : R+ −→ R+ be any strictly concave function whose derivative satisfies1≤χ⁰≤2and such thatχ(−log(ε²+e^−2α))≥2α for any ε >0 small enough (in [De3] and [Pa], χ(t) =t+ log(1 +t) but we will choose other functions picked in [MV]). Let us define the two positive functions (againε is assumed to be small enough)

η_ε :=χ(σ_ε) and λ_ε:=−χ⁰(σ_ε)² χ⁰⁰(σε).

Although this is done carefully in [De3] and [Pa], we check quickly that η_ε and λ_ε fulfill condition (3.1)in Theorem 3.2. It is easy to see that

(3.2) −√

−1d⁰d⁰⁰σ_ε≥√

−1 ε²

|s|²d⁰σ_ε∧d⁰⁰σ_ε−h√

−1Θ(E)s, si ε²+|s|² and it is straightforward that

d⁰ηε=χ⁰(σε)d⁰σε , d⁰⁰ηε=χ⁰(σε)d⁰⁰σε , d⁰d⁰⁰ηε =χ⁰(σε)d⁰d⁰⁰σε+χ⁰⁰(σε)d⁰σε∧d⁰⁰σε. Thus, since χ⁰ is positive,

−√

−1d⁰d⁰⁰η_ε≥ 1 χ⁰(σε)

ε²

|s|² + 1 λε

√−1d⁰η_ε∧d⁰⁰η_ε− χ⁰(σ_ε) ε²+|s|²h√

−1Θ(E)s, si.

If ε is small enough, then for any x ∈ X, η_ε(x) ≥ χ(−log(ε²+e^−2α)) ≥ 2α. Taking into account the curvature assumptions (1.1) and (1.2) in Theorem 1.1 as well as the fact that χ⁰ ≤2, we obtain

ηε(√

−1Θ(L) +√

−1d⁰d⁰⁰log|s|²)≥ ηε

α

√−1Θ(E)≥ χ⁰(σε) ε²+|s|²h√

−1Θ(E)s, si.

Finally, summing up the two latter inequalities, we get ηε(√

−1Θ(L) +√

−1d⁰d⁰⁰log|s|²)−√

−1d⁰d⁰⁰ηε−

√−1

λ_ε d⁰ηε∧d⁰⁰ηε ≥ 1 χ⁰(σ_ε)

ε²

|s|²

√−1d⁰ηε∧d⁰⁰ηε

which proves that(3.1)is fulfilled with τ = 1 χ⁰(σ_ε)

ε²

|s|² andµ=ηε. Now, we can write D⁰⁰fe_ε=d⁰⁰η_ε∧g_ε+θ|s|²

ε²

D⁰⁰fe∞

where

gε:=

1 +|s|² ε²

θ⁰

|s|² ε²

fe∞

χ⁰(σε). A quick computation shows that

(3.3) lim

ε→0

1 ε²

Z

X\Y

θ⁰ |s|²

ε² 2

|fe∞|²dVω =c0

Z

Y

|f|²

|ds|²dVY,ω

for some “universal” constant c₀. Therefore, since θ(ε⁻²|s|²) is supported in {|s| < ε}, and since D⁰⁰fe∞= 0 onY, for any γ >0,

θ

|s|² ε²

D⁰⁰fe∞

≤γ Z

X\Y

1 τ

|g_ε|²

|s|² dVω = γ ε²

Z

X\Y

1 +|s|² ε²

2

θ⁰ |s|²

ε² 2

|fe∞|²dVω

if ε >0is small enough.

Hence, we can apply Theorem 3.2: we findu_ε,γ =√

η_ε+λ_εv_ε,γ and w_ε,γ which satisfy the equation

D⁰⁰u_ε,γ+γ^1/2w_ε,γ =D⁰⁰fe_ε onX\Y and such that

Z

X\Y

|u_ε,γ|²

|s|²(ηε+λε)dVω+ Z

X\Y

|w_ε,γ|²

|s|² dVω ≤ 1 ε²

Z

X\Y

1 +|s|² ε²

2

θ⁰ |s|²

ε²

2 |fe∞|² χ⁰(σε)dVω

≤ 4 ε²

Z

X\Y

θ⁰ |s|²

ε² 2

|fe∞|²dVω. (3.4)

3.3. Regularization of the approximate solution. Recall that Y is a divisor such that E ' O_X(Y). Here we use a trick of Demailly: we consider s⁻¹u_ε,γ (resp. s⁻¹w_ε,γ) as a L² (0, q)-form (resp. (0, q+ 1)-form) with values in the twisted line bundleK_X ⊗L⊗ O_X(−Y) equipped with a smooth Hermitian metric. By Proposition 2.1(c), we can write

D⁰⁰(s⁻¹u_ε,γ) +γ^1/2s⁻¹w_ε,γ =s⁻¹D⁰⁰fe_ε

not only onX\Y but also on X becauses⁻¹uε,γ is locally L²,s⁻¹wε,γ isL² hence locally L¹, ands⁻¹D⁰⁰fe_ε is of classC^k hence locally L¹ (recall thatfe∞, as chosen in Lemma 3.1, is such thats⁻¹D⁰⁰fe∞ is of classC^k,k≥1). However, we do not know much about the regularity of u_ε,γ andw_ε,γ.

But E_c^∞(X,Λ^n,qT_X^? ⊗L⊗ O_X(−Y)) is dense in DomD⁰⁰ for the graph norm, where we consider D⁰⁰ as an operator acting on (n, q) forms on X with values in L⊗ O_X(−Y). More precisely, the density holds when X is endowed with a complete metric. If X = Xc as we assumed above, we can work instead on X_c⁰ for some c⁰ > c, and there exists on X_c⁰ some complete Kähler metric which coincides withω onX_c.

Then, we can find sometε,γ ∈ E^∞(X,Λ^n,qT_X^? ⊗L⊗ O_X(−Y)), which is L², such that (3.5)

Z

X

|t_ε,γ|² ηε+λε

dVω− Z

X

|s⁻¹uε,γ|² ηε+λε

dVω

≤ε

(recall that ηε is bounded by 2α from below), andD⁰⁰(tε,γ −s⁻¹uε,γ) has L² norm bounded byγ^1/2 from above. As a consequence,

D⁰⁰tε,γ =s⁻¹D⁰⁰feε+rε,γ

on X, with rε,γ ∈ E^k(X,Λ^n,q+1T_X^? ⊗L⊗E⁻¹) since D⁰⁰tε,γ and s⁻¹D⁰⁰feε are of class C^k. Moreover,r_ε,γ satisfies

(3.6)

Z

X

|r_ε,γ|²≤C₁²γ

for some positive constantC₁ depending onf, but not onεandγ(see (3.4) and (3.3)). Finally, s⁻¹D⁰⁰fe_ε isD⁰⁰-closed onX\Y, hence on X by Proposition 2.1 (c), and therefore D⁰⁰r_ε,γ = 0.

3.4. The choice of η_ε and λ_ε. Let us come now to the choice of η_ε and λ_ε (see [MV] for more details). For any0< κ≤1, we define for t≥0 the functions

gκ(t) =κ⁻¹e^κt , hκ(t) = Z t

0

1

2e^κy−1dy and χκ(t) = 1 +t+hκ(t).

One checks immediatly that1≤χ⁰_κ ≤2 andχ⁰⁰_κ<0. Moreover,

χκ(−log(ε²+e^−2α))≥1−log(ε²+e^−2α)≥1 + 2α−log(1 +ε²e^2α)≥2α

when εis small enough. Clearly,χ_κ(t)≤1 + 2tand it follows that χκ(t)

gκ(t) ≤ κ(1 + 2t) e^κt ≤2 as is seen from a simple computation. Moreover,

−χ⁰_κ(t)²

χ⁰⁰_κ(t) ≤2gκ(t).

As a consequence, if we fix κ≤1 and take χ=χ_κ, we have

(3.7) ηε+λε≤ 4

κ(|s|²+ε²)^κ.

3.5. Construction of q-cochains via Leray’s isomorphism. Recall that we have fixed a finite open coveringU ={U_j}_j∈J ofΩ. We endow the groupC₂<sup>`</sup>(U,E<sup>1</sup>(Λ<sup>n,p</sup>T<sub>X</sub><sup>?</sup> ⊗L⊗E<sup>−1</sup>))of (alternate) `-cochains with values in E¹(Λ^n,pT_X^? ⊗L⊗E⁻¹)which are L² with the norm

kς^`k² = max

j0<···<j_`

Z

Uj0,...,j`

|ς_j^`

0,...,j`|<sup>2</sup>e<sup>−ϕj0,...,j`</sup>dVω

and for all0< κ≤1and ε >0, we endow C₂^{`</sup>(U,E<sup>1</sup>(Λ<sup>n,p</sup>T<sub>X</sub><sup>?</sup> ⊗L))with the norm kς<sup>`}k²_κ,ε= max

j0<···<j_`

Z

Uj0,...,j`

|ς_j^`

0,...,j`|²

(|s|²+ε²)^1−κe^{−ϕj0,...,j`} dVω.

Remark that in the case when ς⁰ is the0-cocycle associated to a sectionς ofE¹(Λ^n,pT_X^? ⊗L⊗ E⁻¹))(resp. E¹(Λ^n,pT_X^? ⊗L))),

kς⁰k²≤ Z

X

|ς|²dV_ω

resp. kς⁰k²_κ,ε≤ Z

X

|ς|²

(|s|²+ε²)^1−κdV_ω since the ϕj’s are nonnegative.

Now, we construct a (q+ 1)-cocycle in Z^q+1(U,O(K_X ⊗L⊗E⁻¹))corresponding to rε,γ

via Leray’s isomorphism between the Dolbeault and the Čech cohomology groups. In fact, we are mostly interested in the intermediate cochains which appear during the process and the control we have on their norm. The extension fe∞ of f is supposed to be sufficiently regular (i.e. k is large enough in Proposition 3.1) in order that r_ε,γ, and every cochain obtained by solving local D⁰⁰-equations below, is at least of class C¹ (see section 3.3, Lemma 2.2 and use Sobolev lemma: W^m(U)⊂ E¹(U) for any open subset U ⊂X if m >1 +ⁿ₂).

For notational simplicity, we denoter^`_ε,γ,(j

0,...,j`)∈Γ(Uj0,...,j<sub>`</sub>,E¹(Λ^{n,q−`</sup>T<sub>X</sub><sup>?</sup> ⊗L⊗E<sup>−1</sup>)) by r<sup>`}_j

0,...,j`.

First, we solve the equation D⁰⁰r⁰_j =r_ε,γ on the U_j’s, then we solve the equations D⁰⁰r_j^`+1_0,...,j

`+1 = (δr<sup>`</sup>)j0,...,j`+1 on Uj0 ∩ · · · ∩Uj`+1 (0≤`≤q−1)

using each time Lemma 2.2(a). Finally,δrε,γ^q ∈Z^q+1(U,O(K_X⊗L⊗E⁻¹))is a representative in Čech cohomology of [r_ε,γ]∈H^q+1(X, K_X ⊗L⊗E⁻¹).

Lemma 3.3. For any`, we have kr<sub>ε,γ</sub><sup>`</sup> k²≤ (`+ 1). . .2.1

(q+ 1).q . . .(q−`+ 1)kr<sub>ε,γ</sub>k<sup>2</sup>≤ (`+ 1). . .2.1 (q+ 1).q . . .(q−`+ 1)

Z

X

|r_ε,γ|²dV_ω.

Proof. For any`≥1,

kr^`_ε,γk² ≤ 1

q−`+ 1kδr<sup>`−1</sup>_ε,γ k²

≤ `+ 1

q−`+ 1kr<sup>`−1</sup>_ε,γ k² and

kr⁰_ε,γk²≤ 1

q+ 1kr_ε,γk² ≤ 1 q+ 1

Z

X

|r_ε,γ|²dV_ω by the estimate in Lemma 2.2(a).

In a similar manner, we produce aq-cochainζε,γ ∈C^q(U,O(K_X⊗L))corresponding to the

“approximately” D⁰⁰-closed extensionfe_ε−st_ε,γ ∈Γ(X,E¹(Λ^n,qT_X^? ⊗L))of f. More precisely, on theUj’s, we solve the equationD⁰⁰h⁰_j =feε−stε,γ+sr⁰_ε,γ,j, then we solve

D⁰⁰h^`+1_j

0,...,j`+1 = (δh<sup>`</sup>)<sub>j0,...,j`+1</sub>+ (−1)<sup>`+1</sup>sr_j^`+1

0,...,j`+1 on U<sub>j0</sub>∩ · · · ∩U<sub>j`+1</sub> (0≤`≤q−2) using Lemma 2.2(b). This is indeed possible since the right-hand side is D⁰⁰-closed: if

D⁰⁰h<sup>`</sup><sub>j0,...,j`</sub> = (δh<sup>`−1</sup>)<sub>j0,...,j`</sub>+ (−1)^{`</sup>sr<sub>j</sub><sup>`}_0,...,j` then

D⁰⁰(δh<sup>`</sup>)j0,...,j<sub>`+1</sub> = (δ(D⁰⁰h<sup>`</sup>))j0,...,j<sub>`+1</sub> = (δ²h<sup>`−1</sup>)j0,...,j<sub>`+1</sub>+ (−1)^{`</sup>s(δr<sup>`})j0,...,j_`+1

= 0 + (−1)^{`</sup>sD<sup>00</sup>r<sup>`+1}_j

0,...,j`+1.

Finally, letζ_ε,γ :=δh^q−1ε,γ + (−1)^qsrε,γ^q ∈C^q(U,O(K_X⊗L))(in particularD⁰⁰ζ_ε,γ = 0and ζ_ε,γ is actually smooth by ellipticity ofD⁰⁰ in bidegree(0,0)).

In the next proposition, we denote byV the finite Stein covering {V_j}_j∈J ={U_j∩Y}_j∈J of Y.

Proposition 3.4. Let 0< κ≤1. The cochainζ_ε,γ enjoys the following properties:

(a)

kζ_ε,γk²_κ,ε≤ 16(q+ 1)c0

κ Z

Y

|f|²

|ds|²dVY,ω+β(ε, γ)

where β is a positive function such that β(ε, γ)→0 asε, γ →0(recall that for any γ >0, ζε,γ only exists ifε >0 is small enough).

(b) For anyε, γ,ζε,γ|Y ∈Z^q(V,O((K_X⊗L)_|Y))is a representative in Čech cohomology of the cohomology class [f]∈H^q(Y,(K_X ⊗L)_|Y).

Proof. Letχ=χκ be as in section 3.4. We use the corresponding ηε and λε. (a) We have

kζ_ε,γk_κ,ε ≤ kδh^q−1_ε,γ k_κ,ε+ksr^q_ε,γk_κ,ε

≤ p

q+ 1kh^q−1_ε,γ k_κ,ε+e^−ακ2 kr_ε,γk

by Lemma 3.3, since|s|²(|s|²+ε²)^−(1−κ)≤ |s|^κ ≤e^−ακ2 according to assumption(1.3). In the same way, for any `≥1,

kh^`_ε,γk_κ,ε ≤ 1

√q−`(kδh<sup>`−1</sup>_ε,γ k_κ,ε+ksr_ε,γ^` k_κ,ε)

≤

√`+ 1

√q−`kh<sup>`−1</sup>_ε,γ k_κ,ε+

p(`+ 1). . .2.1

p(q+ 1).q . . .(q−`)e^−ακ2 kr_ε,γk

where we also used the estimate in Lemma 2.2(b). Finally, kh⁰_ε,γk_κ,ε ≤ 1

√q(kfe_ε−st_ε,γk_κ,ε+ksr_ε,γ⁰ k_κ,ε)

≤ 1

√qkfe_ε−st_ε,γk_κ,ε+ 1

p(q+ 1).qe^−ακ2 kr_ε,γk.

Collecting all these inequalities, we get kζ_ε,γk_κ,ε≤p

q+ 1kfeε−stε,γk_κ,ε+qe^−ακ2 kr_ε,γk ≤p

q+ 1kfeε−stε,γk_κ,ε+q C1γ^1/2. Now, it is easy to see that

Z

X

|feε−stε,γ|²

(|s|²+ε²)^1−κdVω ≤ Z

X

|st_ε,γ|²

(|s|²+ε²)^1−κdVω+C2ε^2κ

for some constant C₂, as fe_ε is uniformly bounded with support in {|s|< ε}. Finally, by (3.7),

Z

X

|st_ε,γ|²

(|s|²+ε²)^1−κ dVω≤ Z

X

(|s|²+ε²)^κ|t_ε,γ|²dVω≤ 4 κ

Z

X

|t_ε,γ|² η_ε+λ_εdVω

and the desired inequality follows from (3.4), (3.5) and (3.3).

(b) It is clear since onUj0,...,j`+1∩Y, the restriction of(δh<sup>`</sup>)j0,...,j` is alwaysD⁰⁰-closed, hence we construct an “exact” representative in restriction toY.

3.6. Passing to the limit and reversing the process to get the extension. Now, we just have to make γ and ε go to zero and extract a weak limit of ζε,γ. This weak limit ζ is an element of Z^q(U,O(K_X ⊗L)) since D⁰⁰ζ_ε,γ = 0for any ε, γ, and δζ_ε,γ = (−1)^qδ(srε,γ^q ) which, by(3.6)and Proposition 3.3, hasL² norm bounded by some constant times γ^1/2, thus δζ = 0. Moreover,ζ|Y ∈Z^q(V,O((K_X⊗L)|Y))is a representative in Čech cohomology of the cohomology class[f]∈H^q(Y,(KX⊗L)_|Y)since this is the case of anyζε,γ|Y by Proposition 3.4 (b). For all j₀, . . . , j_q, as theϕ_j’s are bounded from above by 1, we obtain

Z

Uj0,...,jq

|ζ|²

|s|^2(1−κ)dVω≤ 16e^q+1(q+ 1)c0

κ

Z

Y

|f|²

|ds|²dVY,ω

if we take the limit in the inequality of Proposition 3.4 (a).

Finally, we construct the desired extension F in the following way. For 0≤`≤q−1, we produce ξ<sup>`</sup> ∈C^{`</sup>(U,E<sup>∞</sup>(Λ<sup>n,q−`−1}T_X^? ⊗L))such that

(i) (δξ^q−1)_j0,...,jq =ζ_j0,...,jq on U_j0 ∩ · · · ∩U_jq, (ii) (δξ<sup>`</sup>)j0,...,j`+1=D⁰⁰ξ_j^`+1_0,...,j

`+1 on Uj0 ∩ · · · ∩Uj`+1 (0≤`≤q−2).

Theseδ-equations are solved by using the partition of unity {σ_j}_j∈J subordinate toU in the following way:

ξ_j^q−1

0,...,jq−1 = P

iσiζi,j0,...,jq−1, ξ_j^`

0,...,j` = P

iσ_iD⁰⁰ξ_i,j^`+1

0,...,j<sub>`</sub> (0≤`≤q−2).

Finally, we set

F =D⁰⁰ξ_j⁰ =q! X

j1<···<jq ji6=j

ζ_jq,...,j1,jd⁰⁰σ_j1 ∧ · · · ∧d⁰⁰σ_jq

onU_j. Then,F defines a D⁰⁰-closed section inΓ(Ω,E^∞(Λ^0,qT_X^? ⊗K_X ⊗L))such that[F_|Y] = [f]∈H^q(Y,(K_X⊗L)_|Y)and the estimate

Z

Ω

|F|²

|s|^2(1−κ) dVω≤ 16e^q+1(q+ 1)c0Cσ|J|!

κ(|J| −q−1)!

Z

Y

|f|²

|ds|²dVY,ω

holds for some constantC_σ depending only on the partition of unity{σ_j}_j∈J and q (one can takeC_σ = maxj∈J|d⁰⁰σ_j|^2q).

References

[De1] J.-P. Demailly, Estimations L² pour l’opérateur ∂¯ d’un fibré vectoriel holomorphe semi-positif au- dessus d’une variété kählérienne complète,Ann. Sci. École Norm. Sup.15, 1982, 457-511 3

[De2] J.-P. Demailly,L² vanishing theorems for positive line bundles and adjunction theory,Transcendental methods in algebraic geometry (Cetraro, 1994), Lecture Notes in Math., 1646, Springer, Berlin, 1996, 1-97 3, 4

[De3] J.-P. Demailly, On the Ohsawa-Takegoshi-ManivelL² extension theorem,Complex analysis and ge- ometry (Paris, 1997), Prog. Math., 188, Birkhäuser, Basel, 2000, 47-82 1, 2, 3, 5, 6, 7

[Ma] L. Manivel, Un théorème de prolongement L² de sections holomorphes d’un fibré vectoriel, Math.

Zeitschrift212, 1993, 107-122 1

[MV] J. D. McNeal and D. Varolin, Analytic inversion of adjunction: L² extension theorems with gain, Ann. Inst. Fourier (Grenoble)57, 2007, 703-718 7, 9

[OT] T. Ohsawa and K. Takegoshi,On the extension of L² holomorphic functions, Math. Z.195, 1987, 197-204 1

[Pa] M. Păun,Extensions with estimates of pluricanonical forms,Notes for the summer school of Grenoble, 2007 6, 7

[Si] Y.-T. Siu, A vanishing theorem for semipositive line bundles over non-Kähler manifolds, J. Diff.

Geom.19, 1984, 431-452 2

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E-mail address: [email protected]