The Hodge theory of the Decomposition Theorem (after de Cataldo and Migliorini)

TL;DR

This paper discusses a new, elementary proof of the Decomposition Theorem in algebraic geometry, connecting Hodge theory and perverse sheaves, with broad implications across multiple mathematical disciplines.

Contribution

It provides a simplified proof of the Decomposition Theorem using classical Hodge theory and perverse sheaves, making the theorem more accessible and easier to apply.

Findings

New proof of the Decomposition Theorem using classical Hodge theory

Connections established between Hodge theory and perverse sheaves

Broader applicability in algebraic geometry and related fields

Abstract

In its simplest form the Decomposition Theorem asserts that the rational intersection cohomology of a complex projective variety occurs as a summand of the cohomology of any resolution. This deep theorem has found important applications in algebraic geometry, representation theory, number theory and combinatorics. It was originally proved in 1981 by Beilinson, Bernstein, Deligne and Gabber as a consequence of Deligne's proof of the Weil conjectures. A different proof was given by Saito in 1988, as a consequence of his theory of mixed Hodge modules. More recently, de Cataldo and Migliorini found a much more elementary proof which uses only classical Hodge theory and the theory of perverse sheaves. We present the theorem and outline the main ideas involved in the new proof.