Birational geometry of irreducible holomorphic symplectic tenfolds of O’Grady type

O’Grady tenfolds as moduli spaces of sheaves

“…In fact, the coinvariant lattice is isometric to the coinvariant lattice with respect to the induced action of on the second integral cohomology lattice of the manifold of type; hence, it does not contain prime exceptional divisors by [16, Theorem 2.2]. The characterization of prime exceptional divisors for manifolds of type (see [36, Proposition 3.1]) thus implies that does not contain any vectors of square . We apply [32, Theorem 1.3], and we conclude that .…”

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

“…The generality assumption can conjecturely be made more precise by saying that V does not contain a plane or a rational cubic scroll (see [LPZ20, Section 5.7] for the case of a rational cubic scroll -the case of a plane is expected to behave similarly), and in these two cases one expects to find examples of the walls of the Kähler cone described in [MZ16] (in the singular setting) and [MO22] (in the desingularisation).…”

The pseudo-effective cone of the known irreducible holomorphic symplectic manifolds

“…The following conjecture holds true for all the known deformation classes of (projective) IHS manifolds (cf. [MO20], [MR20] for the O'Grady-type case, [Mat17] for the Kum n -type and K3 [n] -type case), but it is not known in general.…”

“…Hence we had to restrict to the known cases, where the monodromy and the stably exceptional classes are well known, thanks to recent work (cf. [Mar13], [MO20], [MR20]). In particular, the idea is that a stably exceptional class stays (up to a sign) stably exceptional under the action of the monodromy group, and a stably exceptional class is effective.…”