[Paper Review] On the monodromy of moduli spaces of sheaves on K3 surfaces II

Eyal Markman|ArXiv.org|May 1, 2003

Algebraic Geometry and Number Theory44 references15 citations

TL;DR

This paper establishes that the monodromy representation of the cohomology of Hilbert schemes of points on K3 surfaces arises from geometric correspondences associated with reflections in the Mukai lattice. It proves that these correspondences induce monodromy operators via stratified elementary transformations and birational involutions, linking algebraic geometry to derived categories and Hodge theory.

ABSTRACT

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.

Motivation & Objective

• To establish a geometric realization of the monodromy representation on the integral cohomology of Hilbert schemes of points on K3 surfaces.
• To connect the action of isometries in the Mukai lattice—specifically reflections with respect to $-2$ and $+2$ vectors—to actual correspondences on the moduli space.
• To demonstrate that these correspondences induce monodromy operators via deformation theory and stratified elementary transformations.
• To relate the cohomological action of these isometries to autoequivalences of the derived category $D(S)$, linking geometry to categorification.
• To prove that the induced cohomology endomorphisms are monodromy operators by analyzing degenerations and local systems over curves with involution.

Proposed method

• Constructs a correspondence $\sum_{t=0}^\mu \mathcal{Z}_t \subset S^{[n]} \times S^{[n]}$ as the closure of fiber products over Brill-Noether strata of the Hilbert scheme.
• Uses stratified elementary transformations to deform the correspondence over a curve $C$ tangent to the negative part of the quadratic form at a singular point $\ell$.
• Applies deformation to the normal cone of the smallest stratum to analyze the behavior of the correspondence near the singular fiber.
• Relies on the existence of a birational involution induced by a $+2$ vector, arising from a very ample line bundle $L$ on $S$, to define a relative correspondence.
• Uses the involution $\rho$ on the base curve $C$ to relate the classifying maps $\kappa'$ and $\kappa''$ to the period map, showing $p \circ \kappa' = \rho \circ p \circ \kappa''$.
• Establishes that the induced cohomology action $\mathcal{Z}_*$ is the monodromy operator by analyzing the monodromy of the local system over $\overline{C} \setminus \{\bar{\ell}\}$, quotiented by the involution.

Experimental results

Research questions

• RQ1 How can the monodromy representation of the cohomology of $S^{[n]}$ be geometrically realized via algebraic correspondences?
• RQ2 What is the relationship between reflections in the Mukai lattice (with respect to $-2$ and $+2$ vectors) and autoequivalences of $D(S)$?
• RQ3 How do Brill-Noether stratifications on $S^{[n]}$ induce correspondences that realize isometries in $\Gamma_v$?
• RQ4 Can the cohomological action of such isometries be shown to be a monodromy operator via deformation and local system analysis?
• RQ5 To what extent is the monodromy action independent of the choice of curve $C$ and analytic in the parameter $t$?

Key findings

• The correspondence $\sum_{t=0}^\mu \mathcal{Z}_t$ induces the cohomology endomorphism $\gamma(\tau_n)$, which corresponds to the reflection $\tau_n$ in the Mukai lattice.
• The cohomology class of $\sum_{t=0}^\mu \mathcal{Z}_t$ is expressed in terms of Chern classes of the universal ideal sheaf on $S \times S^{[n]}$, via formula (7).
• The action $\mathcal{Z}_*$ on $H^*(S^{[n]}, \mathbb{Z})$ is identified as a monodromy operator through the monodromy of the local system over $\overline{C} \setminus \{\bar{\ell}\}$.
• The construction is independent of the choice of curve $C$ and varies analytically with $t$, ensuring the monodromy is well-defined.
• The involution $\rho$ induced by the $+2$ reflection corresponds to a birational involution on $S^{[n]}$, sending a length $n$ subscheme $D$ to the complementary subscheme $Z \setminus D$ in a $2n$-point subscheme $Z$.
• The monodromy action arises as the limit $\lim_{t \to \ell} g_{t*}$, where $g_t$ is the isomorphism between fibers over $t \in C \setminus \{\ell\}$, confirming the geometric origin of the monodromy.