Perverse obstructions to flat regular compactifications

Suppose $π:W\to S$ 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 $π$ admits a regular, flat compactification. In other words, when does there exists a flat, proper morphism $\barπ:\overline{W}\to\bar S$ extending $π$ with $\overline{W}$ regular? One interesting recent example of this occurs in the preprint arXiv:1602.05534 of Laza, Sacca and Voisin where $π$ 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 $R^1π_*\mathbb{Q}$ in degree at least $2$ provides an obstruction to finding a $\barπ$. Moreover, non-vanishing in degree $1$ provides an obstruction to finding a $\barπ$ 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.

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