Deligne's duality for de Rham realizations of 1‐motives

Abstract

Let M be a 1-motive over a base scheme S and M its Cartier dual. We show the existence of a canonical duality between the de Rham realizations of M and M ; this generalizes a result in [5]. Furthermore, we study universal extensions of 1-motives and their relation with -extensions.

Universal extension crystals of 1-motives and applications

“…⟨ , ⟩ Id M 0 and ⟨ , ⟩ Id N 0 . (c) It follows from (ii) applied to U = S 0 and Z = S 0 and from Section 5.5 that the pairing ⟨ , ⟩ s 0 (S 0 = S 0 ) coincides with Deligne's pairing (14) on [5,Section 4]. In particular, if s 0 = Id M 0 , it is perfect.…”

“…Since a morphism of crystals over S crys 0 , which is an isomorphism when restricted to S Zar 0 , is an isomorphism, we conclude that ⟨ , ⟩ Id M 0 is a perfect pairing. Let q : S → T be a morphism of schemes and let s : M → N be a morphism of 1-motives over S. Then, s induces Deligne's pairing [5,Section 4], and, hence, a morphism of O S -modules…”

“…Recall now that, if P denotes the Poincaré biextension of M and P E(M) its pull-back to the universal extensions, the biextension P E(M) is endowed with a canonical ♮-structure ([14, 10.2.7.2], [5,Section 4]). More generally, given any G mbiextension P of 1-motives M and N there is a canonical ♮-structure on the pull-back P E of P to E(M) and E(N).…”

“…[14, 10.1.7]). (For the existence of universal extensions see also [1,19,5]). Note that E(M) is endowed with a weight filtration where W 0 …”

“…Deligne's pairing in[5, Section 4] constructed via the canonical ♮-structure ∇ M on P E(M) could be constructed using any ♮-structure on P E(M) . Proof of Corollary 2.6.…”