Speaker：Christine Laurent, Université Joseph Fourier, Institut Fourier

Sponsored by：BCIMR, CNRS, Mount Everest Project

Time：10:00-12:00, From 2015-04-27 to 2015-04-30, 2015-05-05 to 2015-05-08

Venue：Room 29 at Quan Zhai, BICMR

Abstract：The Cauchy-Riemann equation is one of the most important tools in complex analysis. Different approaches are possible to study this equation, here we will focus on the theory of integral representations developped by Grauert, Lieb, Henkin in the 70’s for the local study and use the Grauert’s bumping method to derive global results from the local ones.

The obstruction to the solvability of the Cauchy-Riemann equation on a complex manifold is given by the Dolbeault cohomology groups. We will study these groups in several settings: for smooth forms, for Lp form or for currents. The vanishing of a Dolbeault cohomology group in some setting is equivalent to the solvability of the Cauchy-Riemann equation in the same setting, the finiteness says that the obstruction is rather small.

For some problems, it is natural to introduce support conditions in the study of the Cauchy-Riemann equation, the case of compact supports is of great importance. Serre duality will give some relation between the usual Dolbeaut cohomology and the Dolbeault cohomology with compact support.

Applications to the extension of CR functions and to the solvability of the Cauchy-Riemann equation with prescribe support will be given.

注：课程中会用到微分流形和外微分形式的定义及一些基础知识，比如外微分形式的积分和 Stokes 公式等，请同学们自行预习。