SINGULAR SYMPLECTIC MODULI SPACES YINBANG LIN Abstract. These are notes of a talk given at the NEU-MIT graduate student seminar. It is based on the paper by Kaledin-Lehn-Sorger, showing examples of singular symplectic moduli spaces of sheaves that do not admit a symplectic resolution. 1. Introduction Let X be a projective K3 surface and H be an ample divisor. Let v ∈ H even (X, Z) be the Mukai vector of a sheaf. Let M v be the moduli space of Gieseker semistable sheaves with respect to the polarization H . Suppose v = mv 0 for a primitive v 0 , i.e. not an integral multiple of another Mukai vector, and m ∈ N. When v is primitive, that is m = 1, and H is generic, we know that M v is an irreducible symplectic manifold. This reflects the geometry of the surface. Barbara Bolognese [Bol16] has demonstrated an example that the moduli space is actually a K3 surface. When the moduli space has higher dimension, Isabel Vogt [Vog16] has explained that it is deformation equivalent to Hilbert scheme of points. When v is not primitive, the moduli space M v is singular. However, the stable locus M s v still admits a non-degenerate 2-form. We are interested in the question whether the 2-form can be extended to resolutions of singularities of M v . (Actually, if it extends to one, it extends to all.) Bolognese [Bol16] has shown us O’Grady’s example [O’G99] where the answer is positive. This article is primarily interested in the cases where the 2-form does not extend to a resolution of singularities. These are summarized in Table 2. 1 In this article, we will concentrate on the case where v 0 =(r 0 ,c 0 ,a 0 ) and m satisfy the following conditions. (1) Either r 0 > 0 and c 0 ∈ NS(X ), or r = 0, c 0 ∈ NS(X ) is effective, and a 0 / = 0. (2) m ≥ 3 and ⟨v 0 ,v 0 ⟩≥ 2, or m = 2 and ⟨v 0 ,v 0 ⟩≥ 4. The first condition makes sure that v 0 is the Mukai vector of a coherent sheaf. In the rest of this article, we will assume that v 0 and m satisfy these conditions. We aim to demonstrate the following result. Theorem. If either m ≥ 2 and ⟨v 0 ,v 0 ⟩> 2 or m > 2 and ⟨v 0 ,v 0 ⟩≥ 2, then M mv 0 is a locally factorial singular symplectic variety, which does not admit a proper symplectic resolution. 1 Similar statements also hold for abelian surfaces. 1