Perverse obstructions to flat regular compactifications

TL;DR

This paper investigates obstructions to extending smooth, proper morphisms to regular, flat compactifications, highlighting the role of local intersection cohomology and providing examples involving cubic fourfolds.

Contribution

It identifies intersection cohomology non-vanishing as an obstruction to such compactifications and discusses computational approaches and specific examples.

Findings

Non-vanishing of local intersection cohomology obstructs regular flat compactifications.

Non-vanishing in degree 1 obstructs irreducible fibers in the compactification.

Examples include cubic 4-folds and applications of Brylinski, Beilinson, and Schnell's results.

Abstract

Suppose $\pi:W\to S$ is a smooth, proper morphism over a variety $S$ contained as a Zariski open subset in a smooth, complex variety $\bar S$. The goal of this note is to consider the question of when $\pi$ admits a regular, flat compactification. In other words, when does there exists a flat, proper morphism $\bar\pi:\overline{W}\to\bar S$ extending $\pi$ with $\overline{W}$ regular? One interesting recent example of this occurs in the preprint arXiv:1602.05534 of Laza, Sacca and Voisin where $\pi$ is a family of abelian $5$-folds over a Zariski open subset $S$ of $\bar S=\mathbb{P}^5$. In that paper, the authors construct $\overline{W}$ using the theory of compactified Prym varieties and show that it is a holomorphic symplectic manifold (deformation equivalent to O'Grady's $10$-dimensional example). In this note I observe that non-vanishing of the local intersection cohomology of $R^1\pi_*\mathbb{Q}$ in degree at least $2$ provides an obstruction to finding a $\bar\pi$. Moreover, non-vanishing in degree $1$ provides an obstruction to finding a $\bar\pi$ with irreducible fibers. Then I observe that, in some cases of interest, results of Brylinski, Beilinson and Schnell can be used to compute the intersection cohomology. I also give examples involving cubic $4$-folds, and ask a question about palindromicity of hyperplane sections.