Kollár’s Package for Polystable Locally Abelian Parabolic Higgs Bundles

Abstract

We generalize Kollár’s package (including torsion-freeness, injectivity theorem, vanishing theorem and decomposition theorem) to polystable locally abelian parabolic Higgs bundles twisted by a multiplier ideal sheaf associated with an $\mathbb{R}$-divisor. This gives a uniform treatment for various kinds of Kollár’s package in different topics in complex geometry. As applications, the weak positivity (in the sense of Viehweg) and the generic vanishing property for higher direct image sheaves are deduced.