１. BICMR-CNRS Lectures in Mathematics - Integral Formulas and Serre Duality for the Cauchy-Riemann Equation

Speaker(s)Christine Laurent, Université Joseph Fourier, Institut Fourier
DateFrom 2015-04-27 To 2015-05-08
VenueRoom 29 at Quan Zhai, BICMR

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.

2. BICMR-CNRS Lectures in Mathematics--Entire Holomorphic Curves and Algebraic Differential Equations

Speaker(s)Jean-Pierre Demailly, Université Joseph Fourier,Institut Fourier
DateFrom 2015-04-07 To 2015-06-18
VenueRoom 77201 at #78 courtyard, Beijing International Center for Mathematical 
Research/Room 29 at Quan Zhai, BICMR

Speaker：Jean-Pierre Demailly, Université Joseph Fourier, Institut Fourier

Sponsored by：BCIMR, CNRS, Mount Everest Project

Time： Every Tuesday & Thursday, 14:00-16:00, From 2015-04-7 to 2015-06-18

Venue: Classroom 77201(2015-04-07 to 2015-06-11), Classroom Quan29 (2015-06-16 to 2015-06-18)

Abstract：The geometry of projective algebraic varieties is intimately related to positivity or negativity properties of their curvature tensor. In the introduction, we will first recall some basic concepts of complex differential geometry: 
complex manifolds, Dolbeault cohomology, vector bundles, connections, Chern classes, holomorphic Morse inequalities. These fundamental concepts will then be used to investigate the existence of entire holomorphic curves drawn in projective algebraic varieties. This problem is related to deep conjectures concerning number theory (Diophantine equations) and,in a more algebraic setting, to curvature properties of jet bundles. The ultimate goal will be to prove a recent result of the lecturer in the direction of the Green-Griffiths-Lang conjecture, according to which every entire curve drawn in a variety of general type satisfies algebraic differential equations.