Geometric Invariant Theory and Birational Geometry

TL;DR

This paper surveys recent advances in Geometric Invariant Theory and their applications to Birational Geometry, including factorization theorems and progress on hyperKähler manifolds.

Contribution

It provides a comprehensive overview of recent developments in GIT variation and its applications to birational geometry, highlighting new theorems and open problems.

Findings

Weak Factorization Theorems for nonsingular projective varieties

Progress on birational geometry of hyperKähler manifolds

Discussion of open problems and conjectures in the field

Abstract

In this paper we will survey some recent developments in the last decade or so on variation of Geometric Invariant Theory and its applications to Birational Geometry such as the weak Factorization Theorems of nonsingular projective varieties and more generally projective varieties with finite quotient singularities. Along the way, we will also mention some progresses on birational geometry of hyperK\"ahler manifolds as well as certain open problems and conjectures.