Organizers: Carmen Rovi and Bena Tshishiku
A Lefschetz fibration $M^4 \to S^2$ is a generalization of a surface bundle which also allows finitely many nodal singular fibers. The Arakelov--Parshin rigidity theorem implies that holomorphic Lefschetz fibrations of genus $g \geq 2$ admit only finitely many holomorphic sections. In this talk, we will show that no such finiteness result holds for smooth or symplectic sections by giving examples of genus-$g$ ($g \geq 2$) Lefschetz fibrations with infinitely many homologically distinct sections. This is joint work with Carlos A. Serván.
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The Prym representations of the mapping class group are an important family of representations that come from abelian covers of a surface. They are defined on the level-$\ell$ mapping class group, which is a fundamental finite-index subgroup of the mapping class group. One consequence of our work is that the Prym representations are infinitesimally rigid, i.e. they can not be deformed. We prove this infinitesimal rigidity by calculating the twisted cohomology of the level-$\ell$ mapping class group with coefficients in the Prym representation, and more generally in the $r$-tensor powers of the Prym representation. Our results also show that when $r\ge 2$, this twisted cohomology does not satisfy cohomological stability, i.e. it depends on the genus $g$.
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