Tautological classes on moduli spaces of hyper-Kähler manifolds

Source abstract

We study algebraic cycles on moduli spaces Fh of h-polarized hyper-Kähler manifolds. Following previous work of Marian, Oprea, and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces, we first define the tautological ring on Fh. We then study the images of these tautological classes in the cohomology groups of Fh 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 hyper-Kähler manifolds with n≤2. Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.