Seminar on Complex Analysis and Complex Geometry - Fall 2017

On Wednesday when no extramural speaker is listed, the members of the complex geometry group may meet for a working seminar. For more information, send an email to Gordon Heier.

Wednesday, August 23, 2017, 3-4pm, PGH 646A

Title: Minimal rational curves on moduli space of stable bundles

Speaker: Xiaotao Sun, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstract: I will prove the equivalence of minimal rational curves and Hecke curves. As applications, the main theorem gives another proof of Torelli theorem and structures of automorphism group of moduli spaces.

Wednesday, November 1, 2017, 3-4pm, PGH 646A (in the Colloquium)

Title: Unique ergodicity for foliations

Speaker: Nessim Sibony, Université Paris-Sud (Orsay)

Abstract: Consider the polynomial differential equation in C^2: dz/dt = P(z,w), dw/dt=Q(z,w). The polynomials P and Q are holomorphic, the time is complex. In order to study the global behavior of the solutions, it is convenient to consider the extension as a foliation in the projective plane.

I will discuss some recent results around the following questions. What is the ergodic theory of such systems? How do the leaves distribute in a generic case?

Friday, November 3, 2017, 3-4pm, AH 301

Title: Equidistribution in holomorphic dynamics

Speaker: Nessim Sibony, Université Paris-Sud (Orsay)

Abstract: Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree.

The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. I will discuss some results and tools developed during the last two decades.

Wednesday, November 15, 2017, 3-4pm, PGH 646

Title: Levi Problem and Pfaff Systems

Speaker: Nessim Sibony, Université Paris-Sud (Orsay)

Abstract: Let U be a relatively compact pseudo-convex open set with smooth boundary in a complex manifold M. The question is whether U is Stein. There are famous counter-examples due to Grauert, and partial positive results due to Oka, Grauert, Hirschowitz, Greene-Wu in particular. There is a survey by Siu on the Levi problem.

I will give conditions which imply that U is Stein. They are related to the integrability of a natural Pfaff System. I will also discuss in the above setting, the existence of bounded strictly psh exhaustion functions, with useful estimates. If time permits, I will describe the geometry of pseudo-convex non-Stein domains and develop a foliated version of the above results.