On IHS fourfolds with $b_2 = 23$

Abstract

The present work is concerned with the study of four-dimensional irreducible holomorphic symplectic manifolds with second Betti number 23. We describe their birational geometry and their relations to EPW sextics.Proposition 2.1. Let X be an IHS fourfold with b 2 = 23. The Fujiki constant of X is an integer of the form 3n 2 for some n ∈ N. In particular the minimal degree of the self-intersection H 4 of an ample divisor H ⊂ X is 12 and in this case h 0 (O X (H)) = 6. Proof. First from the H-R-R theorem for IHS fourfolds we infer thatRemark 2.2. For an IHS manifold X with b 2 (X) = 23 to admit an ample divisor with H 4 = 12 there are two possibilities:• The Fujiki invariant is 3, the B-B lattice is even and there exists an h ∈ H 2 (X, Z) with (h, h) = 2. • The Fujiki invariant is 12 and there exists an h ∈ H 2 (X, Z) with (h, h) = 1.It is a natural problem to decide whether the latter case can occur. 3 6

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

“…See [21,22] for extra constraints on each cases; see [35,51] for related results in dimension 6 and the more recent work [33] for a conjectural bound in arbitrary dimension based on [19]. When b 2 = 23, let us mention the work [31,45], which aims at determining the deformation type of IHS fourfolds upon fixing some extra topological data.…”

On the Betti numbers of compact holomorphic symplectic orbifolds of dimension four

“…See [18] and [19] for extra constraints on each cases; see [30] for a related result in dimension 6. When b 2 = 23, let us mention the work [40] [27], which aims at determining the deformation type of IHS fourfolds upon fixing some extra topological data.…”

EPW cubes

“…We call the invariant sets G, Ω, F and P 19 the (projective) orbits of ∧ 3 for P GL(6). See [Kap14,Appendix] for some results about the geometry of Ω and its relations with EPW sextics. For any nonzero vector w ∈ W , denote by…”