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Realizations of 1-motives over a scheme of characteristic 0

Cristiana Bertolin

Abstract

Let S be a connected and smooth scheme of finite type over the complex numbers. We construct functorially the Hodge realization of a 1-motive over S as a torsion-free, polarizable and admissible variation of mixed Hodge structures of type (0,0),(-1,0),(0,-1),(-1,-1). We prove that this construction yields an equivalence between the category of 1-motives over S and the category of such variations of mixed Hodge structures, thereby extending Deligne's equivalence over the complex numbers to the relative case and providing a positive answer to a question of André concerning the geometric origin of admissible variations of mixed Hodge structures of the above type. We also describe the l-adic and de Rham realizations of 1-motives and show that these realizations fit naturally into Deligne's framework of smooth mixed realizations.

Realizations of 1-motives over a scheme of characteristic 0

Abstract

Let S be a connected and smooth scheme of finite type over the complex numbers. We construct functorially the Hodge realization of a 1-motive over S as a torsion-free, polarizable and admissible variation of mixed Hodge structures of type (0,0),(-1,0),(0,-1),(-1,-1). We prove that this construction yields an equivalence between the category of 1-motives over S and the category of such variations of mixed Hodge structures, thereby extending Deligne's equivalence over the complex numbers to the relative case and providing a positive answer to a question of André concerning the geometric origin of admissible variations of mixed Hodge structures of the above type. We also describe the l-adic and de Rham realizations of 1-motives and show that these realizations fit naturally into Deligne's framework of smooth mixed realizations.
Paper Structure (4 sections, 6 theorems, 35 equations)

This paper contains 4 sections, 6 theorems, 35 equations.

Table of Contents

  1. Hodge realization
  2. $\ell$-adic realization
  3. De Rham realization
  4. Compatibility between realizations

Key Result

Proposition 1.1

Let $S$ be a connected and smooth scheme over $\mathbb{C}.$ (1) If $S$ is also proper over $\mathbb{C},$ then (2) If $S$ has a compactification with boundary with normal crossings, then the inclusions eq:exsecTorus are isomorphisms The first isomorphism is the enriched Hodge realization of the isomorphism D89 (3) In general, the inclusions eq:exsecTorus induce isomorphisms

Theorems & Definitions (16)

  • Definition 1
  • Remark 2
  • Proposition 1.1
  • proof
  • Theorem 1.2
  • proof
  • Definition 1.3
  • Definition 1.4
  • Lemma 2.1
  • Lemma 3.1
  • ...and 6 more