Sheng Rao

Several Special Complex Structures and Their Deformation Properties

We introduce a natural map from the space of pure-type complex differential forms on a complex manifold to the corresponding one on the infinitesimal deformations of this complex manifold. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang, and the first author. As direct corollaries, we prove several deformation invariance theorems for Hodge numbers. Moreover, we also study the Gauduchon cone and its relation with the balanced cone in the Kähler case, and show that the limit of the Gauduchon cone in the sense of D. Popovici for a generic fiber in a Kählerian family is contained in the closure of the Gauduchon cone for this fiber.

Extension formulas and deformation invariance of Hodge numbers

Abstract We introduce a canonical isomorphism from the space of pure-type complex differential forms on a compact complex manifold to the one on its infinitesimal deformations. By use of this map, we generalize an extension formula in a recent work of K. Liu, X. Yang and the second author. As a direct corollary of the extension formulas, we prove several deformation invariance theorems for Hodge numbers on some certain classes of complex manifolds, without using the Frolicher inequality or the topological invariance of the Betti numbers.

On the Maslov-type Index for Symplectic Paths with Lagrangian Boundary Conditions and Spectral Flow

In this paper, we consider the Maslov-type index theory for symplectic paths starting from the identity with Lagrangian boundary conditions developed by Chun-gen Liu. We firstly give a brief review and some necessary remarks of this theory and then present an explicit formula describing the connection of this index and the spectral flow. Some new concepts and basic properties of complex symplectic theory are introduced.