The Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces and generalizations

TL;DR

This paper determines the Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces, showing it is isomorphic to Z, and provides a general theorem for orthogonal Shimura varieties.

Contribution

It proves the Picard group of the compactified moduli space of K3 surfaces is isomorphic to Z, contrasting with the more complex structure in the case of smooth curves.

Findings

Picard group of the moduli space of quasi-polarized K3 surfaces is Z

General theorem for orthogonal Shimura varieties with Picard group of dimension 1

Construction of an arithmetic obstruction space related to Heegner divisors

Abstract

In this paper, we investigate the Picard group of the Baily--Borel compactification of orthogonal Shimura varieties. As a key result, we determine the Picard group of the Baily--Borel compactification of the moduli space of quasi-polarized K3 surfaces, proving that it is isomorphic to $\mathbb{Z}$. Notably, this contrasts with the moduli space of smooth curves, where the Picard group exhibits a more complex structure after natural compactification. Our result follows from a general theorem for orthogonal Shimura varieties: for even lattices $M$ of signature $(2,n)$ with $n > 8$ satisfying specific arithmetic conditions (e.g. K3 type or $p$-elementary), the rational Picard group of $\overline{\operatorname{Sh}}_\Gamma(M)$ with $\Gamma$ containing the stable orthogonal group is $1$-dimensional. The core of our proof lies in constructing an arithmetic obstruction space that governs the extension of Heegner divisors to the generic points of the boundary in orthogonal Shimura varieties. We further establish a connection between this obstruction space and the space of theta series, demonstrating that the obstruction space is maximal under our conditions.