Complex symplectic geometry and compact locally hyper-Kählerian manifolds

Abstract

Complex symplectic geometry is the study of complex manifolds admitting a global closed nondegenerate holomorphic two-form. Compact, simply connected complex symplectic manifolds are of interest for various reasons; for example, they always admit a Ricci-flat metric. In this work, the close relationship between the complex geometry of such manifolds and their Riemannian structures is exploited in order to obtain results in both directions: Riemannian techniques are used to obtain results on the complex automorphism group, and the problem of constructing examples of compact locally hyper-Kählerian manifolds with prescribed holonomy groups is discussed. © 1993 American Institute of Physics.