Abstract: We define the residues of a holomorphic connection of a holomorphic line bundle along a simple normal crossing divisor in a complex manifold by only assuming that its curvature extends across the divisor. We then prove a residue formula that localizes the first Chern class to the singular locus of the given holomorphic connection. As applications, we discuss proofs for a nonexistence theorem of Levi flat hypersurfaces with transversely affine Levi foliation in compact Kaehler surfaces and Brunella's conjecture about exceptional minimal sets of codimension one holomorphic foliations with ample normal bundle. This is joint work with Séverine Biard and Judith Brinkschulte.

Abstract: Every automorphism group of a compact Kähler manifold satisfies Tits alternative: (1) either it admits a solvable subgroup of finite index, (2) or it contains a free group of two generators. In the first case, we show that this group cannot be too big. We also study particular algebraic manifolds which allow us to show the existence of manifolds of any dimension (greater or equal to 2) with infinitely many of non-equivalent real forms. This talk is based on my joint works with H.-Y. Lin, V.-A. Nguyen, K. Oguiso, N. Sibony, X. Yu, D.-Q. Zhang.

Abstract: The study of reproducing kernel in several complex variables is fundamental in complex and CR geometry. In this talk, we will focus on the Szegö kernel on embeddable compact strongly pseudoconvex CR manifolds. We will recall the asymptotic expansion obtained by Boutet de Monvel and Sjöstrand via complex-valued phase Fourier integral operator, and discuss how to determine its second coefficient (in the part of pole type singularities). This is a joint work with Chin-Yu Hsiao.

Abstract: We establish an asymptotic version of Bismut's local family index theorem for the Bergman kernel. The key idea is to use the superconnection as in the local family index theorem.