Noncommutative Local Monodromy Theorem

Let  X \to D^* be a smooth projective variety over the formal punctured disk D^*=spec K= spec \bC((t)).

   The Griffiths-Landman-Grothendieck ``Local Monodromy Theorem'' asserts that

the Gauss-Manin connection on the de Rham cohomology H^*_{DR}(X/D^*)  has a regular  singularity at the origin and that the monodromy of this connection is quasi-unipotent.  I will explain a  noncommutative generalization of this result, 

where the de Rham cohomology is replaced by the periodic cyclic homology of a (smooth proper)  DG category over K equipped with the Gauss-Manin-Getzler connection. The proof of the Noncommutative Local Monodromy Theorem is based on the reduction modulo p technique and some ideas of N.Katz and D. Kaledin. Namely, I will prove that for any  smooth proper DG category over F_p((t)) the p-curvature of  the Gauss-Manin-Getzler connection on its 

periodic cyclic homology is nilpotent.If time allows I will also explain a noncommutative generalization of the Katz p-curvature formula relating the p-curvature of the Gauss-Manin-Getzler connection with the Kodaira-Spencer class (which is, in the noncommutative setting, a canonical element  of the second Hochschild cohomology group of the DG category)

 

This  talk is based on a joint work with Dmitry Vaintrob.