On the monodromy of moduli spaces of sheaves on K3 surfaces II

By E. Markman. 2003

103 citations recorded by the source corpus.

Let S be a K3 surface. In part I of this paper, we constructed a representation of the group Aut D(S), of auto-equivalences of the derived category of S. We interpreted this infinite dimensional representation, as the natural action of Aut D(S) on the cohomology of all moduli spaces of stable sheaves (with primitive Mukai vectors) on S. The main result, of part I, is the precise relation of this action with the monodromy of the Hilbert schemes S^[n] of points on the surface. The proof of the above result was reduced, in part I, to the case of two monodromy operators of S^[n], associated with choices of line bundles on the surface S, of degree 2n-4 and 2n respectively. When n=1, the first sequence of monodromy operators specializes to the reflection by a -2 curve. The n=1 case of the second sequence is related to the Galois involution, of a double cover of the projective plane, branched along a sextic. We complete the proof by treating these two sequences of examples.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3d979aaa6dcf98c22d3818fdc2b14c9fd7ee4953

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Fano varieties of cubic fourfolds containing a plane (2009, background): “y H2(F(Y),Z) ∼= H2(M(S′,v′,B′),Z) as in Lemma 3.8, by the main result in [36] (see also [18]), we get a birational map F(Y) 99K M(S′,v′,B′) as required. Notice that here the monodromy calculations in [30, 31] are crucial. Remark 3.9. As suggested by one of the referees, one may give a diﬀerent geome”
A global Torelli theorem for hyperkaehler manifolds (after Verbitsky) (2011, methodology): “The monodromy group Mon(X) has been computed by Markman for X = Hilb(S) and arbitrary n (see [19, 20]).”
Moduli of K3 Surfaces and Irreducible Symplectic Manifolds (2010, background): “Markman first noticed the importance of these operators for the study of irreducible symplectic manifolds, developing his ideas in a series of papers [84], [83], [85].”
Prime exceptional divisors on holomorphic symplectic varieties and monodromy reflections (2009, background): “ in the proof of Proposition 3.18 in [Ma1]. 2) Let w∈ K(S) be a class orthogonal to (1,H2,−1). The equality Z PH0(F) ψ∗(θ(w)) = −(v 0,w) follows by an argument identical to the proof of Lemma 4.11 in [Ma3]. It follows that ψ(PH0(F)) has class (θ(−v0),•) in H2(MH(1,H2,−1),Z). Lemma 10.12. The closure”
Intersection Numbers of Extremal Rays on Holomorphic Symplectic Varieties (2009, background): “[20] E. Markman, On the monodromy of moduli spaces of sheaves on K3 surfaces, J. Algebraic Geom., 17:1 (2008), pp. 29–99.”
Hitchin fibrations, abelian surfaces, and the P=W conjecture (2019, methodology): “We refer to [39] for a parallel theory concerning K3 surfaces.”
Hodge theory and Lagrangian planes on generalized Kummer fourfolds (2010, background): “In light of Markman’s results on the monodromy of Hilbert schemes of K3 surfaces [23, 22], it is natural to ask the following:”
Lagrangian Fibrations of Holomorphic-Symplectic Varieties of K3[n]-Type (2013, methodology): “, The τ O ( Λ ) , given by ρ τ ( λ ) = − λ + ( λ, τ ) τ , belongs to Mon 2 ( X ) by ([Ma1], Corollary 1.8), and it sends α 2 to − α 2 .”