On the Nakano vanishing theorem

Nadel-Nakano vanishing theorems of vector bundles with singular Hermitian metrics

We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generaliz…

Positivity and vanishing theorems for ample vector bundles

In this paper, we study the Nakano-positivity and dual-Nakano-positivity of certain adjoint vector bundles associated to ample vector bundles. As applications, we get new vanishing theorems about ample vector bundles. For example, we prove that if $E$ is an ample vector bundle over a compact Kähler manifold $X$, $S^kE\ts \det E$ is both Nakano-positive and dual-Nakano-positive for any $k\geq 0$. Moreover, $H^{n,q}(X,S^kE\ts \det E)=H^{q,n}(X,S^kE\ts \det E)=0$ for any $q\geq 1$. In particular, if $(E,h)$ is a Griffiths-positive vector bundle, the naturally induced Hermitian vector bundle $(S^k…

Vanishing Theorem for the de Rham Complex of Unitary Local System

We will prove a Kodaira-Nakano type of vanishing theorem for the logarithmic de Rham complex of unitary local system. We will then study the weight filtration on the logarithmic de Rham complex, and prove a stronger statement for the associated graded complex.

态射的非线性霍奇对应

Nonlinear Bochner-Kodaira-Nakano identity and nonlinear Hodge correspondence for morphisms

研究非线性调和丛中希格斯截面与平坦截面关系，证明一般情况下二者等价需次数障碍消失，还通过将态射解释为图子纤维化，把结果扩展到子纤维化和态射。

Some Kollar-Enoki type injectivity and Nadel type vanishing theorems on compact Kahler manifolds

In this paper we will first show some Kollar-Enoki type injectivity theorems on compact Kahler manifolds, by using the Hodge theory, the Bochner- Kodaira-Nakano identity and the analytic method provided by O. Fujino and S. Matsumura in [15, 25, 36, 39]. We have some straightforward corollaries. In particular, we will show that our main injectivity theorem implies several Nadel type vanishing theorems on smooth projective manifolds. Second, by applying the transcendental method, especially the Demailly-Peternell-Schneider equisingular approximation theorem and the Hormander L2 estimates, we wil…

$L^2$ estimates and vanishing theorems for holomorphic vector bundles equipped with singular Hermitian metrics

We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

通过弱一般 Nakano 消没定理研究 L²-Betti 数与规范化 Hodge 数的收敛性

L^2-Betti Numbers and Convergence of Normalized Hodge Numbers via the Weak Generic Nakano Vanishing Theorem

本文利用弱一般 Nakano 消没定理，研究 Abel 覆盖塔中规范化 Hodge 数与多重亏格的增长及收敛，并计算相关非正则簇的 L²-Betti 数，推广 Kollár 的可乘性结果。

特征p>0下的Hodge--Tate分裂与Akizuki--Nakano消失定理

Hodge--Tate splitting and Akizuki--Nakano vanishing in positive characteristic

本文定义正特征下的Hodge--Tate分裂，证明其与de Rham复形分解等价，建立相关准则并构造新簇例子，还将其应用于混合特征下的消失定理证明。

Effective vanishing theorems for ample and globally generated vector bundles

By proving an integral formula of the curvature tensor of $E\ts \det E$, we observe that the curvature tensor of $E\ts \det E$ is very similar to that of a line bundle and obtain certain new Kodaira-Akizuki-Nakano type vanishing theorems for vector bundles. As special cases, we deduce vanishing theorems for ample, nef and globally generated vector bundles by analytic method instead of the Leray-Borel-Le Potier spectral sequence.

通过 Du Bois 复形实现奇异簇的广义消没

Generic Vanishing for Singular Varieties via Du Bois complexes

本文通过 Du Bois 复形证明了奇异簇的广义消没定理，推广了 Green-Lazarsfeld 和 Nakano 型定理，并解释了 Hacon-Kovács 提出的反例。

Fujiki类C的通用消失定理

Generic vanishing theorem for Fujiki class C

本文将Nakano型通用消失定理从紧Kähler流形推广至Fujiki类C流形，使其适用于光滑紧合复代数簇。

正特征Fano三维流形的可提升性与消失定理I

Liftability and vanishing theorems for Fano threefolds in positive characteristic I

本文证明正特征光滑Fano三维流形可提升到Witt向量环，并满足Akizuki-Nakano消失、Hodge到de Rham谱序列E1退化及晶体上同调无挠性，针对$|-K_X||$非常丰沛且Picard群由$\omega_X$生成的情形。

Sommese Vanishing for non-compact manifolds

The Kodaira-Nakano Vanishing Theorem has been generalized to the relative setting by A. Sommese. We prove a version of this theorem for non-compact manifolds. As an apllication, we prove that the cohomology of a fiber of a symplectic contraction is trivial in odd degrees and pure Hodge-Tate in even degrees.

A remark on the decomposition theorem for direct images of canonical sheaves tensorized with semipositive vector bundles

The purpose of this short note is to give a remark on the decomposition theorem for direct images of canonical sheaves tensorized with Nakano semipositive vector bundles. Although our result is a direct consequence of Takegoshi's work, it was not stated explicitly in his article. Here we give the precise statement and the proof.

Bogomolov-Sommese vanishing on log canonical pairs

Let (X, D) be a projective log canonical pair. We show that for any natural number p, the sheaf (Omega_X^p(log D))^** of reflexive logarithmic p-forms does not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds p. This generalizes a classical theorem of Bogomolov and Sommese. In fact, we prove a more general version of this result which also deals with the orbifoldes géométriques introduced by Campana. The main ingredients to the proof are the extension theorem of Greb-Kebekus-Kovács-Peternell, a new version of the Negativity lemma, the Minimal Model Program, and a resid…

Vanishing theorems for locally conformal hyperkaehler manifolds

Let M be a compact locally conformal hyperkaehler manifold. We prove a version of Kodaira-Nakano vanishing theorem for M. This is used to show that M admits no holomorphic differential forms, and the cohomology of the structure sheaf $H^i(O_M)$ vanishes for i>1. We also prove that the first Betti number of M is 1. This leads to a structure theorem for locally conformally hyperkaehler manifolds, describing them in terms of 3-Sasakian geometry. Similar results are proven for compact Einstein-Weyl locally conformal Kaehler manifolds.

正特征域上 Fano 三维簇的提升性与消失定理 II

Liftability and vanishing theorems for Fano threefolds in positive characteristic II

本文研究正特征域上光滑 Fano 三维簇的提升性与上同调性质，证明当 Picard 数或 Fano 指数大于 1 时其具有拟 F-分裂性，从而在除特定极丰沛情形外建立 Akizuki–Nakano 消失、Hodge–de Rham 谱序列 E1-退化及晶体上同调无挠性。

A vanishing theorem of Koll'ar-Ohsawa type

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi. As an application, we prove a vanishing theorem of Koll'ar-Ohsawa type by combining the L^2-method for the dbar-equation.

Vanishing theorems for Mather-Jacobian multiplier ideals on a Gorenstein projective variety

In this paper, we establish several results related to vanishing theorems for Mather-Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffith-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather-Jacobian multiplier ideals.

Kollár--Nadel type vanishing theorem

We prove an analytic generalization of Kollár's vanishing theorem, which contains the Nadel vanishing theorem as a special case.

Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs

We prove the Kodaira vanishing theorem for log-canonical and semi-log-canonical pairs. We also give a relative vanishing theorem of Reid--Fukuda type for semi-log-canonical pairs.

A remark on Kovács' vanishing theorem

We give an alternative proof of Kovács' vanishing theorem. Our proof is based on the standard arguments of the minimal model theory. We do not need the notion of Du Bois pairs. We reduce Kovács' vanishing theorem to the well-known relative Kawamata--Viehweg--Nadel vanishing theorem.

DB pairs and vanishing theorems

The main purpose of this article is to define the notion of DuBois singularities for pairs and proving a vanishing theorem using this new notion. The main vanishing theorem specializes to a new vanishing theorem for resolutions of log canonial singularities.

Variation on Artin's vanishing theorem

We give a proof of Artin's vanishing theorem in characteristic zero, based on Deligne's Riemann-Hilbert correspondence. Just as a curiosity.

Multiplication maps and vanishing theorems for toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.

Cohomology of rational forms and a vanishing theorem on toric varieties

We explicitly describe cohomology of the sheaf of differential forms with poles along a semiample divisor on a complete simplicial toric variety. As an application, we obtain a new vanishing theorem which is an analogue of the Bott-Steenbrink-Danilov vanishing theorem.

Non-vanishing theorem for generalized log canonical pairs with a polarization

We prove that the non-vanishing conjecture holds for generalized lc pairs with a polarization.

Deformations of the trivial line bundle and vanishing theorems

This paper reproves a general form of the Green-Lazarsfeld 'generic vanishing' theorem and more recent strengthenings, as well as giving some new applications.

Vanishing Theorems on Toric Varieties, with Erratum to a Former Paper

In this paper, we clarify the mistakes made in the former article entitled "Vanishing Theorems on Toric Varieties in Positive Characteristic". On the other hand, we use the positive characteristic method to reprove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties over a field of arbitrary characteristic, where concerned Weil divisors are torus invariant.

A vanishing theorem for log canonical pairs after De Fernex-Ein

We extend the main vanishing theorem in a paper of de Fernex and Ein to singular varieties without assuming locally complete intersection.

Non-vanishing theorem for lc pairs admitting a Calabi--Yau pair

We prove the non-vanishing conjecture for lc pairs $(X,Δ)$ when $X$ is of Calabi--Yau type.

Vanishing theorems for parabolic Higgs bundles

This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.

On Saito's vanishing theorem

We reprove Saito's vanishing theorem for mixed Hodge modules by the method of Esnault and Viehweg. The main idea is to exploit the strictness of direct images on certain branched coverings.

Differential equations as embedding obstructions and vanishing theorems

We generalize a vanishing theorem for the cohomology of symmetric powers of the cotangent bundle of subvarieties of projective space due to Schneider. From this we deduce new vanishing results for Green-Griffiths jet differential bundles, generalizing results of Diverio and Pacienza-Rousseau.

On the generic vanishing theorem of Cartier modules

We generalize the Generic Vanishing theorem by Hacon and Patakfalvi in the spirit of Pareschi and Popa. We give several examples illustrating the pathologies appearing in the positive characteristic setting.

On vanishing theorems for local systems associated to Laurent polynomials

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand-Kapranov-Zelevinsky into various directions.

Vanishing and injectivity theorems for Hodge modules

We prove a surjectivity theorem for the Deligne canonical extension of a polarizable variation of Hodge structure with quasi-unipotent monodromy at infinity along the lines of Esnault-Viehweg. We deduce from it several injectivity theorems and vanishing theorems for pure Hodge modules. We also give an inductive proof of Kawamata-Viehweg vanishing for the lowest graded piece of the Hodge filtration of a pure Hodge module using mixed Hodge modules of nearby cycles.

A class of sheaves satisfying Kodaira's vanishing theorem

This paper contains a Kawamata-Viehweg-Kollár type vanishing theorem for vector bundles. In order to formulate and prove this cleanly, we introduce a class of sheaves that automatically satisfies a vanishing theorem. This is obtained by closing up the subclass of sheaves of adjoint type under extensions, direct images, and taking direct summands.

A sharp vanishing theorem for line bundles on K3 or Enriques surfaces

Let $L$ be a line bundle on a K3 or Enriques surface. We give a vanishing theorem for $H^1(L)$ that, unlike most vanishing theorems, gives necessary and sufficient geometrical conditions for the vanishing. This result is essential in our study of Brill-Noether theory of curves on Enriques surfaces (reference [KL1]) and of Enriques-Fano threefolds (reference [KLM]).

Kawamata-Viehweg Vanishing Theorem for del Pezzo Surfaces over imperfect fields in characteristic $p>3$

In this article we prove that the Kawamata-Viehweg vanishing theorem holds for regular del Pezzo surfaces over imperfect ground fields of characteristic $p>3$.

关于正特征面上消失定理和博戈莫洛夫不等式

On Vanishing Theorems and Bogomolov's Inequality on Surfaces in Positive Characteristic

本文研究正特征面上博戈莫洛夫不等式与miyaoka-sakai定理的等价性，并推导出相关消失定理及fujita猜想的reider类型结果。

A vanishing theorem for threefolds in characteristic $p>5$ and applications

In this note we show a Kawamata-Viehweg vanishing theorem for pl-contractions on threefolds in characteristic $p>5$. We deduce several applications for klt threefolds: the vanishing of higher direct images of structure sheaves of Mori fibre spaces, Kollár's result on local vanishing and the rationality of the singularities of the base of a conic bundle.

Non-vanishing theorem for log canonical pairs

We obtain a correct generalization of Shokurov's non-vanishing theorem for log canonical pairs. It implies the base point free theorem for log canonical pairs. We also prove the rationality theorem for log canonical pairs. As a corollary, we obtain the cone theorem for log canonical pairs. We do not need Ambro's theory of quasi-log varieties.

广义对的消灭定理

Vanishing theorems for generalized pairs

本文为 lc 广义对建立了 Kodaira 和 Kawamata-Viehweg 消灭定理，并由此给出了基点自由定理的新证明，进而在不依赖 Kollár 粘合理论的情况下证明了 lc 广义对的收缩定理。

光滑超曲面上Hodge理想的消失定理

Vanishing Theorem for Hodge Ideals on smooth hypersurfaces

本文通过Koszul型消解证明光滑超曲面的弱Bott消失定理，并由此建立有效Cartier除子对应Hodge理想的消失定理，推广了Mustata和Popa的已有工作。

Vanishing theorems for threefolds in characteristic $p>5$

We prove Grauert-Riemenschneider-type vanishing theorems for excellent dlt threefolds pairs whose closed points have perfect residue fields of positive characteristic $p>5$. Then we discuss applications to dlt singularities and to Mori fiber spaces of threefolds.

On semipositivity, injectivity and vanishing theorems

This is a survey article on the recent developments of semipositivity, injectivity, and vanishing theorems for higher-dimensional complex projective varieties.

Vanishing Theorems on Toric Varieties in Positive Characteristic

We use the liftability of the relative Frobenius morphism of toric varieties and the strong liftability of toric varieties to prove the Bott vanishing theorem, the degeneration of the Hodge to de Rham spectral sequence and the Kawamata-Viehweg vanishing theorem for log pairs on toric varieties in positive characteristic. These results generalize those results of Danilov, Buch-Thomsen-Lauritzen-Mehta, Mustata and Fujino to the case where concerned Weil divisors are not necessarily torus invariant.

Vanishing theorem for the cohomology of line bundles on Bott-Samelson varieties

We use the toric degeneration of Bott-Samelson varieties and the description of cohomolgy of line bundles on toric varieties to deduce vanishings results for the cohomology of lines bundles on Bott-Samelson varieties.

A local version of the Kawamata-Viehweg vanishing theorem

We give a criterion for a divisorial sheaf on a log terminal variety to be Cohen-Macaulay. The log canonical case and applications to moduli are also considered.

Vanishing theorems

We prove some injectivity, torsion-free, and vanishing theorems for simple normal crossing pairs. Our results heavily depend on the theory of mixed Hodge structures on compact support cohomology groups. We also treat several basic properties of semi divisorial log terminal pairs.

Fourier transforms, generic vanishing theorems and polarizations of abelian varieties

The purpose of this paper is to give two applications of Fourier transforms and generic vanishing theorems: - we give a cohomological characterization of principal polarizations - we prove that if $X$ an abelian variety and $Θ$ a polarization of type $(1,...,1,2)$, then a general pair $(X,Θ)$ is log canonical

Vanishing Theorems for constructible Sheaves on Abelian Varieties

We show that the hypercohomology of most character twists of perverse sheaves on a complex abelian variety vanishes in all non-zero degrees. As a consequence we obtain a vanishing theorem for constructible sheaves and a relative vanishing theorem for a homomorphism between abelian varieties. Our proof relies on a Tannakian description for convolution products of perverse sheaves, and with future applications in mind we discuss the basic properties of the arising Tannaka groups.

The cone theorem and the vanishing of Chow cohomology

We show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.