Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

On the monodromy of moduli spaces of sheaves on K3 surfaces


Author: Eyal Markman
Journal: J. Algebraic Geom. 17 (2008), 29-99
DOI: https://doi.org/10.1090/S1056-3911-07-00457-2
Published electronically: July 2, 2007
MathSciNet review: 2357680
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Abstract: Let $ S$ be a $ K3$ surface and $ \operatorname{Aut}D(S)$ the group of auto-equivalences of the derived category of $ S$. We construct a natural representation of $ \operatorname{Aut}D(S)$ on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on $ S$. The main result of this paper is the precise relation of this action with the monodromy of the Hilbert schemes $ S^{[n]}$ of points on the surface. A formula is provided for the monodromy representation in terms of the Chern character of the universal sheaf. Isometries of the second cohomology of $ S^{[n]}$ are lifted, via this formula, to monodromy operators of the whole cohomology ring of $ S^{[n]}$.


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Eyal Markman
Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
Email: markman@math.umass.edu

DOI: https://doi.org/10.1090/S1056-3911-07-00457-2
Received by editor(s): December 5, 2005
Published electronically: July 2, 2007
Additional Notes: The author was partially supported by NSF grant number DMS-9802532

American Mathematical Society