1,539 research outputs found
Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties
The aim of this paper is to study the singularities of certain moduli spaces
of sheaves on K3 surfaces by means of Nakajima quiver varieties. The
singularities in question arise from the choice of a non--generic polarization,
with respect to which we consider stability, and admit natural symplectic
resolutions corresponding to choices of general polarizations. For sheaves that
are pure of dimension one, we show that these moduli spaces are, locally around
a singular point, isomorphic to a quiver variety and that, via this
isomorphism, the natural symplectic resolutions correspond to variations of GIT
quotients of the quiver variety.Comment: 40 pages; final version; As pointed out to us by Z. Zhang, we prove
quadraticity and not formality of the Kuranishi family. Quadraticity is all
we need for our main theorem. The current version reflects this correction. A
few other improvements in exposition and correction of typo
Relative Prym varieties associated to the double cover of an Enriques surface
Given an Enriques surface T , its universal K3 cover f : S â T , and a genus g linear system |C| on T, we construct the relative Prym variety PH = Prymv,H(D/C), where C â |C| and D â |fâC| are the universal families, v is the Mukai vector (0, [D], 2â2g) and H is a polarization on S. The relative Prym variety is a (2gâ2)-dimensional possibly singular variety, whose smooth locus is endowed with a hyperk Ìahler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space Mv,H (S). There is a natural Lagrangian fibration η : PH â |C|, that makes the regular locus of PH into an integrable system whose general fiber is a (g â 1)-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if |C| is a hyperelliptic linear system, then PH admits a symplectic resolution which is birational to a hyperk Ìahler manifold of K3[gâ1]-type, while if |C| is not hyperelliptic, then PH admits no symplectic resolution. We also prove that any resolution of PH is simply connected and, when g is odd, any resolution of PH has h2,0-Hodge number equal to one
The Hodge numbers of O'Grady 10 via Ng\^o strings
We determine the Hodge numbers of the hyper-K\"ahler manifold known as
O'Grady 10 by studying some related modular Lagrangian fibrations by means of a
refinement of the Ng\^o Support Theorem.Comment: Revised and final version to appear in Jour. Math. Pur. et App
Poessagio: Poeti Italiani dâAmerica edited by Peter Carravetta and Paola Valesio and special double issue of Gradiva edited by Luigi Fontanella and Paolo Valesio
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