UC Irvine

Let $(L,h) \to (M,\omega)$ be a polarized K\"ahler manifold. We define the Bergman kernel for $H^0(M,L^k)$, holomorphic sections of the high tensor powers of the line bundle $L$. In this thesis, we will study the asymptotic expansion of the Bergman kernel. We will consider the on-diagonal, near-diagonal and far off-diagonal, using $\mathcal{L}^2$ estimates to show the existence of the asymptotic expansion and computation of the coefficients for the on and near-diagonal case, and a heat kernel approach to show the exponential decay of the off-diagonal of the Bergman kernel for noncompact manifolds assuming only a lower bound on Ricci curvature and $C^2$ regularity of the metric.