Characteristic classes on complex manifolds and Chern-number inequalities on compact Kähler surfaces
Abstract of thesis entitled
ON COMPLEX MANIFOLDS
AND CHERN-NUMBER INEQUALITIES
?ON COMPACT KAHLER SURFACES
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.
School:The University of Hong Kong
School Location:China - Hong Kong SAR
Source Type:Master's Thesis
Keywords:euler characteristic complex manifolds inequalities mathematics kahlerian
Date of Publication:01/01/2005