Guillaume Tahar: Meromorphic differentials with prescribed singularities - a refinement of a classical Mittag-Leffler theorem
Time: Wed 2020-02-19 13.15 - 14.15
Lecturer: Guillaume Tahar, Institut de Mathématiques de Jussieu
A classical theorem of Mittag-Leffler asserts that in a given Riemann surface X, for any pattern of multiplicities of poles and any configuration of residues (summing to zero), there is a meromorphic 1-form on X that realize them. The only obstruction is that residues at simple poles should be nonzero.
If we require that the multiplicity of the zeroes is also prescribed, the problem can be reformulated in terms of strata of meromorphic differentials. Using the dictionary between complex analysis and flat geometry, we are able to provide a complete characterization of configurations of residues that are realized for a given pattern of singularities. Two nontrivial obstructions appear concerning the combinatorics of the multiplicity of zeroes and the arithmetics of the residues.
This work can be interpreted as the characterization of nonempty fibers of the isoresidual fibration of the strata. If time allows, I will give some insights about the topology of the generic fiber of this fibration. This is a joint work with Quentin Gendron.