L 2 extension of ∂ ¯ -closed forms on weakly pseudoconvex Kähler manifolds
aa r X i v : . [ m a t h . C V ] S e p L EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEXK ¨AHLER MANIFOLDS JIAN CHEN AND SHENG RAO
One world, one fight
Abstract.
Combining V. Koziarz’s observation about the regularity of some modified sectionrelated to the initial extension with J. McNeal–D. Varolin’s regularity argument, we generalizetwo theorems of McNeal–Varolin for the L extension of ¯ ∂ -closed high-degree forms on a Steinmanifold to the weakly pseudoconvex K¨ahler case under mixed positivity conditions. Contents
1. Introduction: main results and applications 12. Some results used in the proofs 53. Proof of Theorem 1.1 8Appendix A. Alternative proof of Theorem 1.3 17Acknowledgement 18References 191.
Introduction: main results and applications
As well known, the task of the classical Ohsawa–Takegoshi theorem [OT87] is to extend aholomorphic object from some lower or same dimensional analytic subvariety to the ambient spacewith some L estimate involved. In recent years, problems of extending holomorphic sections havebeen treated almost completely in the category of analytic manifolds. So it is natural to ask whetherthese extensions are feasible for ¯ ∂ -closed forms of high degree, which is a natural broadening of theclassical Ohsawa–Takegoshi–Manivel extension theorem for holomorphic sections of line bundles.Many interesting works appear along this line, such as [Ma93, Dm00, Kz11, ZGZ12, Bn12,BPZ15, MV19, ZZ19], etc. The biggest difficulty of this problem is the regularity issue for solutionsof related ¯ ∂ -equation for high degree because ¯ ∂ operator for high degree is no longer hypoelliptic.For solving the regularity issue, mainly two methods were adopted: minimizer method and Leray’sisomorphism method.L. Manivel [Ma93] firstly considered this problem, while his proof has difficulty to complete dueto the use of a singular weight and the failure of regularity for the solution of the related ¯ ∂ -equation.Then J.-P. Demailly [Dm00] suggested an approach to overcome this difficulty, while no one seemsto have implemented his program yet completely. V. Koziarz [Kz11] used the Leray’s isomorphismto reduce the extension of high-degree forms to the classical zero-degree case and thereby deducedextensions of cohomology classes. B. Berndtsson [Bn12] applied the minimizer method by solving a¯ ∂ -equation for a current to get the related extension theorem on compact manifolds. In [MV19], J.McNeal–D. Varolin made use of the Kohn solution, to handle the well-known regularity issues on a Date : September 30, 2020.2010
Mathematics Subject Classification.
Primary 32D15; Secondary 32L10, 32Q15, 32T35.
Key words and phrases.
Continuation of analytic objects in several complex variables; Sheaves and cohomologyof sections of holomorphic vector bundles, general results, K¨ahler manifolds, Exhaustion functions.Both authors are partially supported by NSFC (Grant No. 11671305, 11771339, 11922115) and the FundamentalResearch Funds for the Central Universities (Grant No. 2042020kf1065).
Stein or essentially Stein manifold (i.e., a K¨ahler manifold that becomes Stein after a hypersurfaceis removed from it). Furthermore, L. Zhu–Q. Guan–X. Zhou [ZGZ12] and L. Baracco–S. Pinton–G.Zampieri [BPZ15] also got some results in some special cases.It is natural to ask whether we can establish extension theorems of ¯ ∂ -closed forms of high degreeon general weakly pseudoconvex K¨ahler manifolds. Recall that a complex manifold is weaklypseudoconvex if it admits a smooth plurisubharmonic exhaustion function. Combining Koziarz’sobservation [Kz11] about the regularity of some modified section related to the initial ambientextension with McNeal–Varolin’s regularity argument [MV19] for the extension of high-degreeforms, we generalize the L extension theorems (cf. Theorems 2.3 and 2.5) of McNeal–Varolin[MV19] on a Stein manifold to a weakly pseudoconvex K¨ahler manifold. Theorem 1.1 (Ambient L extension) . On a weakly pseudoconvex n -dimensional K¨ahler manifold ( X, ω ) , let the smooth subvariety Y ι ֒ → X be the zero set of a holomorphic section s ∈ H ( X, E ) ofa smooth Hermitian holomorphic line bundle ( E, λ ) and ( L, ϕ ) a smooth Hermitian holomorphicline bundle. Assume that for any ≤ q ≤ n − , the inequalities hold on X √− ∂ ¯ ∂ ( ϕ − λ ) ∧ ω q ≥ σω ∧ ω q , (1.1) √− ∂ ¯ ∂ ( ϕ − (1 + δ ) λ ) ∧ ω q ≥ , (1.2) | s | e − λ ≤ , (1.3) where σ is some positive lower semi-continuous function and < δ ≤ is some constant. Thenthere is a universal constant C > such that for any smooth section f of the bundle ( K X ⊗ L ⊗ Λ ,q T ⋆X ) | Y → Y , satisfying ¯ ∂ ( ι ∗ f ) = 0 and Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω < ∞ , there exists a smooth ¯ ∂ -closed K X ⊗ L -valued (0 , q ) -form F on X with F | Y = f and Z X | F | ω e − ϕ dV X,ω ≤ Cδ Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω < ∞ . Here we denote by Λ r,s T ⋆X the bundle of differential forms of bidegree ( r, s ) on X and similarly forothers. Note that since ( K X ⊗ L ⊗ Λ ,q T ⋆X ) | Y does not admit a natural notion of ¯ ∂ , the above ι ∗ f is notthe usual pullback of differential forms, but induced as Definition 2.2. And the positivity conditions (1.1) (1.2) hold in the sense of ( q + 1 , q + 1) -forms as follows: a real (1 , θ on X satisfiesthe positivities on X (1.4) θ ∧ ω q > (resp. ≥ ) 0if and only if λ + · · · + λ q +1 > (resp. ≥ ) 0 , where λ · · · λ n are the eigenvalues of θ with respect to ω at any point x ∈ X . Moreover,they are both equivalent to that h [ θ, Λ ω ] β, β i ω is positive (resp. semipositive) at any point x ∈ X for any ( n, q + 1)-form β ( x ) = 0 (see Lemmata 2.9 and 2.10 for a better understanding).For the case of semi-positivity conditions, the error term method of Demailly [Dm00] of solving¯ ∂ -equations is always used to overcome the lack of sufficient positivity. Unfortunately, we find nogood methods to combine the regularity argument of McNeal–Varolin with the error term methodto get a result under the semi-positivity conditions. The strict positivity condition (1.1) is mainlyapplied to force the Dirichlet semi-norm to be a norm on some relatively compact domains of theweakly pseudoconvex K¨ahler manifold and to control well the estimate (3.12). However, sincethere are a variety of vanishing theorems, Theorem 1.1 may not tell us anything valuable if oneconsiders the extension problem of cohomology as L is very highly positively curved.Theorem 1.1 is clearly different from [Bn12, Theorem 3.1] which is in an intrinsic sense (seeDefinition 2.1 for this notion) by the adjunction formula. We give an example satisfying the EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 3 conditions of Theorem 1.1, but not in the setting of [MV19, Theorem 1] or the essentially Steincase of [MV19] (cf. [MV19, p. 425]). Example 1.2.
Let ( B m , ω ) be the unit ball in C m equipped with the Euclidean metric ω = √− ∂ ¯ ∂ ( z + · · · + z m ), and ( Y, ω ) a k -dimensional compact K¨ahler manifold which does nothave any closed complex hypersurfaces. By the heredity property of weakly pseudoconvexity, X := B m × Y is a weakly pseudoconvex K¨ahler manifold equipped with the natural K¨ahler metric ω := π ∗ ω + π ∗ ω . As X admits a compact submanifold Y which contains no hypersurfaces, it isneither a Stein nor an essentially Stein manifold. Apparently, X is not a compact manifold, either.Let L = E = O X , s = z , ϕ = | z | , σ = q + 1 − k q + 1) , and λ ≡ , where z is the first global coordinate function on B m , | z | := z + · · · + z m . Then the above settingsatisfies (1.1), (1.2) and (1.3) as q ≥ k .Varolin suggested that one can take Y as a generic torus of dimension ≥
2. In fact, a torus admitsno divisors if and only if it has algebraic dimension zero, i.e., the only meromorphic functions areconstant (e.g. [EF82, p. 31]). On the other hand, for a very general lattice Γ ⊂ C n the meromorphicfunction field K ( C n / Γ) is trivial (e.g. [Huy05, p. 58]). (cid:3)
Just as [MV19, p. 423], if η is an L -valued (0 , q )-form on Y, the orthogonal projection P : T , X | Y → T , Y induced by the K¨ahler metric ω maps η to the ambient L -valued (0 , q )-form P ∗ η , given by h P ∗ η, ¯ v ∧ · · · ∧ ¯ v q i := h η, ( P ¯ v ) ∧ · · · ∧ ( P ¯ v q ) i in L y for all y ∈ Y ι ֒ → X and v , . . . , v q ∈ T , X,y . Around y , we choose a local coordinate and frame( U, { z , . . . , z n } , σ ) such that Z ∩ U = { z = 0 } and { d ¯ z , d ¯ z , · · · , d ¯ z n } is an ω ( y )-orthonormalbasis of ∧ , T ⋆X,y . At y , set η = P / ∈ J a J d ¯ z J ◦ ι ⊗ σ and then P ∗ η = P / ∈ J a J d ¯ z J ⊗ σ . So P ∗ isan isometry for the pointwise norm of L -valued (0 , q )-forms induced by ω and the metric of L . As ι ∗ P ∗ η = η, we can apply Theorem 1.1 to f = P ∗ u and obtain Theorem 1.3, while a sketch of adirect proof for it is also given in Appendix A. Theorem 1.3 (Intrinsic L extension) . With the setting of Theorem 1.1, there is a universalconstant
C > such that for any smooth ¯ ∂ -closed ( K X ⊗ L ) | Y -valued (0 , q ) -form u on Y satisfying Z Y | u | ω e − ϕ | ds | ω e − λ dV Y,ω < ∞ , there exists a smooth ¯ ∂ -closed K X ⊗ L -valued (0 , q ) -form U on X with ι ∗ U = u and Z X | U | ω e − ϕ dV X,ω ≤ Cδ Z Y | u | ω e − ϕ | ds | ω e − λ dV Y,ω < ∞ . We now present two applications of the main theorems. The first one is a surjectivity theoremfor the restriction maps in Dolbeault cohomology. It is similar to the extension theorem forcohomology classes (without L estimate) recently obtained by Zhou–Zhu [ZZ19, Theorem 1.1,Remark 1.1] on a holomorphically convex manifold with the more general curvature conditions,while our method is rather different from theirs. Corollary 1.4.
Let
X, Y, E, L be as in Theorem 1.3 and also Y compact. Then the restrictionmorphism H ,q ( X, K X ⊗ L ) −→ H ,q (cid:16) Y, ( K X ⊗ L ) | Y (cid:17) is surjective for any ≤ q ≤ n − . JIAN CHEN AND SHENG RAO
In the language of sheaf theory, Corollary 1.4 tells us that the homomorphism H q ( X, O X ( K X ⊗ L )) → H q ( Y, O Y ( K X ⊗ L )) ∼ = H q ( X, O X ( K X ⊗ L ) ⊗ O X / J ) , where J is the ideal sheaf of Y , is surjective by Leray’s isomorphism. The surjectivity is of courseequivalent to the injectivity of the homomorphism H q +1 ( X, O X ( K X ⊗ L ) ⊗ J ) → H q +1 ( X, O X ( K X ⊗ L )) , for any fixed 0 ≤ q ≤ n −
1. Note that Corollary 1.4 is trivial when L is highly positively curvedwhile it may still satisfy the setting of Theorem 1.3.Much inspired by [Dm00, Corollary 4.11], we consider the extension behavior of ¯ ∂ -closed formson bounded pseudoconvex domains as the second application of the main theorems. It tells ussome information about the relationship between smooth “generalized quasi-plurisubharmonic”functions and ¯ ∂ -closed forms. Corollary 1.5.
Let Ω ⊂ C n be a bounded pseudoconvex domain, and Y ⊂ X a nonsingular complexsubvariety defined by a section s of some Hermitian holomorphic line bundle ( E, λ ) . Assume that s is everywhere transverse to the zero section and that | s | on Ω . Let ϕ be an any smoothfunction such that for ≤ q ≤ n − , some δ ∈ (0 , and the Chern curvature form Θ E of ( E, λ ) , √− ∂ ¯ ∂ϕ − Θ E and √− ∂ ¯ ∂ϕ − (1 + δ )Θ E are below-bounded in the sense of (1.4) on Ω , i.e., thesums of their smallest q + 1 eigenvalues with respect to the standard complex Euclidean metric arebelow bounded on Ω . Then there is a constant C > (depending on Ω and the “lower bound” of √− ∂ ¯ ∂ϕ − Θ E and √− ∂ ¯ ∂ϕ − (1 + δ )Θ E on Ω in the sense of (1.4) ), with the following property:for any ¯ ∂ -closed ( n, q ) -form or (0 , q ) -form f on Y with Z Y | f | | ds | − e − ϕ dV Y < + ∞ , there exists a ¯ ∂ -closed extension F of f to Ω with Z Ω | F | e − ϕ dV Ω C Z Y | f | | ds | e − ϕ dV Y . Proof.
Assume first that f is a ¯ ∂ -closed ( n, q )-form on Y . Let L := Ω × C be the trivial bundleequipped with a weight function e − ϕ − A | z | . We can choose a sufficiently large constant A > √− ∂ ¯ ∂ϕ − Θ E and √− ∂ ¯ ∂ϕ − (1 + δ )Θ E on Ω in thesense of (1.4) such that the curvature assumptions (1.1) and (1.2) are satisfied. Then there existsan extension F of f to Ω such that Z Ω | F | e − ϕ e − A | z | dV Ω b C Z Y | f | | ds | e − ϕ e − A | z | dV Y according to Theorem 1.3. Note that e − A | z | has lower and upper bounds which depend on Ω. Sothere exists C which depends on Ω and the “lower bound” of √− ∂ ¯ ∂ϕ − Θ E and √− ∂ ¯ ∂ϕ − (1 + δ )Θ E on Ω in the sense of (1.4), such that Z Ω | F | e − ϕ dV Ω C Z Y | f | | ds | e − ϕ dV Y . When f is a ¯ ∂ -closed (0 , q )-form on Y , the application of the above argument for f ∧ dz ∧ dz ∧· · · ∧ dz n and | dz ∧ dz ∧ · · · ∧ dz n | = 1 (possibly after normalizing the Euclidean metric) completethe proof. (cid:3) One would expect to weaken the pseudoconvexity assumption of X in Theorem 1.1 as theexistence of an upper semi-continuous exhaustion on X and thus Ω in Corollary 1.5 could beweakened to be just a bounded domain in C n admitting an upper semi-continuous exhaustion.However, Varolin provided us with a counterexample to the expectation. EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 5 Example 1.6.
Set the domain Ω := (cid:8) z ∈ C ; < | z | < (cid:9) and the subspace Y := (cid:8) z = (cid:0) z , z (cid:1) ∈ Ω; z = 0 (cid:9) . Then it is easy to construct a continuous exhaustion, such as ρ = | z |− / −| z | ) . Take the function f ( ζ,
0) := ζ − n for any n ∈ N + . Then R Y | f | | dz | − dV Y < + ∞ . So if one assumes the aboveexpectation, then there exists some F ∈ O (Ω) such that F | Y = f . By Hartogs theorem and theidentity theorem, F has a unique extension to the unit ball. The restriction of this extension to Y agrees with f, and this means that f itself has a holomorphic extension to the unit disk D × { } .This is impossible by the identity theorem and that ζ − n blows up near the origin in D . (cid:3) A further interesting topic about the extension of ¯ ∂ -closed forms is the singular metric version ofthe main results here, which is very attractive and full of application prospects. However, it seemsvery difficult for extensions of general ¯ ∂ -closed forms. In [MV19, Remark 1.2], McNeal–Varolintold us that the routine method—taking a regularization of the singular weight first and thenpassing to some kind of limit—cannot get the singular version of their extension theorems at leaston a Stein manifold due to that the minimal extension operator may not exist.The other difficulty of dealing with the singular metric version is that the operator ¯ ∂ -Laplacianwith respect to a singular metric may lose the ellipticity in general. Demailly [Dm00, p. 17]hoped that the Laplacian with respect to a singular metric may have a little “ellipticity” whenthe singularity of the metric involved is mild. The expectation of Demailly seems to be a difficultproblem in PDE. All in all, it seems very difficult to obtain the singular metric version of an L extension theorem of general ¯ ∂ -closed forms.The paper is organized as follows. We list some results in Section 2 to be used in the proof ofTheorem 1.1. Then, we prove Theorem 1.1 in Section 3. At last, we give a sketch of a direct proofof Theorem 1.3 in Appendix A. Notation 1.7.
Unless otherwise stated, we will always adopt the notations in Section 1 in thelatter sections and in particular, use | s | or | s | e − λ/ to denote the pointwise norm of s .2. Some results used in the proofs
In this section, we collect several results to be used in the proofs of our main results. Let ι : Y ֒ → X be the natural inclusion of a smooth complex hypersurface Y in a complex manifold X and L a line bundle on X .It is noteworthy that when q ≥
1, there are two natural choices for the restriction to Y of an L -valued (0 , q )-form on X :(i) one can pull back the L -valued differential form on X via the natural inclusion Y ι ֒ → X toproduce an L -valued (0 , q )-form on Y , i.e., a section of L | Y ⊗ Λ ,q T ⋆Y → Y , which we call the intrinsic restriction , or(ii) one can view an L -valued (0 , q )-form as an abstract section of an abstract bundle L ⊗ Λ ,q T ⋆X and pullback the section. That is to say, the restriction is a section of the restricted vectorbundle ( L ⊗ Λ ,q T ⋆X ) | Y → Y . We call a section of the vector bundle ( L ⊗ Λ ,q T ⋆X ) | Y → Y an ambient form , and the restriction of an L -valued (0 , q )-form on X to Y an ambient restriction .More precisely, one has the following definitions. Definition 2.1 ([MV19, Definition 3.1]) . (i) An L | Y -valued (0 , q )-form η on Y is called the intrinsic restriction of an L -valued (0 , q )-form θ on X if ι ∗ θ = η. (ii) A section ξ of the vector bundle ( L ⊗ Λ ,q T ⋆X ) | Y is called the ambient restriction of an L -valued(0 , q )-form θ on X if θ ( y ) = ξ ( y )for all y ∈ Y , that is θ | Y = ξ . Note that | θ ( y ) | = | ξ ( y ) | on Y when X and L are equippedwith some Hermitian metrics in this case. JIAN CHEN AND SHENG RAO
Since ( L ⊗ Λ ,q T ⋆X ) | Y is not a holomorphic vector bundle, it does not admit a natural notion of¯ ∂ . Then one needs: Definition 2.2 ([MV19, p. 422]) . Let ι : Y ֒ → X be the natural inclusion of a smooth complexhypersurface Y in a complex manifold X and L a line bundle on X . Then for any ambient form ξ , ι ∗ ξ is defined to be an L | Y -valued (0 , q )-form on Y by h ι ∗ ξ, ¯ v , . . . , ¯ v q i := h ξ, dι ( y )¯ v , . . . , dι ( y )¯ v q i in L y ( y ∈ Y ) , v , . . . , v q ∈ T , Y,y . Note that ¯ ∂ ( ι ∗ ξ ) is now well defined naturally.In [MV19], McNeal–Varolin established the following two extension theorems and we will usethem to construct some smooth extensions locally on a weakly pseudoconvex K¨ahler manifold inSubsection 3.1 and Appendix A. Theorem 2.3 (Ambient L extension) . Let X be an n -dimensional Stein manifold with K¨ahlerform ω and ι : Y ֒ → X a smooth hypersurface. Let L → X be a holomorphic line bundle withsmooth Hermitian metric e − ϕ . Assume that the line bundle L Y → X associated to the smoothdivisor Y has a section s ∈ H ( X, L Y ) and a smooth Hermitian metric e − λ such that Y is thedivisor of s and sup X | s | e − λ ≤ . Assume also that for any ≤ q ≤ n − , √− (cid:0) ∂ ¯ ∂ ( ϕ − λ ) + Ricci( ω ) (cid:1) ∧ ω q ≥ and √− (cid:0) ∂ ¯ ∂ ( ϕ − (1 + δ ) λ ) + Ricci( ω ) (cid:1) ∧ ω q ≥ for some constant < δ ≤ . Then there is a constant
C > such that for any smooth section ξ of the vector bundle (cid:0) L ⊗ Λ ,q T ⋆X (cid:1)(cid:12)(cid:12) Y → Y satisfying ¯ ∂ ( ι ∗ ξ ) = 0 and Z Y | ξ | ω e − ϕ | ds | ω e − λ ω n − < + ∞ , there exists a smooth ¯ ∂ -closed L -valued (0 , q ) -form Ξ on X with Ξ | Y = ξ and Z X | Ξ | ω e − ϕ ω n n ! ≤ Cδ Z Y | ξ | ω e − ϕ | ds | ω e − λ ω n − ( n − . The constant C is universal, i.e., it is independent of all the data. Remark 2.4.
On the above theorem, there is a typo on [MV19, Theorem 1] which only requires δ >
0. In fact, we can conclude that the first two lines in [MV19, p. 438] cannot be true if δ > e v in [MV19, p. 438] by | s | here, and then e v ( τ + A ) = | s | (2 + 2 e a − + log(2 e a − − ≥ | s | · e a − = 2 e γ − | s | ( ε + | s | ) δ . As δ >
1, the above is 2 e γ − · | s | ε + | s | · ε + | s | ) δ − , which cannot be bounded when ε → | s | → | s | = ε → Theorem 2.5 (Intrinsic L extension) . With the hypotheses of Theorem 2.3, there is a universalconstant
C > such that for any smooth ¯ ∂ -closed L -valued (0 , q ) -form η on Y satisfying Z Y | η | ω e − ϕ | ds | ω e − λ dV Y,ω < + ∞ , EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 7 there exists a smooth ¯ ∂ -closed L -valued (0 , q ) -form Π on X such that ι ∗ Π = η and Z X | Π | ω e − ϕ dV ω ≤ Cδ Z Y | η | ω e − ϕ | ds | ω e − λ dV Y,ω . Consider the modified ¯ ∂ -operators T := ¯ ∂ ◦ √ τ + A and S := √ τ · ¯ ∂ acting on ( n, q )-forms with values in a vector bundle, where τ , A are positive smooth functions.Then S ◦ T = 0. In solving ¯ ∂ -equation, the basic estimate about the modified ¯ ∂ operator is alwaysneeded to construct some bounded linear functionals. On different occasions, several classical basicestimates have been established in [Os95, Bn96, Mc96, Si96, Dm00], for example. Here we adoptthe following one. Lemma 2.6 (Twisted basic estimate) . Let ( X, ω ) be a K¨ahler manifold and E a holomorphicline bundle with a smooth Hermitian metric e − ϕ over X . Assume that τ and A are smooth andpositive functions on X . Fix a smoothly bounded domain Ω ⊂⊂ X such that its boundary ∂ Ω is pseudoconvex (e.g. [Va10, Section 1.5] for this notion and its effect). Then for any smooth E -valued ( n, q ) -form u in the domain of ¯ ∂ ∗ ϕ , one has the estimate Z Ω ( τ + A ) (cid:12)(cid:12) ¯ ∂ ∗ ϕ u (cid:12)(cid:12) ω e − ϕ dV ω + Z Ω τ (cid:12)(cid:12) ¯ ∂u (cid:12)(cid:12) ω e − ϕ dV ω ≥ Z Ω (cid:28)(cid:20) √− (cid:18) τ ∂ ¯ ∂ϕ − ∂ ¯ ∂τ − ∂τ ∧ ¯ ∂τA (cid:19) , Λ ω (cid:21) u, u (cid:29) ω e − ϕ dV ω , where Λ ω is the dual Lefschetz operator.Proof. The proof is the same as that of [MV19, Lemma 2.2] which is in the context of the Stein set-ting. However, the estimate attributes essentially to the usual twisted Bochner–Kodaira–Morrey–Kohn identity and the pseudoconvexity of ∂ Ω. (cid:3) Furthermore, the ellipticity of the twisted ¯ ∂ -Laplacian (cid:3) := T T ∗ + S ∗ S is needed. Lemma 2.7 ([MV19, Proposition 2.3]) . Assume that the functions τ , A and the Hermitian metric e − ϕ of the holomorphic line bundle E are smooth, and that τ and τ + A are positive, then theoperator (cid:3) is second order (interior) elliptic with smooth coefficients. Lemma 2.8 ([Dm82, Lemma 6.9]) . Let Ω be an open subset of C n and Y an analytic subset of Ω .Assume that v is a ( p, q − -form with L loc coefficients and w is a ( p, q ) -form with L loc coefficientssuch that ¯ ∂v = w on Ω \ Y (in the sense of distribution theory). Then ¯ ∂v = w on Ω . (A moregeneral version for the first order differential operator can be found in [Bk18, Proposition 4.8] .) Lemma 2.9 ([Dm12, Chapter VI-(5.8) Proposition]) . Let ( X, ω ) be an n -dimensional Hermitianmanifold and γ a real (1 , -form. Then there exists an ω -orthogonal basis ( ζ , ζ , . . . , ζ n ) in T , X which diagonalizes both forms ω and γ : ω = √− X j n ζ ⋆j ∧ ¯ ζ ⋆j , γ = √− X j n γ j ζ ⋆j ∧ ¯ ζ ⋆j , γ j ∈ R . For every form u = P u J,K ζ ⋆J ∧ ¯ ζ ⋆K , one has [ γ, Λ ω ] u = X J,K X j ∈ J γ j + X j ∈ K γ j − X j n γ j u J,K ζ ⋆J ∧ ¯ ζ ⋆K . Lemma 2.10 ([Dm12, Chapter VIII-(6.4)]) . Let θ be a smooth real (1 , -form on an n -dimensionalHermitian manifold ( X, ω ) . If λ ( x ) · · · λ n ( x ) are the eigenvalues of θ with respect to ω forall x ∈ X and λ + · · · + λ q > , then for arbitrary ( n, q ) -form g on X , h [ θ, Λ ω ] g, g i ω > ( λ + · · · + λ q ) | g | ω JIAN CHEN AND SHENG RAO and Z X (cid:10) [ θ, Λ ω ] − g, g (cid:11) ω dV ω Z X λ + · · · + λ q | g | ω dV ω . Lemma 2.11.
Let ( X, ω ) be an n -dimensional Hermitian manifold and θ a continuous (1 , -form.Then for any ( n, q ) -form α , we have (cid:2) √− θ ∧ ¯ θ, Λ ω (cid:3) α = T ¯ θ T ∗ ¯ θ α, where T ¯ θ denotes ¯ θ ∧ • .Proof. The proof is the same as that of [GZ15b, Lemma 4.2]. (cid:3)
From classical knowledge about matrices (e.g. [La07]), we can easily get the following result.
Lemma 2.12.
Assume that A and B are Hermitian matrices of n × n and both of them are positivedefinite. Then A − B > implies that B − − A − > . From Lemma 2.12, we can easily conclude the following comparison theorem.
Lemma 2.13.
Let θ and θ be smooth real (1 , -forms on an n -dimensional Hermitian manifold ( X, ω ) . Assume that ( θ − θ ) ∧ ω r > , θ ∧ ω r > and θ ∧ ω r > . Then [ θ , Λ ω ] − − [ θ , Λ ω ] − is positive definite on ( n, r + 1) -forms. Remark 2.14.
The above lemma is a little bit different from and stronger than the classicalresult about the non-increasing of [ θ, Λ ω ] − with respect to θ (see [Dm82, Lemma 3.2] or [Bk18,Propositon 5.2]). Here we know nothing about the comparison information between θ and θ andonly know the comparison data between θ ∧ ω r and θ ∧ ω r . However, from a point of view ofpositive definite transformation we can easily get the above lemma.The following observation is useful to give a direct proof of Proposition 3.3. Lemma 2.15 ([GZ15a, p. 609], or [Bk18, Lemma 9.20]) . If g is an integrable function near ∈ R d , then there exists a sequence x j → in R d such that | g ( x j ) | = o (cid:16) | x j | − d (cid:17) . Proof of Theorem 1.1
We split the proof of Theorem 1.1 in several steps.3.1.
Construction of a smooth extension e f ∞ . Let { W α } be the Stein coordinate patches of X , biholomorphic to polydiscs, and admit the following property: if we denote the correspondingcoordinates by ( z α , w α ) ∈ ∆ × ∆ n − , where w α = (cid:0) w α , . . . , w n − α (cid:1) , then W α ∩ Y = { z α = 0 } .On each W α , we fix some holomorphic σ α ∈ Γ ( W α , K X ⊗ L ) to trivialize K X ⊗ L . Let { θ α } be apartition of unity subordinate to { W α } . For arbitrary α , (1.1),(1.2),(1.3) and − Ricci( ω ) = Θ( K X )along with Theorem 2.3 can conclude that there exists an ambient ¯ ∂ -closed extension f α of f on W α . Set e f ∞ := P α θ α · f α . Then¯ ∂ e f ∞ = ¯ ∂ X α θ α · ( f α − f β ) = X α ¯ ∂θ α · ( f α − f β ) on W β . So ¯ ∂ e f ∞ = 0 along Y . Note that e f ∞ can also be obtained by the same method as [Dm00, (4.4)] or[Kz11, Proof of Lemma 3.1].For the nice regularity of g ε in Subsection 3.3, we need some regularity of ¯ ∂ e f ∞ twisted by s − .On this, Koziarz [Kz11, Lemma 3.1] had provided us with a useful trick to raise the regularity of s − ¯ ∂ e f ∞ and still preserve the ambient extension property of e f ∞ as follows. EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 9 We proceed by induction to get the regularity of s − ¯ ∂ e f ∞ . One says that e f ∞ enjoys the property ( P k ) for k ≥
1, if on each W α (3.1) ¯ ∂ e f ∞ = z α e f α ( z α , w α ) + ¯ z kα d ¯ z α ∧ X | I | = q a I ( w α ) σ α d ¯ w Iα + X | I ′ | = q +1 b I ′ ( w α ) σ α d ¯ w I ′ α + ¯ z k +1 α h α ( z α , w α )for some e f α , h α ∈ E ∞ (cid:0) W α , Λ n,q +1 T ⋆X ⊗ L (cid:1) and a I , b I ′ ∈ E ∞ (cid:0) ∆ n − , C (cid:1) with the increasing multi-indices I, I ′ .Note that for k ≥ ¯ z k z is of class E k − on a complex plane, where z is the complex analyticcoordinate. Then e f ∞ enjoying the property ( P k ) implies that s − ¯ ∂ e f ∞ ∈ E k − (cid:0) X, Λ n,q +1 T ⋆X ⊗ L ⊗ O X ( − Y )) , for k ≥ . In mathematical analysis, one has an observation in the spirit of the Taylor expansion that, asmooth function f of one real variable which vanishes at the origin can be written as f ( x ) = x · g ( x )for a smooth function g . It can be proved by the L’Hospital’s rule to show that g is of class E k for any positive k . Then since ¯ ∂ e f ∞ = 0 along Y ∩ W α = { z α = 0 } and ¯ ∂ e f ∞ is smooth, one canexpand ¯ ∂ e f ∞ along the coordinate function z α such that¯ ∂ e f ∞ = z α · g + ¯ z α · g , where g and g are the corresponding smooth forms. So expanding g along the coordinatefunction z α again similarly can conclude that e f ∞ enjoys the property ( P ).A direct calculation shows¯ ∂ (cid:16) ¯ ∂ e f ∞ (cid:17) = z α ¯ ∂ e f α ( z α , w α ) + k ¯ z k − α d ¯ z α ∧ X | I ′ | = q +1 b I ′ ( w α ) σ α d ¯ w I ′ α + ¯ z kα h ′ α ( z α , w α )for some h ′ α ∈ E ∞ (cid:0) W α , Λ n,q +2 T ⋆X ⊗ L (cid:1) . Then all the above b I ′ vanish identically as ¯ ∂ (cid:16) ¯ ∂ e f ∞ (cid:17) = 0.So we take e f ′∞ = e f ∞ − X α θ α ¯ z k +1 α k + 1 X | I | = q a I ( w α ) σ α d ¯ w Iα to get ¯ ∂ e f ′∞ = ¯ ∂ e f ∞ − X α (cid:18) ¯ z k +1 α k + 1 ¯ ∂θ α + θ α ¯ z kα d ¯ z α (cid:19) ∧ X | I | = q a I ( w α ) σ α d ¯ w Iα − X α θ α ¯ z k +1 α k + 1 ¯ ∂ X | I | = q a I ( w α ) σ α d ¯ w Iα = X α θ α ¯ ∂ e f ∞ − ¯ z kα d ¯ z α ∧ X | I | = q a I ( w α ) σ α d ¯ w Iα + X α ¯ z k +1 α h ′′ α ( z α , w α )= X α z α θ α e f α ( z α , w α ) + X α ¯ z k +1 α (cid:0) θ α h α ( z α , w α ) + h ′′ α ( z α , w α ) (cid:1) for some h ′′ α ∈ E ∞ (cid:0) W α , Λ n,q +1 T ⋆X ⊗ L (cid:1) . Then ¯ ∂ e f ′∞ satisfies (3.1) for k + 1 on each W α , possiblyafter some transformations of coordinates. Then e f ′∞ enjoys the property ( P k +1 ). Apparently, e f ′∞ is still the ambient extension of f while ¯ ∂ e f ′∞ still vanishes along Y .In conclusion, we have proved the following results. Proposition 3.1.
For any k ≥ , there exists a smooth section e f ∞ ∈ E ∞ ( X, Λ n,q T ⋆X ⊗ L ) such that ( a ) e f ∞ is the ambient extension of f , ( b ) ¯ ∂ e f ∞ = 0 at every point of Y , ( c ) s − ¯ ∂ e f ∞ ∈ E k ( X, Λ n,q +1 T ⋆X ⊗ L ⊗ O X ( − Y )) = E k ( X, Λ n,q +1 T ⋆X ⊗ L ⊗ E ∗ ) . From now on, we fix k ≥ Construction of special weights and twist factors.
For the sake of completeness, wewrite the following constructions concretely, which is almost the same as [MV19, § e − ψ := e − ϕ + λ . Next we turn to the choices of the functions A and τ as in [Va08, MV07] and more originally[MV07].Let h ( x ) := 2 − x + log (cid:0) e x − − (cid:1) , v := log | s | and a := γ − δ log (cid:0) | s | + ε (cid:1) , where 0 < δ ≤ x >
1, and γ > a > a ≥ γ − δ log(1 + ε ) ≥ γ − δε due to (1.3). That is to say, for fixed γ >
1, there exists ε > a − ε when ε < ε . It is easy to see that h ′ ( x ) = (2 e x − − − ∈ (0 ,
1) and h ′′ ( x ) = − e x − (2 e x − − < . Define τ := a + h ( a ) and A := (1 + h ′ ( a )) − h ′′ ( a ) . Then A = 2 e a − . Furthermore, (3.2) gives(3.3) τ − (1 + h ′ ( a )) > σ ′ for some constant σ ′ > ε when ε < ε . Moreover, these choices guaranteethat − ∂ ¯ ∂τ − A − ∂τ ∧ ¯ ∂τ = (cid:0) h ′ ( a ) (cid:1) ( − ∂ ¯ ∂a ) . Finally, a straightforward calculation yields − ∂ ¯ ∂a = δ∂ ¯ ∂ log (cid:0) e v + ε (cid:1) = δ | s | | s | + ε ∂ ¯ ∂v + 4 δε ∂ | s | ∧ ¯ ∂ | s | ( | s | + ε ) = − δ | s | | s | + ε ∂ ¯ ∂λ + 4 δε ∂ | s | ∧ ¯ ∂ | s | ( | s | + ε ) , where the last equality follows from the Lelong–Poincar´e equation and | s | [ Y ] = 0 due to Supp[ Y ] = Y . EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 11 A direct calculation together with (1.1), (1.2) and (3.3) yields √− (cid:0) τ ( ∂ ¯ ∂ψ ) − ∂ ¯ ∂τ − A − ∂τ ∧ ¯ ∂τ (cid:1) ∧ ω q = √− (cid:0) τ (cid:0) ∂ ¯ ∂ ( ϕ − λ ) (cid:1) + (cid:0) h ′ ( a ) (cid:1) ( − ∂ ¯ ∂a ) (cid:1) ∧ ω q = (cid:18) τ − (cid:0) h ′ ( a ) (cid:1) (cid:18) | s | | s | + ε (cid:19)(cid:19) · √− ∂ ¯ ∂ ( ϕ − λ ) ∧ ω q + √− (cid:0) h ′ ( a ) (cid:1) | s | | s | + ε (cid:0) ∂ ¯ ∂ ( ϕ − λ ) − δ∂ ¯ ∂λ (cid:1) ∧ ω q + √− (cid:0) h ′ ( a ) (cid:1) (cid:18) δε ∂ | s | ∧ ¯ ∂ | s | ( | s | + ε ) (cid:19) ∧ ω q > √− δ (cid:18) ε ∂ | s | ∧ ¯ ∂ | s | ( | s | + ε ) (cid:19) ∧ ω q + σ ′ σω ∧ ω q > . (3.4)Set B ε := √− (cid:0) τ ( ∂ ¯ ∂ψ ) − ∂ ¯ ∂τ − A − ∂τ ∧ ¯ ∂τ (cid:1) . Then (3.4) tells us that(3.5) B ε ∧ ω q > √− δ (cid:18) ε ∂ | s | ∧ ¯ ∂ | s | ( | s | + ε ) (cid:19) ∧ ω q and(3.6) B ε ∧ ω q ≥ σ ′ σω ∧ ω q hold on X .3.3. Solving twisted ¯ ∂ -Laplace equations with estimates. Recall that in Theorem 1.1, f isa smooth section of the vector bundle (cid:0) K X ⊗ L ⊗ ∧ ,q T ⋆X (cid:1)(cid:12)(cid:12) Y → Y for any 0 ≤ q ≤ n − ∂ ( ι ∗ f ) = 0 and Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω < ∞ . And one has obtained an ambient extension e f ∞ of f in Proposition 3.1. Let 0 < c ≪ θ ∈ E ∞ ([0 , + ∞ )) a cutoff function with values in [0 ,
1] such that θ | [0 ,c ] ≡ θ | [1 , + ∞ ) ≡ | θ ′ | ≤ c . For ε >
0, define g ε := s − ¯ ∂ ( θ (cid:0) ε − | s | (cid:1) e f ∞ ) = s − ¯ ∂ ( θ ( ε − | s | )) ∧ e f ∞ + s − θ ( ε − | s | ) ¯ ∂ e f ∞ . The first term in g ε can be easily written as(3.7) g (1) ε = ¯ ∂ | s | ∧ | s | ( ε − | s | + 1) θ ′ ( ε − | s | ) | s | + ε s − e f ∞ . We also denote the second term s − θ ( ε − | s | ) ¯ ∂ e f ∞ in the above expression of g ε by g (2) ε .From the smoothness of g (1) ε and the regularity information of g (2) ε by Proposition 3.1.(c), weknow that g ε ∈ E k (cid:0) X, Λ n,q +1 T ⋆X ⊗ L ⊗ E (cid:1) . Moreover, g ε is ¯ ∂ -closed outside Y according to thedefinition of g ε . So g ε is ¯ ∂ -closed on X due to the continuity of ¯ ∂g ε .Assume that φ is a smooth plurisubharmonic exhaustion function of the weakly pseudoconvexK¨ahler manifold X . Due to the Sard’s theorem, we can always assume that Ω j := { φ < j } , j =1 , , . . . , satisfy Ω j ⊂⊂ Ω j +1 and lim j →∞ Ω j = [ j ≥ Ω j = X, and every ∂ Ω j is smooth and pseudoconvex. From now on, we will work on Ω j instead of X untilthe end of the penultimate step. For any smooth K X ⊗ L ⊗ E ∗ -valued (0 , q + 1)-form β in the domain of T ∗ and S , we infer theinequality |h β, g ε i| L ≤ h [ B ε , Λ ω ] − g ε , g ε i L h [ B ε , Λ ω ] β, β i L ≤ h [ B ε , Λ ω ] − g ε , g ε i L ( || T ∗ β || + || Sβ || )(3.8)from B ε ∧ ω q >
0, Cauchy–Schwarz inequality and Lemma 2.6. By the variant of Cauchy–Schwarzinequality h α + α , α + α i ≤ h α , α i + h α , α i + c h α , α i + 1 c h α , α i we have(3.9) h [ B ε , Λ ω ] − g ε , g ε i L ≤ (1 + c ) h [ B ε , Λ ω ] − g (1) ε , g (1) ε i L + (1 + 1 /c ) h [ B ε , Λ ω ] − g (2) ε , g (2) ε i L , where c is taken, for convenience, to be the same c as that in the definition of the cutoff function θ . Then according to Lemmata 2.13 and 2.11, (3.5) and (3.7) give the estimate(3.10) (1 + c ) h [ B ε , Λ ω ] − g (1) ε , g (1) ε i L ≤ (1 + c ) Z Ω j ( | s | + ε ) ε δ h ( T ¯ ∂ | s | T ∗ ¯ ∂ | s | ) − g (1) ε , g (1) ε i ω e − ψ dV ω = (1 + c ) Z Ω j ( | s | + ε ) ε δ h T − ∂ | s | g (1) ε , T − ∂ | s | g (1) ε i ω e − ψ dV ω = 1 + cδ Z Ω j ( ε − | s | + 1) θ ′ ( ε − | s | ) ε | e f ∞ | ω e − ϕ dV ω ≤ c )(1 + 2 c ) δ Z Ω j ∩{ ε − | s | ≤ } ε − | e f ∞ | ω e − ϕ dV ω . We denote the right-hand side of the above inequality by C c,ε , whose limit is(3.11) 8 π (1 + c )(1 + 2 c ) δ Z Ω j ∩ Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω as ε → , by the Fubini theorem and Proposition 3.1.(a).It’s turn to estimate the term involving g (2) ε . The relative compactness of Ω j and the lowersemi-continuity of σ imply that ( q + 1) σ ′ σ has a positive lower bound λ j which is independent of ε . Then Lemma 2.10 and (3.6) and the boundedness of θ imply(1 + 1 c ) D [ B ε , Λ ω ] − g (2) ε , g (2) ε E L ≤ (1 + 1 c ) Z Ω j q + 1) σ ′ σ h s − θ (cid:0) ε − | s | (cid:1) ¯ ∂ e f ∞ , s − θ (cid:0) ε − | s | (cid:1) ¯ ∂ e f ∞ i ω e − ψ dV ω ≤ (1 + 1 c ) 1 λ j Z Ω j ∩{ ε − | s | ≤ } | s − ¯ ∂ e f ∞ | ω e − ψ dV ω . As s − ¯ ∂ e f ∞ is of class E k on X and the volume of the integral region of above is ∼ O ( ε ) due tothe relative compactness of Ω j ,(3.12) (1 + 1 c ) D [ B ε , Λ ω ] − g (2) ε , g (2) ε E L ∼ O ( ε ) , which depends on c and j .Denote C c,ε + O ( ε ) by C ε,j,c δ . Then it follows(3.13) |h β, g ε i| L ≤ C ε,j,c δ (cid:16) k T ∗ β k + k Sβ k (cid:17) from (3.8),(3.9),(3.10) and (3.12).Set (cid:3) := T T ∗ + S ∗ S . Then we will solve the equation (cid:3) V ε = g ε in the standard way on the basis of the above estimate (3.13). EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 13 The Dirichlet semi-norm is defined as k β k H := k T ∗ β k + k Sβ k for any smooth K X ⊗ L ⊗ E ∗ -valued (0 , q + 1)-form in the domain of T ∗ and S . Since σ is positivelower semi-continuous and σ ′ >
0, the original norm is dominated by the Dirichlet norm multipliedby some constant on the relatively compact domain Ω j due to Lemmata 2.6 and 2.10 and the strictpositivity (3.6) of B ε on X . So the Dirichlet semi-norm is a norm on Ω j .Let H denote the Hilbert space closure of the set of all smooth K X ⊗ L ⊗ E ∗ -valued (0 , q + 1)-forms in the domain of T ∗ and S . Consider the functional ℓ : H → C , defined by ℓ ( β ) := ( β, g ε ) = Z Ω j h β, g ε i ω e − ϕ + λ dV ω . Recall that the original L -norm is dominated by the Dirichlet norm for smooth K X ⊗ L ⊗ E ∗ -valued (0 , q + 1)-forms in the domain of T ∗ and S . So it is easy to see that (3.13) holds for everyelement of H by taking limits with respect to the graph norm by the density of D T ∗ ∩ D S ∩ E ∞ ( ¯Ω)in D T ∗ ∩ D S . So ℓ is a bounded linear functional on H , and the H ∗ -norm of ℓ is no more than δ − C ε,j,c . Then there exists V ε,j ∈ H (here we omit the subscript c for V ε,j due to that the c << k V ε,j k H = k ℓ k H ∗ ≤ δ − C ε,j,c and ( g ε , β ) = ( T ∗ V ε,j , T ∗ β ) + ( SV ε,j , Sβ )by the Riesz representation theorem. The latter defines the meaning of (cid:3) V ε,j = g ε in the weak sense, as β goes through all the smooth compactly supported forms. Moreover, Lemma2.7 and g ε ∈ E k tell us that (cid:3) V ε,j = g ε in the usual sense and that V ε,j is of class E k +2 on Ω j . As S ◦ T = 0 and Sg ε = √ τ ¯ ∂g ε = 0, we find that0 = ( S (cid:3) V ε,j , SV ε,j ) = k S ∗ SV ε,j k and thus k SV ε,j k = ( S ∗ SV ε,j , V ε,j ) = 0.Now in conclusion we obtain an L ⊗ E ∗ -valued ( n, q + 1)-form V ε,,j of class E k +2 such that (cid:3) V ε,j = g ε and k V ε,j k H ≤ C ε,j,c δ . Set v ε,j := T ∗ V ε,j . It follows that T v ε,j = (cid:3) V ε,j = g ε . Then we have proved the following theorem.
Theorem 3.2.
The equation
T v ε,j = g ε has a solution v ε,j ∈ E k +1 (Ω j , L ⊗ E ∗ ⊗ Λ n,q T ⋆X ) satisfyingthe L -estimate Z Ω j | v ε,j | ω e − ϕ + λ dV ω ≤ C ε,j,c δ . Construction of an E k +1 extension on Ω j with uniform L bound. Set u ε,j := θ (cid:0) ε − | s | (cid:1) e f ∞ − √ τ + Av ε,j ⊗ s. Then u ε,j ∈ E k +1 (Ω j , L ⊗ Λ n,q T ⋆X ) , u ε,j | Y = f and ¯ ∂u ε,j = s ⊗ ( g ε − T v ε,j ) = 0 . Since θ (cid:0) ε − | s | (cid:1) is bounded and supported on a set whose measure tends to 0 with ε → j is relatively compact, there exists ε j > ε ≤ ε j , one has Z Ω j | u ε,j | ω e − ϕ dV ω = (1 + o (1)) Z Ω j ( τ + A ) | v ε,j | ω | s | e − ϕ dV ω = (1 + o (1)) Z Ω j ( e v ( τ + A )) | v ε,j | ω e − ϕ + λ dV ω , where the infinitesimaali above is as ε → e v ( τ + A ) = | s | (2 e a − + 2 + log(2 e a − − ≤ | s | · e a − ≤ e γ − , where 0 < δ ≤
1. It follows that for some sufficiently small ε j , the estimate(3.14) Z Ω j | u ε,j | ω e − ϕ dV ω ≤ (1 + o (1))4 e γ − C ε,j,c δ ≤ Cδ Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω holds for some universal
C >
0, as soon as 0 < ε ≤ ε j , due to (3.11) and (3.12). Thus for any such ε > , u ε,j gives the extension with the desired L estimate in Ω j . Write u j := u ε j ,j . In conclusion, for each j we have found an L ⊗ K X -valued (0 , q )-form u j of class E k +1 on Ω j suchthat(3.15) ¯ ∂u j = 0 , u j | Y ∩ Ω j = f, and Z Ω j | u j | ω e − ϕ dV ω ≤ Cδ Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω . In particular, the right-hand side is independent of j .3.5. A kind of minimization problem for E k +1 extensions. First we give an overview of thissubsection. From the above process we conclude that as j → ∞ , ε is necessarily constrained to besmaller and smaller. In accordance with the previous practice in L extension theory, we wouldlike to take the limit as ε → τ as ε →
0. That is to say, the twisted¯ ∂ -operators T and S become singular as ε →
0, and thereby create a loss of control on the constant C in the statement Theorem 1.1. Furthermore, as the weak limit as ε → f , we must look for better ambient extensions on Ω j to ensure that the weak limit ofthe sequence of ambient extensions is a smooth extension.Here we adopt the method of McNeal–Varolin [MV19, § τ .To attack this problem, we define a subspace which is introduced in [MV19, § V q (Ω j ) denote the Hilbert space closure of the set of all smooth K X ⊗ L ⊗ E ∗ -valued¯ ∂ -closed (0 , q )-forms β on Ω j satisfying Z Ω j | β | ω e − ϕ + λ dV ω < + ∞ , and then V q (Ω j ) consists precisely of all those forms in L (cid:0) ω, e − ϕ + λ (cid:1) that are ¯ ∂ -closed in theweak sense. Let B q (Ω j ) denote the closed unit ball in V q (Ω j ), i.e., β ∈ B q (Ω j ) ⇐⇒ β ∈ V q (Ω j ) and Z Ω j | β | ω e − ϕ + λ dV ω ≤ . Let us define the affine ball B j := u j + s B q (Ω j ) := (cid:8) u j + sβ ; β ∈ B q (Ω j ) (cid:9) ⊂ L (cid:0) ω, e − ϕ (cid:1) . Now there are three problems to solve:(i) Every E k +1 form in B j is an ambient extension of f .(ii) There exists a minimizer in B j .(iii) The minimizer is smooth. EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 15 Note that we can of course adopt the method of McNeal–Varolin [MV19, (12),(13)] to grasp thedata along Y of any continuous bundle-valued form by taking wedge with the current of integration[ Y ] due to Supp[ Y ] = Y . Here we use a simpler direct method but not the current method toverify the ambient extension relationship. Proposition 3.3.
Every continuous form in B j is an ambient extension of f .Proof. Note that verifying the extension or restriction is essentially a local problem. That is tosay, it suffices to prove that ( u j + sβ )( x ) = f ( x ) for any x ∈ Ω j ∩ Y and any continuous u j + sβ in B j . For any fixed x ∈ Ω j ∩ Y , take any local coordinate ( U, z , z , . . . , z n ) of X around x . Fixholomorphic local frames σ of K X ⊗ L and θ of E over U , respectively, such that s = z ⊗ θ . Notethat Y ∩ U = { z = 0 } , β | U = X | K | = q λ K d ¯ z K ⊗ σ ⊗ θ ∗ , where the multi-index K is increasing. Then it suffices to provelim z → z λ K ( z , z , . . . , z n ) = 0 for any increasing multi-index K, since u j is the ambient extension of f .According to the definition of B q , we know that β and thereby λ K ( z , z , . . . , z n ) is L integrable(possibly after shrinking the domain). Then, by the integrability part of Fubini theorem (e.g.[Ru87, 8.8.(c) Theorem]), | λ K ( z , z , . . . , z n ) | is L integrable with respect to z for ( z , . . . , z n )a.e.. Due to Lemma 2.15, there exists a sequence { z ,ν } such that λ K ( z ,ν , z , . . . , z n ) ∼ o ( | z ,ν | − )for any fixed ( z , . . . , z n ) a.e.. Thenlim z → z λ K ( z , z , . . . , z n ) = 0for ( z , . . . , z n ) a.e. due to the continuity of z λ K ( z , z , . . . , z n ). The continuity of z λ K ( z , z , . . . , z n )again implies that lim z → z λ K ( z , z , . . . , z n ) = 0for any ( z , . . . , z n ). (cid:3) For the sake of completeness, we will give the following two propositions, which can be found in[MV19, § Proposition 3.4.
There exists an element of minimal norm U j ∈ B j and U j is orthogonal to s B q (Ω j ) .Proof. By the Fatou’s lemma and Lemma 2.8, B j is a closed subset of the Hilbert space L ( ω, e − ϕ ).Furthermore, it is apparently convex. So B j has an element of minimal norm U j .Suppose that there exists β ∈ s B q (Ω j ) such that ( U j , β ) = c = 0. Consider the form α := cβ k β k ∈ s V q and set e U j = U j − α . Then e U j ∈ B j , but (cid:13)(cid:13)(cid:13) e U j (cid:13)(cid:13)(cid:13) = k U j k − | c | k β k . This contradicts the minimalityof k U j k . (cid:3) Proposition 3.5. ¯ ∂ ∗ U j = 0 in the sense of currents.Proof. For any α ∈ E ∞ (Ω j − Y, L ∗ ⊗ Λ n,q − T ⋆X ), s − ¯ ∂α ∈ B q (Ω j ) possibly after shrinking α bysome constant due to the smoothness of s − on Ω j − Y . By Proposition 3.4,( U j , ¯ ∂α ) = ( U j , ss − ¯ ∂α ) = 0 . So ¯ ∂ ∗ U j = 0, in the sense of currents, on Ω j − Y . The minimality of U j implies easily U j ∈ L loc .An adaptation of the proof of Lemma 2.8 (=[Dm82, Lemme 6.9]) to the ¯ ∂ ∗ -equation or a direct application of [Bk18, Proposition 4.8] yields that the above equation can extend across Y , i.e.,¯ ∂ ∗ U j = 0 on Ω j . (cid:3) It follows from Proposition 3.5 that (cid:3) U j = 0in the sense of currents, where (cid:3) = ¯ ∂ ¯ ∂ ∗ + ¯ ∂ ∗ ¯ ∂ denotes the untwisted ¯ ∂ -Laplacian unrelated with τ . So it follows that U j is smooth on Ω j from the ellipticity of the Laplacian (cid:3) due to thesmoothness of the metric e − ϕ . Thus, Proposition 3.3 gives that U j is an extension of f . Moreover,by the estimate (3.15) for u j and the minimality of U j , we have(3.16) Z Ω j | U j | ω e − ϕ dV ω ≤ Cδ Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω . The end of the proof of Theorem 1.1.
Now we construct the desired extension. Sincethere is no something like the Montel property for holomorphic objects now, we cannot derive anypointwise convergence information. Then we use the same method as [MV19, § { U j } naturally satisfying the current equations [MV19, (16)(17)]of the ambient extension. For the sake of completeness, we show the specific procedure (see [MV19,p. 439, § U j is an ambient extension of f , it satisfies the distribution equations(3.17) U j ∧ √− π ∂ ¯ ∂ log | s | = f ∧ √− π ∂ ¯ ∂ log | s | and(3.18) ( ∂∂ ¯ s y U j ) ∧ √− π ∂ ¯ ∂ log | s | = ( ∂∂ ¯ s y f ) ∧ √− π ∂ ¯ ∂ log | s | , which have been derived in [MV19, (12)(13)] due essentially to the idea that we can grasp the dataalong Y of any continuous bundle-valued form by taking wedge with the current of integration[ Y ] since Supp[ Y ] = Y . Note that | s | here is considered as the holomorphic function with respectto some corresponding local trivialization. Of course, f is only defined on Y . However, we canextend it smoothly in an arbitrary way to the object of the same type on Ω j . Then the supportof [ Y ] forces the above current equation to be well defined on Ω j which does not depend on thechoice of the extension of f .According to (3.16), Alaoglu’s Theorem shows that (possibly a subsequence of) { U j } convergesweakly to some U on X with the L estimate(3.19) Z X | U | ω e − ϕ dV ω ≤ lim inf || U j || L ≤ Cδ Z Y | f | ω e − ϕ | ds | ω e − λ dV Y,ω . Then (cid:3) U = 0 due to that (cid:3) U j = 0, and thus U is smooth.Now we claim that U is our desired extension, i.e., U satisfies the distribution equations (3.17)and (3.18). Note that U j ∧ √− π ∂ ¯ ∂ log | s | = − ¯ ∂ ( U j ∧ √− π dss )and ( ∂∂ ¯ s y U j ) ∧ √− π ∂ ¯ ∂ log | s | = ± (cid:18) ¯ ∂ ( ∂∂ ¯ s y U j ) ∧ dss − ¯ ∂ (( ∂∂ ¯ s y U j ) ∧ dss ) (cid:19) . Then it suffices to show that(3.20) U j ∧ √− π dss , ( ∂∂ ¯ s y U j ) ∧ dss and ¯ ∂ ( ∂∂ ¯ s y U j ) ∧ dss are locally integrable uniformly in j , for showing that U satisfies equations (3.17) and (3.18). EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 17 The ellipticity of (cid:3) , the Harnack’s inequality and (3.16) tell us that for every compact set K ,there exist some j = j ( K ) > C K > (cid:13)(cid:13)(cid:13) | U j | e − ϕ/ (cid:13)(cid:13)(cid:13) L ∞ ( K ) ≤ C K for all j ≥ j . The index j is large enough to make sure that U j is defined on Ω j . It follows thatfor any compact subset K ⊂⊂ X and j ≥ j , Z K (cid:12)(cid:12)(cid:12)(cid:12) U j ∧ dss (cid:12)(cid:12)(cid:12)(cid:12) e − ϕ/ dV ω ≤ C K Z K (cid:12)(cid:12)(cid:12)(cid:12) dss (cid:12)(cid:12)(cid:12)(cid:12) dV ω , and thus the L -norm of U j ∧ dss on K is bounded by a constant independent of j . The similarargument shows that the other terms in (3.20) are locally integrable. Now we can take limits inthe distribution equations (3.17) and (3.18) to conclude that U also satisfies these equations. Thus U is our desired extension with the estimate (3.19). Remark 3.6.
Note that we can make a better control on our extension U . In fact, from (3.13),Theorem 3.2 and (3.14), we know that the universal constant C in (3.15) and (3.19) depends on c . If we take a smaller c in (3.9), we can get a smaller C ε,j,c in (3.13) according to (3.10),(3.11)and (3.12). Then we can get a smaller universal constant C in (3.14) and thereby a better controlin (3.19) for our extension U . However, the constant in (3.19) is not a sharp one and then takinga smaller c may not give more useful information. Appendix A. Alternative proof of Theorem 1.3
Theorem 1.1 can easily imply Theorem 1.3 as stated in Section 1. Now we present a sketch ofa direct proof of Theorem 1.3.Just as in Subsection 3.1, we can glue local extensions from Theorem 2.5 and use the samemethod of raising the regularity of s − ¯ ∂ e f ∞ , to obtain: Proposition A.1 ([Kz11, Lemma 3.1]) . For any k ≥ , there exists a smooth section e f ∞ ∈ E ∞ ( X, Λ n,q T ⋆X ⊗ L ) such that ( a ) e f ∞ is the intrinsic extension of u , ( b ) | e f ∞ | ω,L = | u | ω,L at every point of Y , ( c ) ¯ ∂ e f ∞ = 0 at every point of Y , ( d ) s − ¯ ∂ e f ∞ ∈ E k ( X, Λ n,q +1 T ⋆X ⊗ L ⊗ O X ( − Y )) . From now on, we fix k ≥ j , wecan obtain an L ⊗ K X -valued (0 , q )-form u j of class E k +1 on Ω j such that¯ ∂u j = 0 , ι ∗ u j = u, and Z Ω j | u j | ω e − ϕ dV ω ≤ Cδ Z Y | u | ω e − ϕ | ds | ω e − λ dV Y,ω . In particular, the right-hand side is independent of j .Let V q (Ω j ) denote the Hilbert space closure of the set of all smooth K X ⊗ L ⊗ E ∗ -valued¯ ∂ -closed (0 , q )-forms β on Ω j satisfying Z Ω j | β | ω e − ϕ + λ dV ω < + ∞ , and then V q (Ω j ) consists precisely of all those forms in L (cid:0) ω, e − ϕ + λ (cid:1) that are ¯ ∂ -closed in theweak sense. Let B q (Ω j ) denote the closed unit ball in V q (Ω j ), i.e., β ∈ B q (Ω j ) ⇐⇒ β ∈ V q (Ω j ) and Z Ω j | β | ω e − ϕ + λ dV ω ≤ . Let us define the affine ball B j := u j + s B q (Ω j ) := (cid:8) u j + sβ ; β ∈ B q (Ω j ) (cid:9) ⊂ L (cid:0) ω, e − ϕ (cid:1) . Then we have the following proposition.
Proposition A.2.
Every continuous form in B j is an intrinsic extension of u .Proof. The proof is quite similar with that of Proposition 3.3. It suffices to prove that ι ∗ ( u j + sβ )( x ) = u ( x ) for any x ∈ Ω j ∩ Y and any continuous u j + sβ in B j . For any fixed x ∈ Ω j ∩ Y , takeany local coordinate ( U, z , z , . . . , z n ) of X around x . Fix holomorphic local frames σ of K X ⊗ L and θ of E over U , respectively, such that s = z ⊗ θ . Note that Y ∩ U = { z = 0 } , β | U = X | J | = q − , / ∈ J λ (1) J d ¯ z ∧ d ¯ z J ⊗ σ ⊗ θ ∗ + X | K | = q, / ∈ K λ (2) K d ¯ z K ⊗ σ ⊗ θ ∗ , where the multi-indices J and K are increasing.Then ι ∗ β = X | K | = q, / ∈ K λ (2) K (0 , z , . . . , z n ) d ¯ z K ⊗ σ ◦ ι ⊗ θ ∗ ◦ ι. So it suffices to prove lim z → z λ (2) K ( z , z , . . . , z n ) = 0 for any multi-index K, since u j is the intrinsic extension of u . The leftover argument proceeds just with λ K ( z , z , . . . , z n )in the proof of of Proposition 3.3 replaced by λ (2) K ( z , z , . . . , z n ) here. (cid:3) Remark A.3.
The current equation method in the proof of [MV19, Proposition 4.4] can also workto prove Proposition A.2. In fact, the local expressions of β and ι ∗ β in the above proof can beused to deduce a current equation characterizing the intrinsic restriction due to that Supp[ Y ] = Y .That is, a continuous K X ⊗ L -valued (0 , q )-form U on X is the intrinsic extension of a smooth K X ⊗ L | Y -valued (0 , q )-form u on Y if and only if locally(A.1) U ∧ √− ∂ ¯ ∂ log | s | = u ∧ √− ∂ ¯ ∂ log | s | as K X ⊗ L -valued (1 , q + 1)-currents of order 0. Here we adopt the definition of order of currentsin [Dm12, § | s | here is considered as the holomorphic function withrespect to the corresponding local trivialization. Despite u is only defined on Y , the above currentequation is well defined on X due to the same reason as (3.17) and (3.18). The proof of (A.1) issimilar to the proof of [MV19, (12),(13)] characterizing the ambient restriction. Then we can alsouse this characterization (A.1) to verify Proposition A.2 as the proof of [MV19, Proposition 4.4].Using Proposition A.2 and the similar results to Propositions 3.4 and 3.5, we can obtain anelement U j of minimal norm in B j satisfying (cid:3) U j = 0. Then U j is smooth and thereby theintrinsic extension of u on Ω j by Proposition A.2. Of course, U j satisfies (A.1). At last, by thesame method of extracting weak limits of some subsequence of U j as j → ∞ as in Subsection 3.6,we can obtain our desired extension. Acknowledgement
Both authors would like to thank Professor Dror Varolin, for explaining us the details on theirarticle [MV19], and pointing out an important mistake in our draft, and giving many useful sugges-tions, especially on Examples 1.2 and 1.6. Furthermore, we also thank Professors Bo Berndtsson,Vincent Koziarz for explaining us details on their articles [Bn12, Kz11], respectively, and Profes-sor Langfeng Zhu for his help in learning L theory, and also Professors Chen-Yu Chi and J.-P.Demailly for their two discussions during the second author’s visit to Institut Fourier, Universit´eGrenoble Alpes in April 2019 on this topic of extension theorem. Moreover, the first author wouldlike to thank Runze Zhang for many discussions on L extension theory, as well as Yongpan Zouand Houwang Liu for many discussions on a seminar about L methods. EXTENSION OF ¯ ∂ -CLOSED FORMS ON WEAKLY PSEUDOCONVEX K ¨AHLER MANIFOLDS 19 References [BPZ15] L. Baracco, S. Pinton, G. Zampieri,
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E-mail address : [email protected] Sheng Rao, School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People’sRepublic of China; Universit´e de Grenoble-Alpes, Institut Fourier (Math´ematiques) UMR 5582 duC.N.R.S., 100 rue des Maths, 38610 Gi`eres, France
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