Description
Analysis seminar.
Speaker: Jaikrishnan Janardhanan.
Affiliation: IIT Madras.
Date and Time: Wednesday 11 September, 3:30 pm - 4:30 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Holomorphic mappings into the symmetric product of a Riemann surface.
Abstract: The symmetric product is an interesting and important
construction that is studied in Algebraic Geometry, Complex Geometry,
Topology and Theoretical Physics. The symmetric product of a complex
manifold is, in general, only a complex space. However, in the case of a
one-dimensional complex manifold (i.e., a Riemann surface), it turns out
that the symmetric product is always a complex manifold. The study of the
symmetric product of planar domains and Riemann surfaces has recently
become very important and popular.
In this talk, we present two of our recent contributions to this study.
The first work (joint with Divakaran, Bharali and Biswas) gives a precise
description of the space of proper holomorphic mappings from a product of
Riemann surfaces into the symmetric product of a bordered Riemann
surface. Our work extends the classical results of Remmert and Stein. Our
second result gives a Schwarz lemma for mappings from the unit disk into
the symmetric product of a Riemann surface. Our result holds for all
Riemann surfaces and yet our proof is simpler and more geometric than
earlier proved special cases where the underlying Riemann surface was the
unit disk or, more generally, a bounded planar domain. This simplification
was achieved by using the pluricomplex Green's function. We will also
highlight how the use of this function can simplify several well-know and
classical results.