Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex Kähler manifolds

TL;DR

This work extends logarithmic Akizuki–Nakano vanishing to open, weakly pseudoconvex Kähler manifolds with divisors that may have infinitely many components. It develops an L^2–based framework on relatively compact sublevel sets, using Poincaré-type metrics and Runge approximation to derive local vanishing on each sublevel, then globalizes via Leray coverings and Remmert reduction. The main results show H^q(X, Ω^p_X(log D) ⊗ F) = 0 for p+q ≥ n+1 on holomorphically convex X (with F positive and D possibly infinite), and H^q(X, K_X ⊗ O_X(D) ⊗ F) = 0 for q ≥ 1 on weakly pseudoconvex X, with corollaries for higher direct image sheaves. The approach blends L^2 estimates, log geometry, and Runge-type approximation to bridge finite-component and infinite-divisor cases in a non-compact setting.

Abstract

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

Theorems & Definitions (10)

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• proof : The proof of Theorem \ref{['main-finite']}
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• proof : The proof of Theorem \ref{['q2']}
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