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
Full-text PDF
Abstract |
References |
Additional Information
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 $S^{[n]}$.
References
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- Yoshioka, K.: Some examples of Mukai reflections on K$3$ surfaces. J. Reine Angew. Math. 515 (1999), 97–123. MR 1717621 (2000h:14028)
- Yoshioka, K.: Irreducibility of moduli spaces of vector bundles on K$3$ surfaces. math.AG/9907001
- Yoshioka, K.: Moduli spaces of stable sheaves on abelian surfaces. Math. Ann. 321 (2001), no. 4, 817–884. MR 1872531 (2002k:14020)
- Yoshioka, K.: A Note on Fourier-Mukai transform. Eprint arXiv:math.AG/0112267 v3.
Additional Information
Eyal Markman
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003
MR Author ID:
355854
Email:
markman@math.umass.edu
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