Etale covering between resolution of isolated singularities and Gaffney-Lazarsfeld Theorem for Homogeneous Spaces

Abstract

We discuss two topics in the thesis. One is the resolution of isolated singularities and its application in CR geometry, and the other one is extending the Gaffney-Lazarsfeld theorem to homogeneous spaces which is a theorem dealing with dimension bounds for branched loci. We explain the background of these problems and their significance in the beginning, including the minimal model program and branched coverings. In the second chapter, first we provide some technical results in order to carry out the proofs of main theorems, then prove the multiplicativity of the canonical volume defined for isolated singularities. This gives that any etale covering map between resolutions of a variety having only finitely many isolated singularities is an isomorphism. We also explain some results of CR geometry and show that any non-constant CR morphisms between embeddable strongly pseudo-convex CR manifolds is a biholomorphism. In the third chapter, we begin with the notions of local degrees and higher ramification loci in order to obtain the Gaffney-Lazarsfeld theorem on projective spaces and Grassmannians. After some fundamental properties of homogeneous spaces, we give a generalization of the Gaffney-Lazarsfeld theorem for homogeneous spaces of Picard number one.