Universal covering calabi-yau manifolds of the Hilbert schemes of n points of Enriques surfaces

TL;DR

This paper studies the universal covering Calabi-Yau manifolds of Hilbert schemes of points on Enriques surfaces, exploring their deformations, automorphisms, and classification of isomorphism classes.

Contribution

It establishes relationships between deformations of Hilbert schemes and their universal covers, analyzes automorphisms, and classifies Hilbert schemes with a fixed universal cover.

Findings

Deformation of Hilbert schemes relates to deformation of their universal cover.

Automorphisms of Hilbert schemes are characterized.

Number of isomorphism classes with a fixed universal cover is determined.

Abstract

Throughout this paper, we work over ${\mathbb C}$, and $n$ is an integer such that $n\geq 2$. For an Enriques surface $E$, let $E^{[n]}$ be the Hilbert scheme of $n$ points of $E$. By Oguiso and Schr\"oer, $E^{[n]}$ has a Calabi-Yau manifold $X$ as the universal covering space, $\pi :X\rightarrow E^{[n]}$ of degree $2$. The purpose of this paper is to investigate a relationship of the small deformation of $E^{[n]}$ and that of $X$ $({\rm Theorem}\ 1.1)$, the natural automorphism of $E^{[n]}$ $({\rm Theorem}\,1.2)$, and count the number of isomorphism classes of the Hilbert schemes of $n$ points of Enriques surfaces which has $X$ as the universal covering space when we fix one $X$ $({\rm Theorem}\,1.3)$.