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Tannakian duality and Gauss-Manin connections for a family of curves

Phùng Hô Hai, Võ Quôc Bao, Trân Phan Quôc Bao

Abstract

Let $X/S$ be a smooth family of smooth projective varieties, where $S$ is a smooth affine curve over a field $k$ of characteristic $0.$ We relate the differential fundamental groupoid scheme of $X/k$ with the differential fundamental groupoid scheme of $S/k$ and the relative differential fundamental group of $X/S$ in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least $1,$ we prove that these maps are isomorphisms thus give an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we show that, as a surface over $k$, $X$ after a little shrinking becomes de Rham $K(π,1).$

Tannakian duality and Gauss-Manin connections for a family of curves

Abstract

Let be a smooth family of smooth projective varieties, where is a smooth affine curve over a field of characteristic We relate the differential fundamental groupoid scheme of with the differential fundamental groupoid scheme of and the relative differential fundamental group of in a short exact sequence. This yields natural maps from the group cohomology of the geometric relative fundamental group to the Gauss-Manin connections. For families of curves of genus at least we prove that these maps are isomorphisms thus give an interpretation of the Gauss-Manin connection in terms of cohomology of the differential fundamental group. As a consequence we show that, as a surface over , after a little shrinking becomes de Rham
Paper Structure (54 sections, 58 theorems, 254 equations)

This paper contains 54 sections, 58 theorems, 254 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notations and Conventions
  4. Flat connections and de Rham cohomology
  5. Connections
  6. Flat connections
  7. De Rham cohomology
  8. The Gauss-Manin connection
  9. Absolute connections
  10. The inflation functor
  11. The Gauss-Manin connection
  12. Expression in terms of hyper-cohomology
  13. Finiteness of de Rham cohomology
  14. Fundamental groups and groupoids
  15. Categories of consideration
  16. ...and 39 more sections

Key Result

Theorem 1

Let $f:X\longrightarrow S$ be a smooth projective relative curve of genus $g\geq 1$ over a smooth affine curve $S,$ equipped with a section $\eta:S\longrightarrow X$. Let $(\mathcal{V},\nabla)$ be a vector bundle with flat connection on $X/k.$ Then:

Theorems & Definitions (115)

  • Theorem 1: Theorem \ref{['thm-gmt']}, Lemma \ref{['lem_comparison_functors']} and Theorem \ref{['thm-comparison']}, Corollary \ref{['cor-114']}
  • Proposition 2: Proposition \ref{['pro_surf_kpi1']}
  • Lemma 2.2.1
  • Lemma 2.3.6
  • proof
  • Lemma 2.3.7
  • proof
  • Remark 2.3.8
  • Definition 3.1.1: Special sub-quotients
  • Theorem 4.1.1: Zhang, dos Santos
  • ...and 105 more