On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces

Abstract

In this paper we prove a conjecture of Markman about the shape of the monodromy group of irreducible holomorphic symplectic manifolds of OG10 type. As a corollary, we also compute the locally trivial monodromy group of the underlying singular symplectic variety.

O’Grady tenfolds as moduli spaces of sheaves

“…We prove that X is birational to 𝑀 𝑣 (𝑆, 𝜃) by showing that there exists a Hodge isometry between 𝐻 2 (𝑋, Z) and 𝐻 2 ( 𝑀 𝑣 (𝑆, 𝜃), Z). In fact, when such an isometry exists, birationality is implied by the maximality of Mon 2 (OG10) for OG10 type manifolds; see [43] and [37,Theorem 5.2(2)]. Observe that we have a Hodge isometry…”

“…We prove that X is birational to by showing that there exists a Hodge isometry between and . In fact, when such an isometry exists, birationality is implied by the maximality of for type manifolds; see [43] and [37, Theorem 5.2(2)]. Observe that we have a Hodge isometry and two finite index embeddings We want to lift the Hodge isometry to a Hodge isometry so that the following diagram commutes: Observe that, as abstract lattices, and are both isomorphic to .…”

Symplectic rigidity of O’Grady’s tenfolds

“…By [17,Theorem 5.4], the monodromy group of a manifold of type OG10 is the whole group O + (L). Since L G is negative definite, we have by [6,Lemma 2.3] that G ⊂ O + (L), so that the first condition of the Hodge-theoretic Torelli theorem is satisfied.…”

Nonsymplectic automorphisms of prime order on O’Grady’s sixfolds

“…Recently, the author together with Onorati and Veniani [19] classified symplectic birational transformations on manifolds of OG6 type in the case of finite cyclic groups, hence this paper completes the classification of automorphisms of manifolds of OG6 type. The classification of nonsymplectic automorphisms on manifolds of OG10 type was started by Brandhorst-Cattaneo [11], and recent progress by Onorati [34] about the monodromy group and the wall divisors for this deformation class constitutes a starting point for the study of the symplectic case.…”