Hall 123

Holomorphic mappings into the symmetric product of a Riemann surface

Jaikrishnan Janardhanan

IIT Madras

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.