Tautological classes on moduli spaces of hyper-Kähler manifolds
Résumé
We study algebraic cycles on moduli spaces F h of h-polarized hyperkähler 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 F h. We then study the images of these tautological classes in the cohomology groups of F h 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 man-ifolds with n ≤ 2. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperkähler mani-folds.
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