Tautological classes on moduli spaces of hyper-Kähler manifolds

Abstract

In this paper, we discuss the cycle theory on moduli spaces F h of h-polarized hyperkähler manifolds. Firstly, we construct the tautological ring on F h following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups and prove that most of them are linear combinations of Noether-Lefschetz cycle classes. In particular, we prove the cohomological version of the tautological conjecture on moduli space of K3 [n] -type hyperkähler manifolds with n ≤ 2. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperkähler manifolds.

Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type

“…This is extended to certain families of hyper-Kähler varieties in [16] (cf. also [6]), and most notably for the square of the Fano variety of lines on a smooth cubic fourfold. The aim of this section is to generalize the latter to the Fano variety of lines on a smooth cubic hypersurface of any dimension.…”

“…Proof For the case (0, 9), the family T → B is constructed as the quotient where S → B is as in Notation 7.9(i) , and is as in Theorem 7.6. The quotient ℙ∕⟨ ⟩ can be identified with the weighted projective space ℙ � ∶= ℙ(2, 2, 2, 3, 3) , and quotients of the form T b = S b ∕⟨ b ⟩ can be identified with weighted complete intersections of degree (6,6) in ℙ ′ . It follows that T →B is the same as the universal family of weighted complete intersections of degree (6,6) in ℙ ′ .…”

Motives and the Pfaffian–Grassmannian equivalence

“…be smooth dimensionally transverse intersections, where U ⊂ ∧ 2 V is a codimension k linear subspace. Assume k 6 or (n, k) = (7,7). Assume also that (n, k) satisfies condition (NL), and that H * tr (X, Q) = 0.…”

“…Let X and Y be as above smooth dimensionally transverse intersections. Assume k < 6 is odd, or (n, k) = (7,7). Assume also that the transcendental cohomology H * tr (X, Q) is non-zero.…”

Cohomology of moduli space of cubic fourfolds I

“…But so far it is only known for genus less than 23. The cohomology ring of the moduli spaces F g of quasi-polarised K3 surface of fixed degree 2g − 2 has also many progress recently (see [5], [32] [4]). The tautological ring of moduli space F g involves more ingredients than that of M g .…”