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Noncommutative local monodromy theorem

SPEAKER: Vadim Vologodsky

TITLE: Noncommutative local monodromy theorem

ABSTRACT: I will speak on a joint work (in progress) with Dmitry Vaintrob. Let $S$ be a smooth curve over $C$, $s_0in S$ a point, and let $pi: Xto S^circ $ be a smooth proper scheme over $S^circ =S -s_0$. The Griffiths-Landman-Grothendieck “Local Monodromy Theorem” asserts that the Gauss-Manin connection on the relative de Rham cohomology $H^*_{DR}(X/S^circ)$ has a regular singularity at $s_0$ and that its local monodromy around $s_0$ is quasi-unipotent. The Noncommutative Local Monodromy Theorem (which is an invention of Dmitry Vaintrob and myself) is a generalization of this result, where the de Rham cohomology is replaced by the periodic cyclic homology of a smooth proper DG algebra over $S^circ$ equipped with the Gauss-Manin-Getzler connection. I will sketch a proof of this result based on the reduction to characteristic $p$ technique and ideas of Katz and Kaledin.