Jie Liu

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.