arXiv subjects
Publications and source records attributed to Xiankui Meng.
In this paper, we establish a logarithmic Bogomolov--Sommese vanishing theorem in terms of numerical dimension for pseudo-effective line bundles on compact K\"ahler manifolds. As an application, we obtain a rigidity result with vanishing second Chern class for logarithmic cotangent bundles by combining the vanishing theorem with a structure theorem of Iwai and Matsumura.
Several vanishing theorems for pseudo-effective line bundles are presented. All related results in the present note can be derived from already known theorems, in particular from the Steenbrink vanishing theorem and Boucksom's generalization of Bogomolov's vanishing theorem. No originality or priority is claimed for these statements; they are listed here merely as conjugate forms of the Kawamata--Viehweg vanishing theorem, for convenience of reference and comparison with existing literature.
In the present paper, we establish a general Kawamata-Viehweg-Koll\'ar-Nadel type vanishing theorem for higher direct images in terms of numerical dimension for closed positive currents on compact K\"ahler manifolds, unifying a number of important vanishing theorems.
In this paper, we present a converse to a version of Skoda's $L^2$ division theorem by investigating the solvability of $\bar{\partial}$ equations of a specific type.
Let $φ$ be a quasi-psh function on a complex manifold $X$ and let $S\subset X$ be a complex submanifold. Then the multiplier ideal sheaves $\mathcal{I}(φ|_S)\subset\mathcal{I}(φ)|_{S}$ and the complex singularity exponents $c_{x}\left(φ|_{S}\right)\leqslant c_{x}(φ)$ by Ohsawa-Takegoshi $L^{2}$ extension theorem. An interesting question is to know whether it is possible to get equalities in the above formulas. In the present article, we show that the answer is positive when $S$ is chosen outside a measure zero set in a suitable projective space.
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.