Relative Perversity

TL;DR

This paper introduces a new relative perverse t-structure for schemes, extending the classical theory to a relative setting and establishing properties like exactness, stability, and a notion of good reduction for perverse sheaves.

Contribution

It defines a relative perverse t-structure associated with finitely presented morphisms, linking it to nearby cycles and establishing its properties and applications.

Findings

Defines a relative perverse t-structure for schemes.

Shows the t-structure preserves universally locally acyclic sheaves.

Establishes a notion of good reduction for perverse sheaves.

Abstract

We define and study a relative perverse $t$-structure associated with any finitely presented morphism of schemes $f: X\to S$, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of $f$. The existence of this $t$-structure is closely related to perverse $t$-exactness properties of nearby cycles. This $t$-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category $\mathrm{Perv}^{\mathrm{ULA}}(X/S)$ with many of the same properties familiar in the absolute setting (e.g., noetherian, artinian, compatible with Verdier duality). For $S$ connected and geometrically unibranch with generic point $\eta$, the functor $\mathrm{Perv}^{\mathrm{ULA}}(X/S)\to \mathrm{Perv}(X_\eta)$ is exact and fully faithful, and its essential image is stable under passage to subquotients. This yields a notion of "good reduction" for perverse sheaves.