Delignes conjecture on extensions of 1-motives

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.

💡 Research Summary

The paper addresses a long‑standing problem in the theory of mixed motives: Deligne’s conjecture concerning extensions of 1‑motives. After recalling the classical definition of a 1‑motive (M=