Tautological classes on moduli spaces of hyper-Kähler manifolds

Nicolas Bergeron (1) , Zhiyuan Li (2)

1 IMJ - Institut de Mathématiques de Jussieu
2 LMVI - Key Laboratory of Molecular Virology & Immunology

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.

Domaines: Mathématiques [math]; Géométrie algébrique [math.AG]

Nicolas Bergeron, Zhiyuan Li. Tautological classes on moduli spaces of hyper-Kähler manifolds. Duke Mathematical Journal, 2019, 168 (7), pp.1179-1230. ⟨10.1215/00127094-2018-0063⟩. ⟨hal-02794005⟩