Xiaojun Wu
In this note, we give a new proof of a vanishing result originally due to Bogomolov, and later generalised by Mourougane and Boucksom. The statement holds for arbitrary pseudoeffective line bundles over compact Kähler manifolds, under an assumption on the numerical dimension of the line bundle.
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A logarithmic Bogomolov--Sommese vanishing theorem on compact Kähler manifolds
本文在紧Kähler流形上建立对数Bogomolov–Sommese消失定理，并结合Iwai和Matsumura的结构定理得到对数余切丛第二陈类为零的刚性结果。
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。
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.
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。
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。
We classify all minimal models X of dimension n, Kodaira dimension n-1 and with vanishing Chern number $c_1^{n-2}c_2(X)=0$. This solves a problem of Kollár. Completing previous work of Kollár and Grassi, we also show that there is a universal constant $ε>0$ such that any minimal threefold satisfies either $c_1c_2=0$ or $-c_1c_2>ε$. This settles completely a conjecture of Kollár.
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following t…
An orbifold version of Bogomolov decomposition theorem is established for compact Kähler spaces with quotient singularities and first Chern class zero.The proof is a direct adaptation of the classical smooth case, using Ricci-flat Kähler metrics, and Cheeger-Gromoll splitting theorem. It implies that normal K3 surfaces are uniformised in the orbifold sense either by normal K3 surfaces or by a complex torus, as conjectured by D.Q. Zhang. It also implies some conjectures on "special" threefolds raised in math.AG/0110051
In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by Hörmander's $L^2$-estimates. By using this isomorphism and the construction of smooth Hermitian metrics, we obtain a number of new vanishing theorems for sheaves of logarithmic differential forms on compact Kähler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.
On a vanishing theorem for surfaces
本文提出了一种新的曲面消失定理表述，该定理在log曲面的最小模型理论和丰富定理推导中具有重要应用价值。
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.
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。
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.
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。
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。
Using $L^2$-methods, we prove a vanishing theorem for tame harmonic bundles over quasi-compact Kähler manifolds in a very general setting. As a special case, we give a completely new proof of the Kodaira type vanishing theorems for Higgs bundles due to Arapura. To prove our vanishing theorem, we construct a fine resolution of the Dolbeault complex for tame harmonic bundles via the complex of sheaves of $L^2$-forms, and we establish the Hörmander $L^2$-estimate and solve $(\bar{\partial}_E+θ)$-equations for Higgs bundles $(E,θ)$.
In this paper we establish a Nadel-type vanishing theorem on a projective manifold $X$ concerning the asymptotic multiplier ideal sheaf.
On the Kodaira--Saito vanishing theorem of weakly ample divisor
本文在光滑射影簇上，针对层上同调意义下弱丰沛的既约有效除子，引入并证明了相应的Kadaira--Saito消没定理。
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。
Let $f:X\rightarrow Y$ be a smooth fibration between two complex manifolds $X$ and $Y$, and let $L$ be a pseudo-effective line bundle on $X$. We obtain a sufficient condition for $R^{q}f_{\ast}(K_{X/Y}\otimes L)$ to be reflexive and hence derive a Kollár-type vanishing theorem.
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.
A decomposition theorem for singular Kähler spaces with trivial first Chern class of dimension at most four
本文证明了具有klt奇点和消失第一陈类的紧Kähler四重折叠存在Beauville-Bogomolov分解，并通过提出Lipman-Zariski猜想的新版本，证明了其具有小射影形变及射影基本群。
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…
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.
Steenbrink vanishing theorem for big line bundles
该研究将复射影簇上丰富线丛的Steenbrink消失定理推广到带有乘子理想层的紧复空间上的大线丛，拓展了其适用范围。
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.
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。
On the Artin vanishing theorem for Stein spaces
本文围绕Stein空间的Artin消没定理展开研究，得到Stein空间相对Runge开子集上同调的Artin消没新结果，并证明了一类等变Stein空间相对不动点轨迹的等变上同调Artin消没定理。
In this article, we first establish an $L^2$-type Dolbeault isomorphism for the sheaf of logarithmic differential forms twisted by the multiplier ideal sheaf. By using this isomorphism and $L^2$-estimates equipped with a singular Hermitian metric, we obtain logarithmic vanishing theorems involving multiplier ideal sheaves on compact Kähler manifolds with simple normal crossing divisors.
Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically -- in a sense to be specified later) only on the curvature of $V$ and on the structure of the fibration $X_k\to X$. When $X$ is a surface and $V=T_X$, we give explicit formulae to write down the above curvature as a product of mat…
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