Desingularizations of moduli spaces of rank 2 sheaves with trivial
determinant -- Young-Hoon Kiem

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I'll first speak about three desingularizations of the moduli space M_0 of
rank 2 bundles over a smooth curve by Seshadri, Narasimhan-Ramanan
and Kirwan respectively and show that they are related by blow-ups. 
Using this, we can compute the cohomology of the desingularizations and
the stringy E-function of M_0. Quite surprisingly the formula looks almost
the same as the E-polynomial of the intersection cohomology. 

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Next I turn to the moduli space M of Higgs bundles with trivial
determinant 
and show that three desingularizations are obtained by blow-ups and downs 
(based on Simpson's and O'Grady's work). This enables us to compute the
stringy E-function 
of M explicitly and deduce that there does not exist a
desingularization 
which is hyperk&#228;hler when the dimension of the moduli space is greater
than 10. 
It is not known whether the desingularizations have moduli theoretic meaning
as above.

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Third, I'll speak about the singular moduli spaces of rank 2 sheaves with 
trivial determinant over K3 and Abelian surfaces and show that the same
results 
as in the Higgs case above hold by O'Grady's analysis. 

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This talk is based on joint works with J. Li, I. Choe, J. Choy and S.-B.
Yoo.