Perverse obstructions to flat regular compactifications

Abstract

Suppose $\pi :W\rightarrow S$ is a smooth, surjective, proper morphism to 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 ${\bar{\pi }}:\overline{W}\rightarrow {\bar{S}}$ extending $\pi$ with $\overline{W}$ regular? One interesting recent example of this occurs in the preprint of Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) where $\pi$ is a family of abelian fivefolds 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 (Astérisque 140–141:3–134, 251, 1986) Beilinson (Regulators 571:19–23, 2012) and Schnell (Math Ann 354(2):727–763, 2012) can be used to compute the intersection cohomology. I also give examples involving cubic fourfolds motivated by Laza et al. (Acta Mathematica. arXiv:1602.05534, 2016) and ask a question about palindromicity of hyperplane sections.