20 jun 2016 -- 14:30
Aula Tricerri, DiMaI, Firenze
INdAM Intensive Period "Hypercomplex Function Theory and Applications"
Abstract.
The Bergman kernel and metric on a bounded complex domain D have proved to be, since their introduction by S. Bergman at the beginning of the 20th century, a powerful tool both in complex analysis and geometry. One of the most important results proved on this subject is the celebrated theorem by C. Fefferman which shows that the Bergman kernel has an asymptotic expansion involving a logarithmic function of the distance to the boundary of D, called the log-term. Similar results hold for analogous reproducing kernels which naturally arise in Kähler geometry, like the Szeg ö kernel of the line bundle of a Kähler manifold. Understanding to which extent the behaviour at the boundary of these kernels (and in particular the vanishing of the log-term) determines the domain or the manifold is a fundamental problem which has attracted the interest of many authors. In this talk, after giving an overview of these problems and their connections with other questions and open conjectures in Kähler geometry, we will show vanishing results of the log-term of Bergman and Szegö kernels for the homogeneous Kähler manifolds, which are inspired by and generalize results by M. Englis, G. Zhang and Z. Lu, G. Tian.