The Hodge theory of algebraic maps

Abstract

We give a geometric proof of the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber for the direct image of the intersection cohomology complex under a proper map of complex algebraic varieties. The method rests on new Hodge-theoretic results on the cohomology of projective varieties which extend naturally the classical theory and provide new applications.  2005 Elsevier SAS RÉSUMÉ.-On donne une démonstration géométrique du théorème de décomposition de Beilinson, Bernstein, Deligne et Gabber pour l'image directe, par un morphisme propre de variétés algébriques complexes, du complexe de cohomologie d'intersection. La preuve s'appuie sur des résultats nouveaux concernant la théorie de Hodge des variétés projectives, qui généralisent la théorie classique et donnent de nouvelles applications.

Topology of Hitchin systems and Hodge theory of character varieties: the case A_1

“…The perverse Leray filtration has been implicitly introduced in [7], and it has been studied and employed in [12], [14], [15], [8], [9]. This filtration is the abutment of the perverse Leray spectral sequence which, in turn, is a variant of the classical Leray spectral sequence.…”

“…Definition of the perverse filtration P on H * (M ). We employ freely the language of derived categories and perverse sheaves (see the seminal paper [7], the survey [13], or for example the paper [12]). Standard textbooks on the subject are [21], [37], [38].…”

On the Hard Lefschetz property of stringy Hodge numbers

“…This result is discussed by de Cataldo and Migliorini in Section 2.3 and the beginning of Section 2.4 from [dCM2] for dimensions 3 and 4, but it is not hard to see that their argument works in any dimension. It also follows from their earlier work [dCM1]; see, in particular, Section 2.4 therein. We note that the choice of the space …”

Support Theorems for Algebraic Maps

“…For every α, there are local systems L i α , for i = 1, · · · , k α on Y α and integers n i α such that Support Theorems for Algebraic Maps 37 Support Theorems for Algebraic Maps 17 culmination of his formidable theory of "mixed Hodge modules," [37,38,39]. Another proof, relying on classical Hodge theory, was given in [6] for semi-small maps, and subsequently in [7] in the general case. The papers [8,9,10,12,13] investigate several different aspects of the decomposition theorem.…”