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Motivic realizations of matrix factorizations I

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Anthony Blanc
Tue, 2017-02-28 14:00 - 15:00
MPIM Lecture Hall
Given a Landau--Ginzburg model over the complex numbers, one can associate to it a dg-category of
matrix factorizations. By a result of Efimov, the periodic cyclic homology of this dg-category identifies
with the vanishing cohomology of the singular fiber. In this situation, certain Landau-Ginzburg models
appears as mirrors of certain symplectic manifolds, and this result allows an interpretation of homological
mirror symmetry via the associated noncommutative Hodge structures. 

In these two talks, I want to report on a joint work with Robalo, Toën and Vezzosi, which adresses a
similar situation above, but where we replace the complex field by a complete discrete valuation ring
(e.g. the ring of p-adic integers). This involves constructing a nice cohomology theory of noncommutative
spaces over such a ring, using the theory of noncommutative motives. 
I: In the first part we will recall the analog of our result over the complex numbers by Efimov. This will
involve preliminaries on dg-categories and cyclic homology. From there we will set up more preliminaries
for the sequel, on motives and noncommutative motives, and the method of taking motivic realizations. 
II: In the second part we will review Orlov comparison theorem with the singularity category, the sheaf
of vanishing cycles over a dvr and present some elements of the proof that the l-adic realization of
matrix factorization is given by vanishing cohomology. Finally we hope to put the result in perspective
regarding Toën--Vezzosi approach to the conductor formula. 
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