Morse Inequalities And Bergman Kernels

This thesis consists of two parts. In part I, we prove equivariant Morse inequal-ities via Bismut-Lebeau’s analytic localization techniques. As an application, we obtain degenerate Morse inequalities on a compact manifold with nonempty bound-ary by applying equivariant Morse inequalities to the doubling manifold. In part II, we calculate the second coefficient of the asymptotic expansion of the Bergman kernel of the Hodge-Dolbeault operator associated to high powers of a Hermitian line bundle with non-degenerate curvature, using the method of formal power series developed by Ma and Marinescu.