Vanishing Theorems for Sheaves of Logarithmic Differential Forms on Compact Kähler Manifolds

Abstract

Abstract In this paper, we first establish an $L^2$-type Dolbeault isomorphism for logarithmic differential forms by Hrmander’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, Girbau’s vanishing theorem, Le Potier’s vanishing theorem, and a version of the Kawamata–Viehweg vanishing theorem.