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.
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