Topology, birational geometry and vanishing theorem for complex algebraic varieties

In this proposal, we focus on three aspects of algebraic varieties. Firstly, we want to study two algebro-geometric properties of smooth algebraic varieties: the linearity of  the set of holomorphic 1-forms with zeros on smooth complex projective varieties, which reflects deep topological and birational nature of algebraic varieties; the surjectivity of quasi-Albanese map for smooth quasiprojective varieties, which is a crucial property for quasiprojective varieties and involves both topological nature and birational aspects of them. Both of these two projects (Project A and D in this proposal) are based on generic vanishing theorems for line bundles and local systems. Secondly, we want to study the crucial topological invariant involving first and second Chern classes of good minimal models motivated by a conjecture of Kollar, for which we want to understand the index of good minimal models using birational geometry techniques.  Thirdly, we want to establish Kodaira-Akizuki-Nakano type vanishing theorems for Higgs bundles over projective V-manifolds, which admit mild singularties in birational geometry, based on the vanishing theorem for Higgs bundles on smooth projective varieties by D. Arapura, Y. Deng, H. Li, and the applicant.