### Bogomolov-65: slides of the talks

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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.

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.

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

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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