# Tautological classes on moduli space of hyperk\"ahler manifolds

## TL;DR

This paper develops the tautological ring on moduli spaces of hyperk"ahler manifolds, showing most classes are combinations of Noether-Lefschetz classes and proving related conjectures for specific cases.

## Contribution

It constructs the tautological ring on moduli spaces of hyperk"ahler manifolds and proves the cohomological tautological and Franchetta conjectures in certain cases.

## Key findings

- Most tautological classes are linear combinations of Noether-Lefschetz classes.
- Proved the cohomological tautological conjecture for K3$^{[n]}$-type hyperk"ahler manifolds with n≤2.
- Established the cohomological generalized Franchetta conjecture for these moduli spaces.

## Abstract

In this paper, we discuss the cycle theory on moduli spaces $\cF_h$ of $h$-polarized hyperk\"ahler manifolds. Firstly, we construct the tautological ring on $\cF_h$ following the work of Marian, Oprea and Pandharipande on the tautological conjecture on moduli spaces of K3 surfaces. We study the tautological classes in cohomology groups 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 hyperk\"ahler manifolds with $n\leq 2$. Secondly, we prove the cohomological generalized Franchetta conjecture on universal family of these hyperk\"ahler manifolds.