the Shafarevich conjecture on holomorphic convexity: Is the universal covering of a compact Kähler manifold (or a smooth projective variety) holomorphically convex?
Specifically, I am interested in understanding the holomorphic convexity of intermediate coverings over compact Kähler manifolds, with given properties such as reductive, nilpotent, linear, etc.
J.P. Serre's problem: Can we characterize the fundamental groups of compact Kähler manifolds (or smooth projective varieties) among all finitely presentable groups.
Specifically, I am interested in applying the Bieri-Neumann-Strebel invariant for Kähler groups.
On the holomorphic convexity of reductive Galois coverings over compact Kähler surfaces. [Advances in Mathematics 431 (2023): 109245.]
Cartan’s method and its applications in sheaf cohomology. [Expositiones Mathematicae 41, no. 2 (2023): 288-298.]
Joint with Ya Deng, Quasi-finiteness of morphisms of character varieties. [arXiv preprint, submitted for publication]
On the holomorphic convexity of nilpotent coverings over compact Kähler surfaces. [arXiv preprint, submitted for publication]
Joint with Yongqiang Liu, Bieri-Neumann-Strebel-Renz invariants and tropical varieties of integral homology jump loci. [arXiv preprint, submitted for publication]
The application of the Bieri-Neumann-Strebel invariant on Kähler groups. [arXiv preprint, submitted for publication]
Lecture notes edited:
Introduction to Cartan's Theorem A and B. (101 pages): This is a course taught by Prof. M. Ramachandran during Fall 2021 at UB.