Duke Mathematical Journal

Tautological classes on moduli spaces of hyper-Kähler manifolds

Nicolas Bergeron and Zhiyuan Li

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Abstract

We study algebraic cycles on moduli spaces Fh 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 Fh. We then study the images of these tautological classes in the cohomology groups of Fh 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 n2. 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
Received: 10 April 2017
Revised: 4 December 2018
First available in Project Euclid: 27 April 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1556330420

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

Mathematical Reviews number (MathSciNet)
MR3953432

Zentralblatt MATH identifier
07078882

Subjects
Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14C25: Algebraic cycles 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Keywords
K3 surfaces hyper-Kähler manifolds tautological classes

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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