SINGULAR SYMPLECTIC MODULI SPACES

. Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O’Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension ≥ 4 they do no admit projective symplectic resolutions.

[1]  A. Grothendieck,et al.  Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, (SAG 2) : séminaire de géométrie algébrique, du Bois Marie, 1962 , 2005 .

[2]  J. Drézet.  Variétés de modules extrémales de faisceaux semi-stables sur ℙ2(ℂ) , 1991 .

[3]  Craig Huneke,et al.  Commutative Algebra I , 2012 .

[4]  Takehiko Yasuda,et al.  HIGHER DIMENSIONAL ALGEBRAIC GEOMETRY , 2006 .

[5]  Y. Kiem.  On the existence of a symplectic desingularization of some moduli spaces of sheaves on a K3 surface , 2005, Compositio Mathematica.

[6]  A. Rapagnetta.  Topological invariants of O’Grady’s six dimensional irreducible symplectic variety , 2004, math/0406026.

[7]  M. Lehn,et al.  Local structure of hyperkaehler singularities in O'Grady's examples , 2004, math/0405575.

[8]  K. Yoshioka.  Twisted Stability and Fourier–Mukai Transform I , 2001, Compositio Mathematica.

[9]  K. O’Grady.  A new six-dimensional irreducible symplectic variety , 2000, math/0010187.

[10]  Eyal Markman,et al.  Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces , 2000, math/0009109.

[11]  K. Yoshioka.  Moduli spaces of stable sheaves on abelian surfaces , 2000, math/0009001.

[12]  A. Beauville.  Symplectic singularities , 1999, math/9903070.

[13]  K. O’Grady.  Desingularized moduli spaces of sheaves on a K3, I , 1997, alg-geom/9708009.

[14]  D. Huybrechts,et al.  The geometry of moduli spaces of sheaves , 1997 .

[15]  Carlos Simpson,et al.  Moduli of representations of the fundamental group of a smooth projective variety I , 1994 .

[16]  H. Nakajima.  Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras , 1994 .

[17]  J. Drézet.  Points non factoriels des variétés de modules de faisceaux semi-stables sur une surface rationnelle. , 1991 .

[18]  H. Flenner.  Extendability of differential forms on non-isolated singularities , 1988 .

[19]  J. Drézet.  Groupe de picard des variétés de modules de faisceaux semi-stables sur ℙ2 , 1987 .

[20]  S. Mukai.  Symplectic structure of the moduli space of sheaves on an abelian or K3 surface , 1984 .

[21]  A. Beauville,et al.  Variétés Kähleriennes dont la première classe de Chern est nulle , 1983 .

[22]  Gianfranco Niesi,et al.  On Serre’s conditions in the form ring of an ideal , 1981 .

[23]  P. Deligne.  La conjecture de Weil. I , 1974 .

[24]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[25]  A. Grothendieck,et al.  Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2) , 2005, math/0511279.

[26]  Alexander Grothendieck,et al.  Elements de geometrie algebrique III: Etude cohomologique des faisceaux coherents , 1961 .