Moduli spaces of sheaves on K3 and abelian surfaces: their
symmetries and monodromy -- Eyal Markman

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We describe how equivalences of derived categories,
of K3 and abelian surfaces, act on the cohomology of the collection
of all smooth and compact moduli spaces of stable sheaves on such
surfaces. Due to the effort of many authors, the action has an elegant
description in terms of a groupoid G. The objects of G are the data
needed to construct such a moduli space, namely a triple (S,v,H),
consisting of a K3 or abelian surface S, an indivisible class v of its
topological K-group (rank and chern classes) with c_1(v) of
Hodge type (1,1), and a v-generic ample line-bundle H on S.
A morphism of G from (S,v,H) to (S',v',H') is an equivalence
f:D(S) -> D(S') of the bounded derived categories of coherent sheaves,
which maps v to v'.

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We construct a representation of the groupoid G; a functor F from G
to the category of algebras. F sends an object (S,v,H)
to the cohomology of the corresponding moduli space M(S,v,H).
Morphisms of G are sent by F to monodromy operators.

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The construction enables us to compute the weight 2 monodromy group of
the Hilbert scheme of length n zero dimensional subschemes of a K3
surface. The monodromy group is smaller than expected,
if n-1 is not a prime power.
One gets counter examples to a common version of the Torelli question,
for hyperk&auml;hler 
varieties deformation equivalent to the Hilbert scheme.

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References: <a href=http://arxiv.org/abs/math/?0406016>math.AG/0406016</a>, <a href=http://arxiv.org/abs/math/?0305042>math.AG/0305042</a>