刘   杰

The Hassett-Keel program relates different birational compactifications of moduli of curves as variation of log canonical models of \(\overline{M}_g\) which are connected by elementary birational transformations.
In this talk, we will discuss the higher dimensional generalization, with a particular focus on the case of K3 surfaces with anti-symplectic involutions. Specifically, we will introduce the Hassett-Keel-Looijenga program and show how the wall crossing for K-moduli of log del Pezzo pairs comes into play. I will also survey some recent progress on wall-crossing of moduli spaces of K3 surfaces.

Given a local system on a complex algebraic variety, what are the subvarieties on which the monodromy drops? The talk will discuss these monodromy special loci, a natural generalisation of (the positive period dimension components of) the Hodge loci.

A classical result by Hassett and Tschinkel shows that there exist infinitely many inequivalent equivariant compactifications of vector groups into projective spaces of dimension at least 6. In this talk，we will give a non-commutative analog of this result for the Heisenberg groups. We prove that there exist infinitely many inequivalent equivariant compactifications of Heisenberg groups into odd dimensional projective spaces. This is a joint work with Zhijun Luo.

Lazarsfeld raised the problem that any surjective morphism \(f\) from a rational homogeneous space \(S\) of Picard number 1 to a projective manifold \(X\) must satisfy that either \(X\) is a projective space or \(f\) is an isomorphism. This problem was completely resolved in the affirmative by Hwang and Mok. In this talk, we will provide a partial extension of Lazarsfeld's problem to Fano manifolds of Picard number 1 that have big tangent bundles. As an application, we study the bigness of the tangent bundles of Fano manifolds with Picard number 1. This talk is based on my joint work with Guolei Zhong.