Holomorphic Vector Fields on Fano Manifolds

Professor Ngaiming Mok
The University of Hong Kong

Abstract

We study holomorphic vector fields on Fano manifolds X of Picard number 1 in the general
context of the geometric theory of Fano manifolds in terms of varieties of minimal rational tangents that the speaker has developed in collaboration with Jun-Muk Hwang. Fixing a
space of rational curves of minimal degree the variety of minimal rational tangents  through a generic point x is the collection of all vectors at the point tangent to some minimal rational curve. We show that the geometry of  has strong bearings on the structure of the Lie algebra of holomorphic vector fields on X. Here are the topics to be covered:

• Varieties of minimal rational tangents
• Distributions spanned by minimal rational tangents
• Bounds on vanishing orders of holomorphic vector fields
• Bounds on dimensions of automorphism groups
• The symbolic Lie algebra of principal terms of holomorphic vector fields
• Deformation rigidity of irreducible compact Hermitian symmetric spaces
• Deformation rigidity of the (non-symmetric) Grassmannian of isotropic k-planes on a symplectic vector space of dimension 2n, n > k.