We study algebraic cycles on moduli spaces of -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 . We then study the images of these tautological classes in the cohomology groups of 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-type hyper-Kähler manifolds with . Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.
"Tautological classes on moduli spaces of hyper-Kähler manifolds." Duke Math. J. 168 (7) 1179 - 1230, 15 May 2019. https://doi.org/10.1215/00127094-2018-0063