Hyper-Kähler fourfolds and Grassmann geometry

Abstract

Abstract. We construct a new 20-dimensional family of algebraic hyper-Kähler fourfolds and prove that they are deformationequivalent to the second punctual Hilbert scheme of a K3 surface of degree 22.

Moduli of double EPW-sextics

“…By the self-duality of S F F2 that we discussed above it suffices to prove that if µ(A, λ) ≥ 0 for an ordering 1-PS λ with m ∈ {0, 1} then A satisfies one of Items (1)- (4). In other words it suffices to check that if none of Items (1)- (4) is satisfied then µ(A, λ) < 0 for all ordering 1-PS λ with m ∈ {0, 1}.…”

“…a double cover of a particular kind of sextic hypersurface (an EPW-sextic, first introduced by Eisenbud-Popescu-Walter in [5]) and hence it has an explicit description. We should point out that the generic double EPW-sextic is not isomorphic (nor birational) to the Hilbert square of a K3 surface, and that only a handful of explicit locally complete families of hyperkähler varieties of dimension greater than 2 have been constructed, see [2,4,12,13,16] for the other families.…”

Chow Rings, Decomposition of the Diagonal, and the Topology of Families

“…[n] of length n subschemes of an algebraic K3 surface S. Such hyper-Kähler varieties admit projective deformations that are not obtained by the same construction, but they are not well understood except in a few cases, namely the four different families of hyper-Kähler fourfolds constructed in the papers [10], [29], [55], [77]. The varieties constructed by Beauville and Donagi are obtained as Fano varieties of lines of smooth cubic fourfolds in P 5 .…”

“…Let l = c top 1 (L) ∈ H 2 (X , Q) and decompose as above l = l + π * k, where l has the same image as l in H 0 (B, R 2 π * Q) and k belongs to H 2 (B, Q). Denoting by n the dimension of the fibers, we get 29) Recall now that the decomposition is multiplicative. The class l n l i thus belongs to a direct summand of H 2n+2 (X , Q) isomorphic to H 0 (B, R 2n+2 π * Q) = 0.…”

Deformations of singular symplectic varieties and termination of the log minimal model program

“…For instance, there is a very interesting line of research started by Greb, Kebekus and Peternell on varieties with numerically trivial canonical divisor and singularities that appear in the minimal model program (MMP); see [GKP11]. On the other hand, singular symplectic varieties also play an important role in the study of smooth symplectic varieties; see, for example, [DV10,OG99,OG03]. Given the importance of Huybrechts' theorem to the theory of irreducible symplectic…”