Jonathan Woolf

Witt groups of sheaves on topological spaces

This paper investigates the Witt groups of triangulated categories of sheaves (of modules over a ring R in which 2 is invertible) equipped with Poincare-Verdier duality. We consider two main cases, that of perfect complexes of sheaves on locally compact Hausdorff spaces and that of cohomologically constructible complexes of sheaves on polyhedra. We show that the Witt groups of the latter form a generalised homology theory for polyhedra and continuous maps. Under certain restrictions on the ring R, we identify these constructible Witt groups of a finite simplicial complex with Ranickis free symmetric L-groups. Witt spaces are the natural class of spaces for which the rational intersection homology groups have Poincare duality. When the ring R is the rationals we identify the constructible Witt groups with the 4-periodic colimit of the bordism groups of PL Witt spaces. This allows us to interpret L-classes of singular spaces as stable homology operations from the constructible Witt groups to rational homology.

The decomposition theorem and the intersection cohomology of quotients in algebraic geometry

Suppose a connected reductive complex algebraic group G acts linearly on an irreducible complex projective variety X. We prove that if 1→N→G→H→1 is a short exact sequence of connected reductive groups and Xss the open set of semistable points for the action of N on X then IHH∗(XssN) is (non-canonically) a direct summand of IHG∗(Xss). The inclusion is provided by the decomposition theorem and certain resolutions of the action allow us to define projections.

Some metric properties of spaces of stability conditions

We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped with half the hyperbolic metric. We also make two observations about the way in which the heart changes as we move through the space of stability conditions. Firstly, hearts of stability conditions in the same component of the space of stability conditions are related by finite sequences of tilts. Secondly, if each of a convergent sequence of stability conditions has the same heart then the heart of the limiting stability condition is obtained from this by a right tilt.

The Kirwan Map for Singular Symplectic Quotients

Let M be a Hamiltonian K -space with proper moment map

INTERSECTION COHOMOLOGY OF SYMPLECTIC QUOTIENTS BY CIRCLE ACTIONS

\mu

The Kirwan map, equivariant Kirwan maps, and their kernels

. The symplectic quotient

An introduction to intersection homology theory.

X=\mu^{-1}(0)/K

Stability conditions, torsion theories and tilting

is a singular stratified space with a symplectic structure on the strata. In this paper we generalise the Kirwan map, which maps the K equivariant cohomology of

Cohomology pairings on singular quotients in geometric invariant theory

\mu^{-1}(0)

The fundamental category of a stratified space

to the middle perversity intersection cohomology of X , to this symplectic setting. The key technical results which allow us to do this are Meinrenkens and Sjamaars partial desingularisation of singular symplectic quotients and a decomposition theorem, proved in Section 2 of this paper, exhibiting the intersection cohomology of a ‘symplectic blowup’ of the singular quotient X along a maximal depth stratum as a direct sum of terms including the intersection cohomology of X .