## Duke Mathematical Journal

### Tautological classes on moduli spaces of hyper-Kähler manifolds

#### Abstract

We study algebraic cycles on moduli spaces $\mathcal{F}_{h}$ 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 $\mathcal{F}_{h}$. We then study the images of these tautological classes in the cohomology groups of $\mathcal{F}_{h}$ 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 hyper-Kähler manifolds with $n\leq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on a universal family of these hyper-Kähler manifolds.

#### Article information

Source
Duke Math. J., Volume 168, Number 7 (2019), 1179-1230.

Dates
Revised: 4 December 2018
First available in Project Euclid: 27 April 2019

https://projecteuclid.org/euclid.dmj/1556330420

Digital Object Identifier
doi:10.1215/00127094-2018-0063

Mathematical Reviews number (MathSciNet)
MR3953432

Zentralblatt MATH identifier
07078882

#### Citation

Bergeron, Nicolas; Li, Zhiyuan. Tautological classes on moduli spaces of hyper-Kähler manifolds. Duke Math. J. 168 (2019), no. 7, 1179--1230. doi:10.1215/00127094-2018-0063. https://projecteuclid.org/euclid.dmj/1556330420

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