Hilbert schemes of K3 surfaces are dense in moduli

Abstract

We prove that the locus of Hilbert schemes of n points on a projective K3 surface is dense in the moduli space of irreducible holomorphic symplectic manifolds of that deformation type. The analogous result for generalized Kummer manifolds is proven as well. Along the way we prove an integral constraint on the monodromy group of generalized Kummer manifolds.

Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds

“…We denoteSO + (L) := g ∈Õ + (L) | det(g) = 1 . It follows from Proposition 4.1 and from results by Markman [16], [18] and Mongardi [21] that Γ M,j is an arithmetic subgroup of O(N ) also for K3 [n] -type with n ≥ 3 and for generalized Kummer manifolds. Indeed, Mon 2 (L) is respectively isomorphic toÔ…”

On the monodromy of irreducible symplectic manifolds

“…We will apply this idea to two of the known examples: deformations of generalised Kummers and of O'Grady's ten dimensional manifold [13]. In the first case, thanks to previous computations by Markman [9], we have a lower bound for the monodromy group and the upper bound we obtain coincides with it, therefore we obtain the expected monodromy group. In the second case, as stated in [8,Conjecture 10.7], the monodromy group is expected to coincide with all orientation preserving isometries.…”

Integral cohomology of the generalized Kummer fourfold

“…The isometry τ is also constructed in [Shi78,Lemma 3], and it is shown there that it preserves the period. Moreover, by [MM17,Proposition 4.6], we have either…”

“…Moreover, since n + 1 is a prime power, by [MM17,Lemma 4.3], we can find ν ∈ N (K n (A)) = Mon(K n (A)) such that f • ϕ −1 • ψ • ν(δ) = δ, where δ is half the class of the diagonal divisor in K n (A). Hence, we can exchange the mark ψ for the mark ψ :…”

“…As in [MM17,Section 4], we now consider the dual complex torus A * of A. Then H 1 (A * , Z) is isomorphic to H 1 (A, Z) * .…”