O'Grady tenfolds as moduli spaces of sheaves

Abstract

We give a lattice-theoretic characterization for a manifold of OG10 type to be birational to some moduli space of (twisted) sheaves on a K3 surface. We apply it to the Li-Pertusi-Zhao variety of OG10 type associated to any smooth cubic fourfold. Moreover we determine when a birational transformation is induced by an automorphism of the K3 surface and we use this to classify all induced birational symplectic involutions.

On the period of Li, Pertusi, and Zhao’s symplectic variety

“…From this point of view, the present work can be thought as a natural continuation of the aforementioned works. Recently, similar results of the ones in the present paper appeared in [FGG23].…”

Symplectic rigidity of O’Grady’s tenfolds

“…Put n := rk(L g ). Note that ℓ ≤ ℓ 2 ((L g ) ♯ ) ≤ rk L g = 24 − n and ℓ ≤ ℓ 2 (L ♯ g ) ≤ rk L g = n. Thus, we obtain the following upper bound: (4) |det(L g )| ≤ 3 • 2 min(n,24−n) .…”

“…On the other hand, Theorem 1.1 does not hold for birational transformations. Indeed, it is known for instance that manifolds of type OG10 can admit symplectic birational involutions (see Remark 2.3 for a simple lattice theoretical argument, [1, §7.3] for a geometrical example, [12] for a complete classification, and [4] for induced symplectic birational involutions).…”