We prove the existence of a global family with natural universal properties over every component of the moduli space of marked irreducible holomorphic symplectic manifolds. The analogous result follows for Teichmüller spaces.
I thank Zhiyuan Li for his interesting talk, and for asking whether universal families exist over Teichmüller spaces of holomorphic symplectic manifolds, at the workshop “Hyper-Kähler Manifolds, Hodge Theory, and Chow Groups” at the Tsinghua Sanya International Mathematics Forum in December 2016. I thank the organizers, Radu Laza, Kieran O’Grady, and Claire Voisin for the invitation to this interesting and instructive workshop. I thank Daniel Huybrechts and Sukhendu Mehrotra for helpful communications and for their comments on an earlier draft of this article. I thank Jenia Tevelev for Remark 3.8. Finally, I thank the referees for numerous insightful comments and suggestions.
The author’s work was partially supported by Simons Foundation grant 427110.
"On the existence of universal families of marked irreducible holomorphic symplectic manifolds." Kyoto J. Math. 61 (1) 207 - 223, April 2021. https://doi.org/10.1215/21562261-2019-0075