Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex K\"{a}hler manifolds

TL;DR

This paper proves a new logarithmic vanishing theorem for weakly pseudoconvex Kähler manifolds with divisors having infinitely many components, extending previous results and deriving corollaries for direct image sheaves.

Contribution

It generalizes Norimatsu's vanishing theorem to non-compact, weakly pseudoconvex Kähler manifolds with complex divisors.

Findings

Established a logarithmic vanishing theorem for weakly pseudoconvex Kähler manifolds.

Derived vanishing results for direct image sheaves.

Extended classical theorems to more general geometric settings.

Abstract

In this paper, we establish a logarithmic vanishing theorem on weakly pseudoconvex K\"ahler manifolds, where the divisor may have infinitely many irreducible components. This result serves as a generalization of Norimatsu's findings on compact K\"ahler manifolds. We derive vanishing theorems for certain direct image sheaves as a direct corollary.