On a Bogomolov type vanishing theorem

TL;DR

The paper proves a Bogomolov-type vanishing theorem on compact Kähler manifolds for pseudoeffective line bundles with singular metrics, showing $H^n(X,\mathcal{O}(\Omega^p_X\otimes L)\otimes\mathcal{I}(h))=0$ when $p\ge n-\operatorname{nd}(L,h)+1$. The approach centers on defining the numerical dimension $\operatorname{nd}(L,h)$ via a cohomological product of curvature currents and leveraging $L^2$-estimates for the $\bar{\partial}$-equation, a Čech cohomology vanishing criterion, and analytic regularization with openness results. The authors connect $\operatorname{nd}(L,h)$ to the Kodaira–Iitaka dimension $\kappa(L)$ (with $\kappa(L)\le \operatorname{nd}(L,h)$) and show that the theorem recovers Bogomolov’s classical result and Watanabe’s big-line-bundle case as special instances. Overall, the work extends vanishing theorems to non-projective, Kähler settings using multiplier ideals and numerical dimension theory, offering tools for future investigations in complex geometry.

Abstract

Let $X$ be a compact Kähler manifold and $(L,h)\rightarrow X$ be a pseudoeffective line bundle, such that the curvature $iΘ_{L,h}\geq 0$ in the sense of currents. The main result of the present paper is that $H^n(X,\mathcal{O}(Ω^p_X\otimes L)\otimes \mathcal{I}(h))=0$ for $p\geq n-nd(L,h)+1$. This is a generalization of Bogomolov's vanishing theorem.

Theorems & Definitions (7)

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