Characteristic classes on complex manifolds and Chern-number inequalities on compact Kähler surfaces

by Yang, Chen

(Uncorrected OCR) Abstract of thesis entitled CHARACTERISTIC CLASSES ON COMPLEX MANIFOLDS AND CHERN-NUMBER INEQUALITIES ?ON COMPACT KAHLER SURFACES submitted by Yang Chen for the degree of Master of Philosophy at The University of Hong Kong in August 2004 The Euler characteristic is a fundamental topological invariant of a compact oriented differentiable manifold. Hopf in his thesis calculated the Euler characteristic by the index of a generic vector field on the differentiable manifold, which is the Poincar?-Hopf theorem. The Euler-Poincar? characteristic is a topological invariant for vector bundles over compact differentiable manifolds. The representation of the Euler-Poincar? characteristic by characteristic classes can be viewed as a generalization of the Poincar?-Hopf theorem, which is the Hirzebruch? Riemann-Roch theorem. Definitions of characteristic classes were given in three forms, namely the ? singular cohomology, the Cech cohomology, and the de Rham cohomology. The form of Chern classes represented by curvature tensors of vector bundles was used to calculate a number of interesting Chern-number inequalities. After representing the proof of Riemann-Roch theorem in [Hir], deformations of complex structures of [Kodaira1] were studied. We used Kuranishi? theory to represent the deformation by Euler-Poincar? characteristics and the Riemann-Roch theorem to calculate the Euler-Poincar? characteristics.