Albanese varieties with modulus and Hodge theory

The unipotent Albanese map and Selmer varieties for curves

We discuss p-adic unipotent Albanese maps for curves of positive genus, extending the theory of p-adic multiple polylogarithms. This construction is then used to relate linear Diophantine conjectures of ‘Birch and Swinnerton-Dyer type’ to non-linear theorems of Faltings-Siegel type. In a letter to Faltings [14] dated June, 1983, Grothendieck proposed several striking conjectural connections bet...

Hodge cycles on abelian varieties

This is a TeXed copy of – Hodge cycles on abelian varieties (the notes of most of the seminar “Périodes des Intégrales Abéliennes” given by P. Deligne at I.H.E.S., 1978–79; pp9– 100 of Deligne et al. 1982). somewhat revised and updated. See the endnotes1 for more details.

Kuga-satake Varieties and the Hodge Conjecture

Kuga-Satake varieties are abelian varieties associated to certain weight two Hodge structures, for example the second cohomology group of a K3 surface. We start with an introduction to Hodge structures and we give a detailed account of the construction of Kuga-Satake varieties. The Hodge conjecture is discussed in section 2. An excellent survey of the Hodge conjecture for abelian varieties is [...

Hodge Theory and Representation Theory

Hodge Theory and Geometry

This expository paper is an expanded version of a talk given at the joint meeting of the Edinburgh and London Mathematical Societies in Edinburgh to celebrate the centenary of the birth of Sir William Hodge. In the talk the emphasis was on the relationship between Hodge theory and geometry, especially the study of algebraic cycles that was of such interest to Hodge. Special attention will be pl...