國立清華大學 數學系

Abstract: In my work(Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, 101 pages, to appear in CAG), I give for the first time a microlocal study of the complex Witten Laplacian. As an application, we obtain a full asymptotic expansion of the spectral function corresponding to the lower part of the spectrum of the Kodaira Laplacian on high tensor powers of a holomorphic line bundle. From this result, we could deduce many classical results in complex geometry (eg Kodaira embedding and vanishing Theorems, Demailly‘s Morse inequalities, Bergman kernel asymptotics for ample line bundles...). In this talk, I will explain how to obtain these classical results from this result. In time is enough, I will also mention a new result obtained in this work: the existence of the full asymptotics expansion for Bergman kernel for a big line bundle twisted with a multiplier ideal sheaf. As a corollary, we could reprove the shiffman conjecture, asserting that Moishezon manifolds can be characterized in terms of integral Kahler currents. Tea Time: 3:30PM, R707