3 - Motivic Decomposition for Resolutions of Threefolds

Summary

Introduction

This paper has two aims.

The former is to give an introduction to our earlier work and more generally to some of the main themes of the theory of perverse sheaves and to some of its geometric applications. Particular emphasis is put on the topological properties of algebraic maps.

The latter is to prove a motivic version of the decomposition theorem for the resolution of a threefold Y. This result allows to define a pure motive whose Betti realization is the intersection cohomology of Y.

We assume familiarity with Hodge theory and with the formalism of derived categories. On the other hand, we provide a few explicit computations of perverse truncations and intersection cohomology complexes which we could not find in the literature and which may be helpful to understand the machinery. We discuss in detail the case of surfaces, threefolds and fourfolds. In the surface case, our “intersection forms” version of the decomposition theorem stems quite naturally from two well-known and widely used theorems on surfaces, the Grauert contractibility criterion for curves on a surface and the so called “Zariski Lemma,” cf.

The following assumptions are made throughout the paper

Assumption 3.1.1.We work with varieties over the complex numbers. A map f : X → Y is a proper morphism of varieties. We assume that X is smooth. All (co)homology groups are with rational coefficients.