Witt groups of abelian categories and perverse sheaves

TL;DR

This paper investigates the structure of Witt groups of perverse sheaves on stratified spaces, revealing a canonical decomposition and introducing a new splitting relation that generalizes isotropic reduction.

Contribution

It introduces a novel splitting relation for Witt classes of perverse sheaves and demonstrates a canonical decomposition of Witt groups in this context.

Findings

Witt groups decompose as a direct sum of local system Witt groups.

A new splitting relation generalizes isotropic reduction.

Methods are algebraic and applicable to broader triangulated categories.

Abstract

In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another `splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a `splitting relation' which is a generalisation of isotropic reduction. The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin. Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.