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Abstract: A conjecture of Beauville and Catanese from 1980's stated
that the sets of rank one local systems on a compact KÃ¤hler manifold
with prescribed cohomology are special varieties, that is, their
irreducible components are torsion-translated subtori. The conjecture
has finally been fully proved by Botong Wang in 2013. Around the same
time, Wang and I proved that the same holds for all quasi-projective
complex algebraic manifolds. In this talk, we present the recent proof
of a much more subtle case: germ complements of complex analytic sets.
This is a vast generalization of the classical Monodromy Theorem
stating that the eigenvalues of the monodromy on the cohomology of the
Milnor fiber of a germ of a holomorphic function are roots of unity.
The proof uses the Riemann-Hilbert correspondence between D-modules
and perverse sheaves. Joint work with Botong Wang.