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Nonexistence of a crepant resolution of some moduli spaces ofsheaves on a K3 surface

김영훈|최재유
대한수학회 (The Korean Mathematical Society)|2007.01|대한수학회지 (Journal of the KMS)|제44권 제1호|pp. 35-54

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Abstract

LetM c = M (2,0,c) be the moduli space of O(1)-semistablerank 2 torsion-fre sheaves with Chern clases c1 = 0 and c2 = c on aK3 surface X , where O(1) is a generic ample line bundle on X . Whenc = 2 n ≥ 4 is even, M c is a singular projective variety equiped witha holomorphic symplectic structure on the smoth locus. In p articular,M c has trivial canonical divisor. In [2], O’Grady asks if there is anysymplectic desingularization of M 2n forn ≥ 3. In this paper, we showthat here is no crepant resolution of M 2n for n ≥ 3. This obviouslyimplies that here is no symplectic desingularization.

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