Such a bundle admits a curvature current (1) ΘE,h:=i∂∂ϕ=i∂∂ϕ0+i∂∂u which is locally the sum of a positive (1,1)-current i∂∂ϕ0 and a smooth (1,1)-form i∂∂u

ON NON REDUCED ANALYTIC SUBSPACES

JUNYAN CAO⁽¹⁾, JEAN-PIERRE DEMAILLY⁽²⁾, SHIN-ICHI MATSUMURA⁽³⁾

In memory of Professor Lu QiKeng (1927–2015)

Abstract. The main purpose of this paper is to generalize the celebratedL²extension theorem of Ohsawa-Takegoshi in several directions : the holomorphic sections to extend are taken in a possibly singular hermitian line bundle, the subvariety from which the extension is performed may be non reduced, the ambient manifold is K¨ahler and holomorphically convex, but not necessarily compact.

1. Introduction and preliminaries

The purpose of this paper is to generalize the celebrated L² extension theorem of Ohsawa- Takegoshi [OT87] under the weakest possible hypotheses, along the lines of [Dem15b] and [Mat16b]. Especially, the ambient complex manifoldXis a K¨ahler manifold that is only assumed to beholomorphically convex, and is not necessarily compact; by the Remmert reduction theo- rem, this is the same as a K¨ahler manifoldX that admits a proper holomorphic mapπ:X→S onto a Stein complex space S. This allows in particular to consider relative situations over a Stein base. We consider a holomorphic line bundleE→X equipped with a singular hermitian metric h, namely a metric which can be expressed locally as h = e^−ϕ where ϕ is a quasi-psh function, i.e. a function that is locally the sumϕ =ϕ₀+u of a plurisubharmonic function ϕ₀ and of a smooth functionu. Such a bundle admits a curvature current

(1) Θ_E,h:=i∂∂ϕ=i∂∂ϕ₀+i∂∂u

which is locally the sum of a positive (1,1)-current i∂∂ϕ0 and a smooth (1,1)-form i∂∂u. Our goal is to extend sections that are defined on a (non necessarily reduced) complex subspace Y ⊂ X, when the structure sheaf O_Y :=O_X/I(e^−ψ) is given by the multiplier ideal sheaf of a quasi-psh function ψ with neat analytic singularities, i.e. locally on a neighborhood V of an arbitrary pointx0 ∈X we have

(2) ψ(z) =clogX

|g_j(z)|²+v(z), gj ∈ O_X(V), v ∈C^∞(V).

Let us recall that the multiplier ideal sheafI(e^−ϕ) of a quasi-psh functionϕ is defined by

(3) I(e^−ϕ)_x0 =

f ∈ O_X,x0; ∃U 3x₀, Z

U

|f|²e^−ϕdλ <+∞

with respect to the Lebesgue measure λ in some local coordinates near x₀. As usual, we also denote by KX = ΛⁿT_X^∗ the canonical bundle of an n-dimensional complex manifold X. As is well known, I(e^−ϕ) ⊂ O_X is a coherent ideal sheaf (see e.g. [Dem-book]). Our main result is given by the following general statement.

Date: April 2, 2017, version 2.

2010Mathematics Subject Classification. Primary 32L10; Secondary 32E05.

Key words and phrases. Compact K¨ahler manifold, singular hermitian metric, coherent sheaf cohomology, Dolbeault cohomology, plurisubharmonic function,L²estimates, Ohsawa-Takegoshi extension theorem, multiplier ideal sheaf.

This work was supported by the Agence Nationale de la Recherche grant “Convergence de Gromov-Hausdorff en g´eom´etrie k¨ahl´erienne” (ANR-GRACK), the European Research Council project “Algebraic and K¨ahler Ge- ometry” (ERC-ALKAGE, grant No. 670846 from September 2015), the Japan Society for the Promotion of Science Grant-in-Aid for Young Scientists (B) (JSPS, grant No. 25800051) and the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers.

1

Theorem 1.1. Let E be a holomorphic line bundle over a holomorphically convex K¨ahler mani- fold X. Let h be a possibly singular hermitian metric on E, ψ a quasi-psh function with neat analytic singularities onX. Assume that there exists a positive continuous function δ >0onX such that

(4) Θ_E,h+ (1 +αδ)i∂∂ψ≥0 in the sense of currents, for all α∈[0,1].

Then the morphism induced by the natural inclusionI(he^−ψ)→ I(h) (5) H^q(X, K_X⊗E⊗ I(he^−ψ))→H^q(X, K_X ⊗E⊗ I(h))

is injective for everyq ≥0. In other words, the morphism induced by the natural sheaf surjection I(h)→ I(h)/I(he^−ψ)

(6) H^q(X, KX ⊗E⊗ I(h))→H^q(X, KX ⊗E⊗ I(h)/I(he^−ψ)) is surjective for everyq ≥0.

Remark 1.2. If h is smooth, we have I(h) = O_X and I(h)/I(he^−ψ) = O_X/I(e^−ψ) := O_Y where Y is the zero subvariety of the ideal sheaf I(e^−ψ). Then for q = 0, the surjectivity statement can be interpreted an extension theorem for holomorphic sections, with respect to the restriction morphism

(7) H⁰(X, KX ⊗E)→H⁰(Y,(KX⊗E)_|Y).

In general, the quotient sheaf I(h)/I(he^−ψ) is supported in an analytic subvariety Y ⊂ X, which is the zero set of the quotient ideal

J_Y :=I(he^−ψ) :I(h) =

f ∈ O_X; f · I(h)⊂ I(he^−ψ) ,

and (6) can be considered as a restriction morphism toY.

The crucial idea of the proof is to prove the results (say, in the form of the surjectivity statement), only up to approximation. This is done by solving a∂-equation

∂uε+wε=v

where the right hand sidev is given andw_ε is an error term such that kw_εk=O(ε^a) as ε→0, for some constanta >0. A twisted Bochner-Kodaira-Nakano identity introduced by Donnelly and Fefferman [DF83], and Ohsawa and Takegoshi [OT87] is used for that purpose, with an additional correction term. The version we need can be stated as follows.

Proposition 1.3. (see [Dem15b, Prop. 3.12]) Let X be a complete K¨ahler manifold equipped with a(non necessarily complete) K¨ahler metricω, 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 curvature operator

B =B^n,q_E,h,ω,η,λ= [ηΘE,h−i ∂∂η−iλ⁻¹d∂η∧∂η,Λω]∈C^∞(X,Herm(Λ^n,qT_X^∗ ⊗E)) satisfiesB+εI >0 for someε >0 (so thatB can be just semi-positive or even slightly negative;

here I is the identity endomorphism). Given a section v∈L²(X,Λ^n,qT_X^∗ ⊗E) such that ∂v= 0 and

M(ε) :=

Z

X

h(B+εI)⁻¹v, vidVX,ω<+∞,

there exists an approximate solution fε∈L²(X,Λ^n,q−1T_X^∗ ⊗E) and a correction term wε ∈ L²(X,Λ^n,qT_X^∗ ⊗E) such that∂uε=v−wε and

Z

X

(η+λ)⁻¹|u_ε|²dVX,ω+1 ε

Z

X

|w_ε|²dVX,ω ≤M(ε).

Moreover, ifv is smooth, thenuε and wε can be taken smooth.

In our situation, the main part of the solution, namely uε, may very well explode as ε→ 0.

In order to show that the equation ∂u = v can be solved, it is therefore needed to check that the space of coboundaries is closed in the space of cocycles in the Fr´echet topology under consideration (here, the L²_loc topology), in other words, that the related cohomology group H^q(X,F) is Hausdorff. In this respect, the fact of considering ∂-cohomology of smooth forms equipped with theC^∞topology on the one hand, or cohomology of formsu∈L²_locwith∂u∈L²_loc on the other hand, yields the same topology on the resulting cohomology groupH^q(X,F). This comes from the fact that both complexes yield fine resolutions of the same coherent sheaf F, and the topology of H^q(X,F) can also be obtained by using ˇCech cochains with respect to a Stein covering U of X. The required Hausdorff property then comes from the following well known fact.

Lemma 1.4. Let X be a holomorphically convex complex space and F a coherent analytic sheaf overX. Then all cohomology groupsH^q(X,F) are Hausdorff with respect to their natural topology(induced by the Fr´echet topology of local uniform convergence of holomorphic cochains).¹ In fact, the Remmert reduction theorem implies that X admits a proper holomorphic map π : X → S onto a Stein space S, and Grauert’s direct image theorem shows that all direct images R^qπ∗F are coherent sheaves on S. Now, as S is Stein, Leray’s theorem combined with Cartan’s theorem B tells us that we have an isomorphism H^q(X,F) ' H⁰(S, R^qπ∗F). More generally, ifU ⊂S is a Stein open subset, we have

(8) H^q(π⁻¹(U),F)'H⁰(U, R^qπ∗F)

and when U b S is relatively compact, it is easily seen that this a topological isomorphism of Fr´echet spaces since both sides are O_S(U) modules of finite type and can be seen as a Fr´echet quotient of some direct sumO_S(U)^⊕N by looking at local generators and local relations ofR^qπ∗F. ThereforeH^q(X,F)'H⁰(S, R^qπ∗F) is a topological isomorphism and the space of sections in the right hand side is a Fr´echet space. In particular, H^q(X,F) is Hausdorff.

The isomorphism (8) shows that it is enough to prove Theorem 1.1 locally over X, i.e., we can replaceX by X⁰ =π⁻¹(S⁰) bX where S⁰ bS. Therefore, we can assume thatδ > 0 is a constant rather than a continuous function.

2. Proof of the extension theorem

In this section, we give a proof of Theorem 1.1 based on a generalization of the arguments of [Dem15b, Th. 2.14]. We start by proving the special case of the extension result for holomorphic sections (q= 0).

Theorem 2.1. Let (X, ω) be a holomorphically convex K¨ahler manifold and ψ be a quasi- psh function with neat analytic singularities. Let E be a line bundle with a possibly singu- lar metric h, and Y the support of the sheaf I(h)/I(he^−ψ), along with the structure sheaf O_Y :=I(he^−ψ) :I(h). Assume that there is a continuous functionδ >0 such that

iΘ_E,h+ (1 +αδ)i∂∂ψ ≥0 in the sense of currents, for allα ∈[0,1].

Then the restriction morphism

H⁰(X,O_X(K_X⊗E)⊗ I(h))→H⁰(Y,O_X(K_X⊗E)⊗ I(h)/I(he^−ψ)_|Y) is surjective.

Proof. (a)Let us first assume for simplicity that h is smooth. We will explain the general case later. ThenI(h) =O_X andI(h)/I(he^−ψ) =O_Y =O_X/I(e^−ψ). After possibly shrinkingXinto a relatively compact holomorphically convex open subset X⁰ = π⁻¹(S⁰) b X, we can suppose

1It was pointed out to us by Prof. Takeo Ohsawa that this result does not hold under the assumption thatX is weakly pseudoconvex, i.e., if we only assume thatX admits a smooth psh exhaustion. A counter-example can be derived from [Kaz84]. As a consequence, it is unclear whether the results of the present paper extend to the K¨ahler weakly pseudoconvex case, although the mainL² estimates are still valid in that situation.

thatδ >0 is a constant and that ψ≤0, after subtracting a large constant toψ. Also, without loss of generality, we can assume thatψ admits a discrete sequence of “jumping numbers”

(9) 0 =m₀ < m₁ <· · ·< m_p<· · · such that I(mψ) =I(m_pψ) for m∈[m_p, m_p+1[.

Since ψ is assumed to have analytic singularities, this follows from using a log resolution of singularities, thanks to the Hironaka desingularization theorem (by the much deeper result of [GZ15] on the strong openness conjecture, one could even possibly eliminate the assumption thatψ has analytic singularities). We fix herepsuch that mp ≤1< mp+1, and in the notation of [Dem15b], we let Y = Y^(mp) be defined by the non necessarily reduced structure sheaf O_Y =O_X/I(e^−ψ) =O_X/I(e^−mpψ).

Step 1(Construction of a smooth extension). Take

f ∈H⁰(Y,O_X(KX ⊗E)|Y) =H⁰(X,O_X(KX ⊗E)⊗ O_X/I(e^−mpψ)).

LetU = (U_i) be a Stein covering of X and let (ρ_i) be a partition of unity subordinate to (U_i).

Thanks to the exact sequence

(10) 0→ I(e^−ψ)→ O_X → O_X/I(e^−ψ)→0, we can find afei ∈H⁰(Ui,O_X(KX ⊗E)) such that

fe_i|_Y∩Ui =f|_Y∩Ui. Then (10) implies that

(11) fe_i−fe_j ∈H⁰(U_i∩U_j,O_X(K_X ⊗E)⊗ I(e^−ψ)).

As a consequence, the smooth section fe:= P

i

ρ_i·fe_i is a smooth extension of f and satisfies

∂fe=P

i

(∂ρ_i)·(fe_i−fe_j) onU_j, hence (12)

Z

X

|∂fe|²_ω,he^−ψdV_X,ω= Z

X

X

j

ρ_j

X

i

(∂ρ_i)·(fe_i−fe_j)

2

ω,he^−ψdV_X,ω <+∞.

Step 2 (L²-estimates). We follow here the arguments of [Dem15b, proof of th. 2.14, p. 217].

Let t ∈ Z⁻ and let χ_t be the negative convex increasing function defined in [Dem15b, (5.8∗), p. 211]. Putη_t:= 1−δ·χ_t(ψ) and λ_t:= 2δ^(χ_χ^2t00^(ψ))2

t(ψ) . We set Rt := ηt(ΘE,h+i∂∂ψ)−i∂∂ηt−λ⁻¹_t i∂ηt∧∂ηt

= ηt(ΘE,h+ (1 +δη⁻¹_t χ⁰_t(ψ))i∂∂ψ) + δ·χ⁰⁰_t(ψ)

2 i∂ψ∧∂ψ.

Note thatχ⁰⁰_t(ψ)≥ ¹₈ onW_t={t < ψ < t+ 1}. The curvature assumption (4) implies Θ_E,h+ (1 +δη_t⁻¹χ⁰_t(ψ))i∂∂ψ ≥0 on X.

As in [Dem15b], we find

(13) R_t≥0 on X

and

(14) R_t≥ δ

16i∂ψ∧∂ψ on W_t={t < ψ < t+ 1}.

Let θ : [−∞,+∞[ → [0,1] be a smooth non increasing real function satisfying θ(x) = 1 for x≤0,θ(x) = 0 for x≥1 and|θ⁰| ≤2. By applying the L² estimate (Proposition 1.3), for every ε >0 we can find sectionsut,ε,wt,ε satisfying

(15) ∂u_t,ε+w_t,ε=v_t:=∂ θ(ψ−t)·fe and

(16) Z

X

(ηt+λt)⁻¹|u_t,ε|²_ω,he^−ψdVX,ω+1 ε

Z

X

|w_t,ε|²_ω,he^−ψdVX,ω ≤ Z

X

h(R_t+εI)⁻¹vt, vtie^−ψdVX,ω ,

where

(17) vt=∂ θ(ψ−t)fe

=θ⁰(ψ−t)∂ψ∧fe+θ(ψ−t)∂f .e Combining (13), (14), (16) and (17), we getR

X|u_t,ε|²_ω,he^−ψdVX,ω<+∞ and (18)

Z

X

|w_t,ε|²_ω,he^−ψdV_X,ω ≤ 128ε δ

Z

{t<ψ<t+1}

|fe|²_ω,he^−ψdV_X,ω+ 2 Z

{ψ<t+1}

|∂fe|²_ω,he^−ψdV_X,ω.

We now estimate the right hand side of (18). Since feis smooth, we have an obvious upper bound of the first term

(19)

Z

{t<ψ<t+1}

|fe|²_ω,he^−ψdV_X,ω≤C₁e^−t,

whereC₁ is the C⁰ norm of f. For the second term, thanks to (9), (11) and (12), we havee (20)

Z

X

|∂fe|²_ω,he^−(1+α)ψdVX,ω <+∞

for anyα ∈]0, mp+1−1[. As a consequence, we get (21)

Z

{ψ<t+1}

|∂fe|²_ω,he^−ψdV_X,ω ≤C₂e^αt

for some constantC2 depending only onα. By takingε=e^(1+α)t, (18), (19) and (21) imply (22)

Z

X

|w_t,ε|²_ω,he^−ψdV_X,ω ≤C₃e^αt=O(ε^1+αα ), for some constantC3, whence the error tends to 0 ast→ −∞and ε→0.

Step 3 (Final conclusion). Putting everything together and redefining u_t =u_t,ε,w_t=w_t,ε for simplicity of notation, we get

(23) ∂(θ(ψ−t)·fe−u_t) =w_t, Z

X

|u_t|²_he^−ψdV_X,ω <+∞

and

(24) lim

t→−∞

Z

X

|w_t|²_ω,he^−ψdVX,ω= 0.

After shrinking X, we can assume that we have a finite Stein covering U = (Ui) where the Ui

are biholomorphic to bounded pseudoconvex domains. The standard H¨ormander L² estimates then provideL² sectionsst,j onUj such that∂st,j =wt on Uj and

(25) lim

t→−∞

Z

Uj

|s_t,j|²_ω,he^−ψdVX,ω = 0.

Then

∂

θ(ψ−t)·fe−ut−X

j

ρjst,j

= −X

j

(∂ρj)·st,j onX

= −X

j

(∂ρj)·(st,j−st,i) on Ui. (26)

As∂(s_t,j −s_t,i) = 0 onU_i∩U_j, the difference is holomorphic and the right hand side of (26) is smooth. Moreover, (25) shows that these differences converge uniformly to 0, hence the right hand side of (26) converges to 0 in C^∞ topology. The left hand side implies that this is a coboundary in the C^∞ Dolbeault resolution of O_X(K_X ⊗E). By applying Lemma 1.4, we conclude that there is aC^∞sectionσt ofKX⊗E converging uniformly to 0 on compact subsets ofX as t→ −∞, such that∂σ_t=P

j

(∂ρ_j)·s_t,j onX. This implies that fe_t:=θ(ψ−t)·fe−u_t−X

j

ρ_js_t,j+σ_t

is holomorphic onX. H¨ormander’s L² estimates also produce local smooth solutions σt,i on Ui

with the additional property that lim

t→−∞

R

Ui|σ_t,i|²_ω,he^−ψdV_X,ω = 0. Therefore fet,i:=θ(ψ−t)·fe−ut−X

j

ρjst,j+σt,i

is holomorphic onU_i and fe_t−fe_t,i converges uniformly to 0 on compact subsets ofU_i. However, by construction,fe_t,i−fe_iis a holomorphic section onU_i that satisfies theL²estimate with respect to the weighte^−ψ, hence fet,i−fei is a section of O_X(KX ⊗E)⊗ I(e^−ψ) on Ui, in other words the image offe_t,i in

H⁰(Ui,O_X(KX ⊗E)⊗ O_X/I(e^−ψ)) coincides withf_|Ui. As a consequence, the image offe_t in

H⁰(X,O_X(KX ⊗E)⊗ O_X/I(e^−ψ)) =H⁰(Y,(KX⊗E)_|Y)

converges to f. By the direct image argument used in the preliminary section, this density property implies the surjectivity of the restriction morphism toY.

(b)We now prove the theorem for the general case whenh=e^−ϕ is not necessarily smooth. We can reduce ourselves to the case when ψ has divisorial singularities (see [Dem15b] or the next section for a more detailed argument). Let us pick a section

f ∈H⁰(X,O_X(KX ⊗E)⊗ I(h)/I(he^−ψ)).

By using the same reasoning as in Step 1, we can find a smooth extensionfe∈ C^∞(X, K_X⊗E) off such that

(27)

Z

X

|∂fe|²_ω,he^−ψdVX,ω <+∞.

For everyt∈Z⁻ fixed, asψ has divisorial singularities, we still have Θ_E,h+ (1 +δη⁻¹_t χ⁰_t(ψ))(i∂∂ψ)_ac≥0 on X,

where (i∂∂ψ)ac is the absolutely continuous part of i∂∂ψ. The regularization techniques of [DPS01] and [Dem15a, Th. 1.7, Remark 1.11] (cf. also the next section) produce a family of singular metrics{h_t,ε}^+∞_k=1 which are smooth in the complementXrZt,ε of an analytic set, such thatI(h_t,ε) =I(h),I(h_t,εe^−ψ) =I(he^−ψ) and

ΘE,ht,ε + (1 +δη_t⁻¹χ⁰_t(ψ))i∂∂ψ≥ −1

2εω on X.

The additional error term−¹₂εω is irrelevant when we use Proposition 1.3, as it is absorbed by taking the hermitian operatorB+εI. Therefore for everyt∈Z⁻, with the adjustment ε=e^αt, α ∈]0, m_k+1−1[, we can find a singular metric h_t = h_t,ε which is smooth in the complement X\Zt of an analytic set, such that I(h_t) =I(h),I(h_te^−ψ) =I(he^−ψ) and ht↑h ast→ −∞, and approximate solutions of the∂-equation such that

∂(θ(ψ−t)·fe−ut) =wt , Z

X

|u_t|²_ω,h

te^−ψdV_X,ω <+∞

and

t→−∞lim Z

X

|w_t|²_ω,hte^−ψdVX,ω = 0.

Proposition 1.3 can indeed be applied sinceXrZtis complete K¨ahler (at least after we shrink X a little bit as X⁰ = π⁻¹(S⁰), cf. [Dem82]). The theorem is then proved by using the same argument as in Step 3; it is enough to notice that the holomorphic sectionsst,j−st,i andfet,i−fei

satisfy theL²-estimate with respect to (ht, ψ) [instead of the expected (h, ψ)], but the multiplier ideal sheaves involved are unchanged. The Hausdorff property is applied to the cohomology groupH¹(X,O_X(KX⊗E)⊗ I(h)) instead ofH¹(X, KX⊗E), and the density property to the morphism of direct image sheaves

π∗ O_X(KX ⊗E)⊗ I(h)

→π∗ O_X(KX ⊗E)⊗ I(h)/I(he^−ψ)

over the Stein spaceS.

Proof of the extension theorem for degree q cohomology classes. The reasoning is extremely similar, so we only explain the few additional arguments needed. In fact, Proposition 1.3 can be applied right away to arbitrary (n, q)-forms with q ≥ 1, and the twisted Bochner- Kodaira-Nakano inequality yields exactly the same estimates. Any cohomology class in

H^q(Y,O_X(K_X ⊗E)⊗ I(h)/I(he^−ψ))

is represented by a holomorphic ˇCechq-cocycle with respect to the Stein coveringU = (U_i), say (ci0...iq), ci0...iq ∈H⁰ Ui0∩. . .∩Uiq,O_X(K_X ⊗E)⊗ I(h)/I(he^−ψ)

.

By the standard sheaf theoretic isomorphisms with Dolbeault cohomology (cf. e.g. [Dem-e-book]), this class is represented by a smooth (n, q)-form

f = X

i0,...,iq

ci0...iqρi0∂ρi1∧. . . ∂ρiq

by means of a partition of unity (ρ_i) subordinate to (U_i). This form is to be interpreted as a form on the (non reduced) analytic subvarietyY associated with the ideal sheaf J =I(he^−ψ) :I(h) and the structure sheaf O_Y = O_X/J. We get an extension as a smooth (no longer ∂-closed) (n, q)-form onX by taking

fe= X

i0,...,iq

ec_i0...iqρ_i0∂ρ_i1∧. . . ∂ρ_iq

whereec_i0...iq is an extension of c_i0...iq from U_i0∩. . .∩U_iq ∩Y toU_i0 ∩. . .∩U_iq. Again, we can find approximateL² solutions of the∂-equation such that

∂(θ(ψ−t)·fe−u_t) =w_t , Z

X

|u_t|²_ω,h

te^−ψdV_X,ω <+∞

and

t→−∞lim Z

X

|w_t|²_ω,h

te^−ψdV_X,ω = 0.

The difficulty is that L² sections cannot be restricted in a continuous way to a subvariety. In order to overcome this problem, we play again the game of returning to ˇCech cohomology by solving inductively∂-equations for w_t onU_i0∩. . .∩U_ik, until we reach an equality

(28) ∂ θ(ψ−t)·fe−eut

=wet:=− X

i0,...,iq−1

st,i0...iq∂ρi0∧∂ρi1 ∧. . . ∂ρiq

with holomorphic sectionss_t,I =s_t,i0...iq on U_I =U_i0 ∩. . .∩U_iq, such that

t→−∞lim Z

UI

|s_t,I|²_ω,h

te^−ψdV_X,ω = 0.

Then the right hand side of (28) is smooth, and more precisely has coefficients in the sheaf C^∞⊗O I(he^−ψ), and we_t → 0 in C^∞ topology. A priori, eu_t is an L² (n, q)-form equal to u_t plus a combination P

ρist,i of the local solutions of ∂st,i = wt, plus P

ρist,i,j ∧∂ρj where

∂st,i,j =st,j−st,i, plus etc. . . , and is such that Z

X

|ue_t|²_ω,h

te^−ψdV_X,ω <+∞.

SinceH^q(X,O_X(KX⊗E)⊗ I(he^−ψ)) can be computed with theL²_locresolution of the coherent sheaf, or alternatively with the∂-complex of (n,^•)-forms with coefficients inC^∞⊗_OI(he^−ψ), we may assume thateut∈ C^∞⊗_OI(he^−ψ), after playing again with ˇCech cohomology. Lemma 1.4 yields a sequence of smooth (n, q)-forms σ_t with coefficients inC^∞⊗_OI(h), such that ∂σ_t=we_t andσt→0 inC^∞-topology. Thenfet=θ(ψ−t)·fe−uet−σtis a∂-closed (n, q)-form onX with values inC^∞⊗_OI(h)⊗ O_X(E), whose image inH^q(X,O_X(KX⊗E)⊗ I(h)/I(he^−ψ)) converges

to{f} in C^∞ Fr´echet topology. We conclude by a density argument on the Stein space S, by looking at the coherent sheaf morphism

R^qπ∗ O_X(K_X⊗E)⊗ I(h)

→R^qπ∗ O_X(K_X ⊗E)⊗ I(h)/I(he^−ψ)

.

3. An alternative proof based on injectivity theorems

We give here an alternative proof based on injectivity theorems, in the case whenXis compact K¨ahler. The case of a holomorphically convex manifold is entirely similar, so we will content ourselves to indicate the required additional arguments at the end.

Proof of Theorem 1.1. First of all, we reduce the proof of Theorem 1.1 to the case whenψ has divisorial singularities. Sinceψhas analytic singularities, there exists a modificationπ:X⁰ →X such that the pull-backπ^∗ψ has divisorial singularities. For the singular hermitian line bundle (E⁰, h⁰) := (π^∗E, π^∗h) and the quasi-psh functionψ⁰ :=π^∗ψ, we can easily check that

π∗(K_X⁰ ⊗E⁰⊗ I(h⁰e^−ψ0)) =K_X ⊗E⊗ I(he^−ψ), π∗(K_X⁰ ⊗E⁰⊗ I(h⁰)) =KX ⊗E⊗ I(h).

Hence we obtain the following commutative diagram : H^q(X, K_X⊗E⊗ I(he^−ψ))

∼= π^∗

f //H^q(X, K_X ⊗E⊗ I(h))

π^∗

H^q(X⁰, K_X⁰ ⊗E⁰⊗ I(h⁰e^−ψ0)) ^{g //}H^q(X⁰, K_X⁰ ⊗E⁰⊗ I(h⁰)),

where f, g are the morphisms induced by the natural inclusions and π^∗ is the natural edge morphism. It follows that the left edge morphism π^∗ is an isomorphism since the curvature of the singular hermitian metric h⁰e^−ψ0 on E⁰ is semi-positive by the assumption. Indeed, even if h⁰ does not have analytic singularities, we can see that

R^qπ∗(K_X⁰⊗E⁰⊗ I(h⁰e^−ψ0)) = 0 for everyq >0

by [Mat16b, Corollary 1.5]. (In the case ofX being a projective variety, a relatively easy proof can be found in [FM16].) If Theorem 1.1 can be proven whenψ has divisorial singularities, it follows that the morphism g in the above diagram is injective since (E⁰, h⁰) = (π^∗E, π^∗h) and ψ⁰ =π^∗ψsatisfy the assumptions in Theorem 1.1 and ψ⁰ has divisorial singularities. Therefore the morphismf is also injective by the commutative diagram.

Now we explain the idea of the proof of Theorem 1.1. If we can obtain equisingular approxi- mationshε of hsatisfying the following properties :

Θ_E,hε+i∂∂ψ ≥ −εω and Θ_E,hε+ (1 +δ)i∂∂ψ≥ −εω,

then a proof similar to [FM16] works, where ω is a fixed K¨ahler form on X. In the case when ψhas divisorial singularities, we can attain either of the above curvature properties, but we do not know whether we can attain them at the same time. For this reason, we will look for an essential curvature condition arising from the assumptions on the curvatures in Theorem 1.1, in order to use the “twisted” Bochner-Kodaira-Nakano identity.

From now on, we consider a quasi-psh ψ with divisorial singularities. Then there exist an effectiveR-divisorD and a smooth (1,1)-form γ on X such that

i

2π∂∂ψ= [D] + 1 2πγ

in the sense of (1,1)-currents, where [D] denotes the current of the integration over D. For the irreducible decompositionD =PN

i=1a_iD_i and the defining section t_i of D_i, we can take a smooth hermitian metricbi on Di such that

e^ψ =|s|²_b :=|t₁|^2a_b ¹

1 |t₂|^2a_b ²

2 · · · |t_N|^2a_b ^N

N and −γ = Θ_b(D) :=

N

X

i=1

a_iΘ_bi(D_i).

For a positive number 0< c1, we define the continuous functionsσ and η onX by σ=σc:= log(|s|²_b+c) and η=ηc:= 1

c −χ(σ), whereχ(t) :=t−log(−t).

Remark 3.1. (i) We may assume that |s|²_b < 1/5 by subtracting a positive constant from ψ.

Further we may assume thatσ <log(1/5) and 1< χ⁰(σ)<7/4 by choosing a sufficiently small c >0.

(ii) Further, the function η is a continuous function on X with η > 1/c. The function η is smooth on X\D, but it need not be smooth on X since |s|²_b is not smooth in the case when 0< a_i <1 for some i.

Throughout the proof, we fix a K¨ahler form ω on X. The following proposition gives a suitable approximation of a singular hermitian metric h on E, which enables us to use the twisted Bochner-Kodaira-Nakano identity. The proof is based on the argument in [Ohs04], [Fuj13] and the equisingular approximation theorem in [DPS01, Theorem 2.3].

Proposition 3.2. There exist singular hermitian metrics {h_ε}_0<ε1 on E with the following properties:

(a) hε is smooth onX\Zε, where Zε is a proper subvariety on X.

(b) h_ε⁰ ≤h_ε⁰⁰ ≤h holds on X for ε⁰ > ε⁰⁰>0.

(c) I(h) =I(h_ε) and I(he^−ψ) =I(h_εe^−ψ) on X.

(d) η(Θ_hε(E) +γ)−i∂∂η−η⁻²i∂η∧∂η≥ −εω onX\D.

(e) For arbitrary t >0, by taking a sufficiently smallε >0, we have Z

e^−tφ−e^−tφε <∞, where φ (resp. φε) is a local weight ofh (resp. hε).

Proof. We fix a a sufficiently smallc with 7c/4≤δ. Then, by Remark 3.1, we can easily check that

χ⁰(σ)|s|²_b η(|s|²_b +c) ≤ 7

4η ≤ 7 4c≤δ.

In particular, it follows that

Θ_h+

1 + χ⁰(σ)|s|²_b η(|s|²_b +c)

γ≥0 onX

sinceψ has divisorial singularities and satisfies the assumptions in Theorem 1.1. By applying the equisingular approximation theorem ([DPS01, Theorem 2.3]) to h, we can take singular hermitian metrics {h_ε}_0<ε1 on E satisfying properties (a), (b), (e), the former conclusion of (c), and the following curvature property:

Θ_hε+

1 + χ⁰(σ)|s|²_b η(|s|²_b +c)

γ ≥ −εω on X.

Now we check property (d) from the above curvature property. The function η may not be smooth onX, but it is smooth on X\D. Therefore the same computation as in [Ohs04] and [Fuj13] works onX\D. In particular, from a complicated but straightforward computation, we obtain

−i∂∂η=−χ⁰(σ)|s|²_b

|s|²_b +c Θ_b(D) + c

χ⁰(σ)|s|²_b + χ⁰⁰(σ) χ⁰(σ)²

i∂η∧∂η

on X\D (see [Fuj13] for the precise computation). Then, by −γ = Θb(D) on X\D, we can see that

η(Θhε(E) +γ)−i∂∂η− 1

η²i∂η∧∂η

= c

χ⁰(σ)|s|²_b + χ⁰⁰(σ) χ⁰(σ)² − 1

η²

i∂η∧∂η+η

Θ_hε(E) + (1 + χ⁰(σ)|s|²_b η(|s|²_b+c))γ

≥ c

χ⁰(σ)|s|²_b + χ⁰⁰(σ) χ⁰(σ)² − 1

η²

i∂η∧∂η−εηω

on X\D. A straightforward computation yields that χ⁰⁰(σ)/χ⁰(σ)² ≥1/η², and thus the first term is semi-positive. Sinceη is bounded above, we infer that property (d) holds.

Finally we check the last conclusion of property (c) by proving the following lemma, which can be obtained from the strong openness theorem (see [GZ15], [Lem14], [Hie14]) and property (e).

Lemma 3.3. For a quasi-psh functionϕ, we haveI(he^−ϕ) =I(hεe^−ϕ). In particular, we obtain the last conclusion of property (c).

Proof. We have the inclusion I(he^−ϕ)⊂ I(h_εe^−ϕ) by h_ε≤h. To get the converse inclusion, we consider a local holomorphic functiong such that|g|²e^−ϕ−φε is integrable, where φ_ε (resp. φ) is a local weight ofhε (resp.h). Then H¨older’s inequality yields

Z

|g|²e^−φ−ϕ = Z

|g|²e^−ϕ−φεe^−φ+φε

≤Z

|g|^2pe^{−p(ϕ+φε)}1/p

·Z

e^{−q(φ−φε)}1/q

,

wherep,q are real numbers such that 1/p+ 1/q = 1 andp >1. By the strong openness theorem, the function|g|^2pe^{−p(ϕ+φε)} is integrable when p is sufficiently close to one. On the other hand, we have

Z

e^{−q(φ−φε)}−1 = Z

e^qφε e^−qφ−e^−qφε

≤supe^qφε Z

e^−qφ−e^−qφε .

The right hand side is finite for a sufficiently smallεby property (e).

This concludes the proof of Proposition 3.2.

From now on, we proceed to prove Theorem 1.1 by using Proposition 3.2. In the same way as in [FM16, Section 5], one constructs a family of complete K¨ahler forms {ω_ε,δ}_0<δ1 on Y_ε:=X\(Z_ε∪D) with the following properties :

(A) ω_ε,δ is a complete K¨ahler form onYε:=X\(Zε∪D) for everyδ >0.

(B) ω_ε,δ ≥ω on Y_ε for everyδ≥0.

(C) For every point p in X, there exists a bounded function Ψε,δ on an open neighborhood Bp such thatω_ε,δ=i∂∂Ψ_ε,δ on Bp and Ψ_ε,δ converges uniformly to a bounded function that is independent ofε.

For simplicity, we put H := he^−ψ and Hε := hεe^−ψ. We consider a cohomology class β ∈ H^q(X, K_X ⊗E ⊗ I(H)) such that β = 0 ∈ H^q(X, K_X ⊗E ⊗ I(h)). By the De Rham-Weil isomorphism

H^q(X, KX ⊗E⊗ I(H))∼=

Ker∂ :L^n,q₍₂₎(E)_H,ω→L^n,q+1₍₂₎ (E)_H,ω Im∂:L^n,q−1₍₂₎ (E)H,ω →L^n,q₍₂₎(E)H,ω

,

the cohomology classβcan be represented by a∂-closedE-valued (n, q)-formuwithkuk_H,ω <∞ (that is,β ={u}). Here L^n,•₍₂₎(E)H,ω is the L²-space of E-valued (n,•)-forms on X with respect to theL²-normk • k_H,ω defined by

k • k²_H,ω:=

Z

X

| • |²_H,ωdVω,

wheredVω :=ωⁿ/n! andn:= dimX. For the L²-normk • k_Hε,ωε,δ defined by k • k²_ε,δ :=k • k²_Hε,ω

ε,δ :=

Z

X

| • |²_Hε,ω

ε,δdVωε,δ, one can easily check that

kuk_ε,δ ≤ kuk_H,ωε,δ ≤ kuk_H,ω <∞.

(29)

Indeed, the first inequality is obtained from property (b), and the second inequality is obtained from property (B) for ω_ε,δ (for example see [FM16, Lemma 2,4]). In particular, we see that u belongs to theL²-space

L^n,q₍₂₎(E)_ε,δ :=L^n,q₍₂₎(Y_ε, E)_Hε,ωε,δ

ofE-valued (n, q)-forms onYε (notX) with respect tok • k_ε,δ. By the orthogonal decomposition (see for example [Mat16a, Proposition 5.8])

L^n,q₍₂₎(F)_ε,δ = Im∂ ⊕ H^n,q_ε,δ(F) ⊕Im∂^∗_ε,δ, theE-valued formu can be decomposed as follows :

u=∂wε,δ+uε,δ for somewε,δ ∈Dom∂ ⊂L^n,q−1₍₂₎ (E)ε,δ, and uε,δ ∈ H^n,q_ε,δ(E).

(30)

Here∂^∗_ε,δ is (the maximal extension of) the formal adjoint of the∂-operator andH^n,q_ε,δ(E) is the space of harmonic forms onYε, that is,

H^n,q_ε,δ(E) :={w∈L^n,q₍₂₎(E)ε,δ|∂w = 0 and ∂^∗_ε,δw= 0.}.

Proposition 3.4 (resp. Proposition 3.5) can be proved by the same method as in [FM16, Proposition 5.4, 5.6, 5.7] (resp. [FM16, Proposition 5.9, 5.10]), so we omit the proofs here.

Proposition 3.4. If we have

lim

ε→0

lim

δ→0

ku_ε,δk_{K,hε,ωε,δ} = 0,

for every relatively compact setK bX\D, then the cohomology class β is zero in H^q(X, K_X⊗ E⊗ I(H)). Herek • k_{K,hε,ωε,δ} denotes the L²-norm on K with respect tohε (not Hε) andωε,δ. Proposition 3.5. There exists vε,δ ∈L^n,q−1₍₂₎ (E)hε,ω_ε,δ satisfying the following properties:

∂vε,δ =uε,δ and lim

δ→0kv_ε,δk_ε,δ is bounded by a constant independent ofε.

(31)

Remark 3.6. In general, we have

L^n,•₍₂₎(E)ε,δ =L^n,•₍₂₎(E)Hε,ωε,δ $L^n,•₍₂₎(E)hε,ω_ε,δ, and thusvε,δ may not be L²-integrable with respect to Hε.

For the above solution v_ε,δ of the ∂-equation, by using the density lemma, we can take a family of smoothE-valued forms{v_ε,δ,k}^∞_k=1 with the following properties :

v_ε,δ,k →v_ε,δ and ∂v_ε,δ,k→∂v_ε,δ =u_ε,δ inL^n,•₍₂₎(E)_hε,ωε,δ. (32)

Now we consider the level setX_c:={x∈X| − |s|²_b < c}bX\Dfor a negative number c. The set of the critical values of|s|²_b is of Lebesgue measure zero from Sard’s theorem. Hence, for a given relatively compactK bX\D, we can choose−1c <0 such that

K bXc:={x∈X| − |s|²_b < c} and d|s|²_b 6= 0 at every point in∂Xc.

Then, by [Mat16b, Proposition 2.5, Remark 2.6] (see also [FK, (1.3.2) Proposition]), we obtain hh∂v_ε,δ,k, uε,δii_X

d,hε,ωε,δ =hhv_ε,δ,k, ∂^∗_hε,ωε,δuε,δii_X

d,hε,ωε,δ−((vε,δ,k,(∂|s|²_b)^∗uε,δ))_∂X

d,hε,ωε,δ

(33)

for almost alld∈]c−a, c+a[, where ais a sufficiently small positive number. Here∂^∗_hε,ωε,δ is the formal adjoint of the ∂-operator in L^n,•₍₂₎(E)_hε,ωε,δ and ((•,•))_∂X

d,hε,ωε,δ is the inner product on the boundary∂X_d defined by

((a, b))_∂X

d,hε,ωε,δ :=

Z

∂Xd

ha, bi_hε,ωε,δdS_ε,δ,

for smoothE-valued formsa,b, wheredSε,δdenotes the volume form on∂Xddefined bydSε,δ :=

− ∗d|s|²_b/ d|s|²_b

hε,ωε.δ and ∗ denotes the Hodge star operator with respect to ω_ε,δ. Note that dV_ε,δ=dS_ε,δ∧d|s|²_b. One can easily see that

k→∞lim hh∂v_ε,δ,k, uε,δii_X

d,hε,ωε,δ =hh∂v_ε,δ, uε,δii_X

d,hε,ωε,δ =hhu_ε,δ, uε,δii_X

d,hε,ωε,δ

by (32), and thus it is sufficient to show that the right hand side of equality (33) converges to zero. For this purpose, we first prove the following proposition.

Proposition 3.7.

ε→0limlim

δ→0k(∂|s|²_b)^∗u_ε,δk_ε,δ= 0.

Proof of Proposition 3.7. By property (d) and property (B), we have η(Θ_hε(E) +γ)−i∂∂η≥η⁻²i∂η∧∂η−εω

≥η⁻²i∂η∧∂η−εωε,δ.

Sinceuε,δ is harmonic with respect to Hε and ωε,δ, we have∂^∗_ε,δuε,δ = 0 and ∂uε,δ = 0. Further we haveγ =i∂∂ψ on X\D. Therefore we obtain

0≥ −k√

ηD^0∗uε,δk²_ε,δ=k√

η∂uε,δk²_ε,δ+k√

η∂^∗_ε,δuε,δk²_ε,δ− k√

ηD^0∗uε,δk²_ε,δ

=hh ηΘ_Hε−i∂∂η

Λu_ε,δ, u_ε,δii_ε,δ+ 2Rehh∂η∧∂^∗_ε,δu_ε,δ, u_ε,δii_ε,δ

=hh ηΘ_Hε−i∂∂η

Λu_ε,δ, u_ε,δii_ε,δ

≥ hh η⁻²i∂η∧∂η)Λu_ε,δ, u_ε,δii_ε,δ−εqku_ε,δk²_ε,δ.

from the twisted Bochner-Kodaira-Nakano identity (see [Ohs04, Lemma 2.1] or [Fuj13, Propo- sition 2.20. 2.21]). On the other hand, one can easily check that

hh η⁻²i∂η∧∂η)Λuε,δ, uε,δii_ε,δ =kη⁻¹(∂η)^∗uε,δk²_ε,δ=kη⁻¹∗∂η∗uε,δk²_ε,δ,

∂η=−χ⁰(σ)∂σ=− χ⁰(σ)

(|s|²_b +c)∂|s|²_b. By the above arguments, we conclude that

εqku_ε,δk²_ε,δ ≥ k χ⁰(σ)

η(|s|²_b +c)(∂|s|²_b)^∗u_ε,δk²_ε,δ.

It follows that the left hand side converges to zero fromkuk_H,ω ≥ ku_ε,δk_ε,δ. Further the function χ⁰(σ)/η(|s|²_b+c) is bounded below since we have

1

5 >|s|²_b, C > η, χ⁰(σ)>1

for some constantC. This completes the proof.

Finally we prove the following proposition by using Proposition 3.7.

Proposition 3.8. (i) For a relatively compact set K bX\D, we have

ε→0limlim

δ→0lim

k→0hhv_ε,δ,k, ∂^∗_hε,δu_ε,δii_K,h

ε,ωε,δ = 0.

(ii)For almost all d∈]c−a, c+a[, we have lim

ε→0

lim

δ→0

lim

k→0

((v_ε,δ,k,(∂|s|²_b)^∗u_ε,δ))_∂X

d,hε,ωε,δ = 0.

Proof of Proposition 3.8. In general, we have the formula D⁰_gu =G⁻¹∂(Gu) for a smooth her- mitian metric g, where G is a local function representing g. Let G_ε be a local function rep- resenting hε. We remark that Gεe^−ψ is a local function representing Hε. By the definition of

∂^∗_ε,δ =∂^∗_Hε,ωε,δ, we have

0 =∂^∗_ε,δu_ε,δ =− ∗D⁰_Hε ∗u_ε,δ =− ∗(G_εe^−ψ)⁻¹∂(G_εe^−ψ∗u_ε,δ),