Sharp de Rham realization

Abstract

We introduce the "sharp" (universal) extension of a 1-motive (with additive factors and torsion) over a field of characteristic zero. We define the "sharp de Rham realization" by passing to the Lie-algebra. Over the complex numbers we then show a (sharp de Rham) comparison theorem in the category of formal Hodge structures. For a free 1-motive along with its Cartier dual we get a canonical connection on their sharp extensions yielding a perfect pairing on sharp realizations. We thus provide "one-dimensional sharp de Rham cohomology" of algebraic varieties.Comment: 30 page

Mixed Hodge Structures With Modulus

“…Duality and Tate twists extend to mixed Hodge structures with modulus. (2) The category MHSM of mixed Hodge structures with modulus contains a full subcategory MHSM 1 which is equivalent to the category of Laumon 1-motives (the duality functor on mixed Hodge structures with modulus corresponding via this equivalence to Cartier duality). (3) Given a smooth proper variety X and two effective simple normal crossing divisors Y, Z on X such that |Y | ∩ |Z| = ∅, we associate functorially an object H n (X, Y, Z) of MHSM for each n ∈ Z.…”

Universal extension crystals of 1-motives and applications

“…The morphism u is a morphism as fppf sheaves on the category of affine k-schemes. In [3] a sharp universal extension E ♯ (M) of M was introduced as well as its sharp de Rham realization T ♯ (M). These constructions generalize E(M) and T dR (M) for Deligne 1-motives.…”

“…Since the category of tori over S is equivalent to the category of tori over S 0 , the morphism T → T ′ induced by E S (s 0 ) is the morphism γ defined in (4). Notice that we require that the morphism E(A) → E(B), induced by E S (s 0 ), coincides with that of (3). We conclude that the behavior of E S (s 0 ) on the graded pieces is uniquely determined.…”

Albanese varieties with modulus and Hodge theory

“…For each point v of X of codimension one, take a presentation ϕ = lg as in (2) Proof. -Let Y = mod(α F ) be the modulus of the rational map α F : X → Alb F (X) associated with F ∈ Λ.…”