Deligne's conjecture on extensions of 1-motives

TL;DR

This paper introduces the concept of extensions of 1-motives, explores their realizations in various cohomological theories, and proves Deligne's conjecture regarding these extensions.

Contribution

It defines extensions of 1-motives, relates them to Picard stacks, and verifies Deligne's conjecture through cohomological realizations.

Findings

Extensions of 1-motives induce extensions of Picard stacks

Computed Hodge, de Rham, and l-adic realizations of these extensions

Proved Deligne's conjecture on extensions of 1-motives

Abstract

We introduce the notion of extension of 1-motives. Using the dictionary between strictly commutative Picard stacks and complexes of abelian sheaves concentrated in degrees -1 and 0, we check that an extension of 1-motives induces an extension of the corresponding strictly commutative Picard stacks. We compute the Hodge, the de Rham and the l-adic realizations of an extention of 1-motives. Using these results we can prove Deligne's conjecture on extensions of 1-motives.