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# Logarithmic vanishing theorems on compact K\"{a}hler manifolds I

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### Abstract

In this paper, we first establish an $$L^2$$-type Dolbeault isomorphism for logarithmic differential forms by H\"{o}rmander'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 K\"ahler manifolds with simple normal crossing divisors, which generalize several classical vanishing theorems, including Norimatsu's vanishing theorem, Gibrau's vanishing theorem, Le Potier's vanishing theorem and a version of the Kawamata-Viehweg vanishing theorem.

### Author and article information

###### Journal
2016-11-23
###### Article
1611.07671

http://arxiv.org/licenses/nonexclusive-distrib/1.0/

###### Custom metadata
14F17, 32L20
math.AG math.CV

Analysis, Geometry & Topology