Bogomolov-65: slides of the talks

September 1-4, 2011,
Laboratory of Algebraic Geometry,
Steklov Math Institute,

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Jean-Louis Colliot-Thélène (Universite Paris-Sud)
Galois descent on the Brauer group of varieties (joint work with A. N. Skorobogatov)

For a smooth and projective variety X over a field k of characteristic zero we prove the finiteness of the cokernel of the natural map from the Brauer group of X to the Galois-invariant subgroup of the Brauer group of the same variety over an algebraic closure of k. Under further conditions on k, e.g. over number fields, we give estimates for the order of this cokernel. We emphasise the role played by the exponent of the discriminant groups of the intersection pairing between the groups of divisors and curves modulo numerical equivalence.

Tony Pantev (University of Pennsylvania)
Integral transforms in non-abelian Hodge theory

In order to construct integral transforms and Fourier-Mukai functors for variations of twistor structures one must have very strong functoriality properties of the non-abelian Hodge correspondence. I will discuss the problem of compatibility of non-abelian Hodge theory with Grothendieck's six operations and will report on a recent joint work with R.Donagi and C.Simpson. Our main result is an explicit algebraic formula which, in the tamely ramified case, captures the interaction of the Hodge correspondence with pushforwards.

Yura Tschinkel (Courant Institute)
Balanced line bundles

I will explain the notion of "balanced" emerging in arithmetic geometry (joint work with B. Hassett and S. Tanimoto).

Misha Verbitsky (HSE)
Global Torelli theorem for hyperkahler manifolds

The global Torelli theorem for hyperkaehler manifolds is a statement about a diffeomorphism between the "birational moduli space", obtained by gluing together certain birationally equivalent points in the coarse moduli space, and a quotient of the period domain by the mapping class group. I will state the theorem and explain how one computes the mapping class group using the Sullivan's theory of rational homotopy.

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Laboratory of Algebraic Geometry and its Applications