Xiankui Meng, Xiangyu Zhou
In this paper we first prove a version of $L^{2}$ existence theorem for line bundles equipped a singular Hermitian metrics. Aa an application, we establish a vanishing theorem which generalizes the classical Nadel vanishing theorem.
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$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics
本文研究了弱伪凸复空间上 $L^2$-Dolbeault 解析与 $L^2$-估计，推广了 Nadel 消失定理。
A relative Nadel-type vanishing theorem
本文证明了相对Nadel型消失定理，适用于Kähler纤维化结构。
In this paper we establish a Nadel-type vanishing theorem and a Kawamata--Viehweg-type vanishing theorem concerning the asymptotic multiplier ideal sheaf on a compact Kähler manifold $X$. After that, we provide a relative variant.
In this paper we establish a Nadel-type vanishing theorem on a projective manifold $X$ concerning the asymptotic multiplier ideal sheaf.
Griffiths and Nakano positivity and Nakano-Nadel vanishing theorems for singular Hermitian metrics on complex spaces
本文在复空间上引入奇异Griffiths/Nakano正性概念，建立其性质，并应用于获得L²-Dolbeault精细分解、上同构及存在性定理，进而证明弱伪凸复空间上的Nakano-Nadel消失定理。
Nakano-Nadel type, Bogomolov-Sommese type vanishing and singular dual Nakano semi-positivity
本文通过 L² 估计和 Dolbeault 同构，在弱伪凸流形上建立了奇异 Nakano 半正性的消失定理，并应用于 Fujita 猜想型定理。
In this article, we propose a definition of Nakano semi-positivity of singular Hermitian metrics on holomorphic vector bundles. By using this positivity notion, we establish $L^2$-estimates for holomorphic vector bundles with Nakano positive singular Hermitian metrics. We also show vanishing theorems, which generalize both Nakano type and Demailly-Nadel type vanishing theorems. As applications, we specifically construct globally Nakano semi-positive singular Hermitian metrics for several bundles, and prove vanishing theorems associated with them.
We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a vanishing theorem of Demailly-Nadel type for these in the special case where the base manifold is a Riemann surface.
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…
The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.
Vanishing theorems for pseudo-effective line bundles
研究在紧凯勒流形上关于闭正流高阶直像的消失定理，通过数值维数建立一般定理，统一多个重要消失定理。
In this article, we will characterize regular points respectively by the local vanishing, positivity of the Ricci curvature and $L^2$-solvability of the $\overline\partial$-equation together with Skoda's theorem for Nadel-Lebesgue multiplier ideal sheaves associated to plurisubharmonic (psh) functions on any (reduced) complex space of pure dimension. As a by-product, we show that any weak Kähler-Einstein metric on \emph{singular} $\mathbb{Q}$-Fano/Calabi-Yau/general type varieties cannot be smooth, and that in general there exists no \emph{singular} normal Kähler complex space such that the Kä…
In this paper, we introduce a coherent subsheaf of Saito's $S$-sheaf, which is a combination of the $S$-sheaf and the multiplier ideal sheaf. We construct its $L^2$-Dolbeault resolution, which generalizes MacPherson's conjecture on the $L^2$ resolution of the Grauert-Riemenschneider sheaf. We also prove various vanishing theorems for the $S$-sheaf (Saito's vanishing theorem, Kawamata-Viehweg vanishing theorem and some new ones like Nadel vanishing theorem) transcendentally. Finally, we discuss some applications of our results on the relative version of Fujita's conjecture (e.g. Kawamata's conj…
The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension of singular hermitian line bundles. The other is an analytic injectivity theorem for log canonical pairs on surfaces, which can be seen as a partial answer for Fujino's conjecture.
A Le Potier-type isomorphism twisted with multiplier submodule sheaves
本文建立乘子子模层扭曲的 Le Potier 型同构，连接奇异 Hermite 向量丛与重言线丛，并据此推出 Kollár 型内射、Nadel 型消灭及奇异全纯 Morse 不等式。
The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for K"ahler morphisms and singular hermitian metrics with non-algebraic…
We present a generalization of Takegoshi's relative version of the Grauert-Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi's vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.
The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs. Moreover, we give some results on semi-ampleness related to the abundance conjecture in birational geometry (the minimal model program).
Let $X$ be a smooth projective variety over an algebraically closed field $\mathbb{K}$ with arbitrary characteristic. Suppose $L$ is an ample and globally generated line bundle. By Castelnuovo--Mumford regularity, we show that $K_X \otimes L^{\otimes \dim X} \otimes A$ is globally generated and $K_X \otimes L^{\otimes (\dim X+1)} \otimes A$ is very ample, provided the line bundle $A$ is nef but not numerically trivial. On complex projective varieties, by investigating Kawamata-Viehweg-Nadel type vanishing theorems for vector bundles, we also obtain the global generation for adjoint vector bund…
We obtain the Bogomolov-Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with L2-estimates and prove the vanishing theorem which is a generalization of the Bogomolov-Sommese type vanishing theorem to holomorphic vector bundles.
Positivity of Singular Hermitian Metrics for Holomorphic Vector Bundles
本文为全纯向量丛的奇异 Hermite 度量引入 Nakano 和 Demailly 正性，支持消没定理并证明尖锐的 Ohsawa-Takegoshi 型 $L^2$ 延拓定理及 Berndtsson 型正性定理。
The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{ä}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation…
We prove several asymptotic vanishing theorems for Frobenius twists of ample vector bundles in positive characteristic. As an application, we prove a generalization of the Bott-Danilov-Steenbrink vanishing theorem for ample vector bundles on toric varieties.
On a Bogomolov type vanishing theorem
本文证明了在紧Kähler流形上，对于曲率非负的伪有效线丛，当$p\geq n-nd(L,h)+1$时上同调群$H^n(X,\mathcal{O}(Ω^p_X\otimes L)\otimes \mathcal{I}(h))=0$，推广了Bogomolov消没定理。
Motivated by a problem in holomorphic dynamics, we present a certain generalization of the celebrated F. and M. Riesz Theorem.
Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex Kähler manifolds
本文针对弱伪凸凯勒流形提出对数阿基祖-纳科诺消失定理，并推广了诺里马atsu在紧凯勒流形上的研究结果。
A remark on Bogomolov type vanishing theorem
研究在射影流形上建立博戈莫洛夫型消失定理，核心方法是基于数值有效线丛和乘子理想层，给出相关定理。
On vanishing theorems for analytic spaces
本文报告了复解析空间间射影态射的新消失定理，推广了Kollár的挠自由性与消失定理至解析简单正规交叉对，并在复解析极小模型研究中起关键作用。
In this paper we prove four cases of the vanishing conjecture of differential operators with constant coefficients and also a conjecture on the Laurent polynomials with no holomorphic parts, which were proposed in [Zh3] by the third named author. We also give two examples to show that the generalizations of both the vanishing conjecture and the Duistermaat-van der Kallen theorem [DK] to Laurent formal power series do not hold in general.
A solution to Fujita's freeness conjecture via an extension theorem with analytic adjoint ideal sheaves
本文利用伴随理想层对乘子理想层不可积轨迹的精细控制，并结合基于内射性定理技术的扩张定理，直接处理 lc 奇点及其极小 lc 中心，从而在 Angehrn-Siu 思路下解决了藤田有效自由性猜想。
Vanishing theorems for projective morphisms between complex analytic spaces
本文讨论复解析空间射影态射的消失定理，从最小模型理论观点建立了Kollár纲领的终极推广，对最小模型理论起关键作用。
We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use $L^{2}$-harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the res…
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