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Infinitesimal invariants of mixed Hodge structures

Rodolfo Aguilar, Mark Green, Phillip Griffiths

Abstract

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair $(X,Y)$ with $X$ a Fano 3-fold and $Y$ a smooth anticanonical K3 surface and in more detail in the case when $X$ is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

Infinitesimal invariants of mixed Hodge structures

Abstract

We introduce the notion of infinitesimal variations of mixed Hodge structures and invariants associated to them. We describe these invariants in the case of a pair with a Fano 3-fold and a smooth anticanonical K3 surface and in more detail in the case when is a cubic threefold. In this last setting, we obtain a generic global Torelli theorem for pairs.

Paper Structure

This paper contains 34 sections, 32 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Infinitesimal variations of mixed Hodge structures
  3. Definitions
  4. Mixed Hodge structures of length two
  5. Delta Invariant for extensions of Hodge Structures
  6. Fano n-folds
  7. Infinitesimal Torelli for anti-canonical pairs
  8. Fano-K3 pairs
  9. Fano Threefolds and anti-canonical sections
  10. Fano threefolds
  11. Properties of $(X,Y)$
  12. Basic exact sequences and interpretation
  13. Cubic form
  14. Cubic form for Fano threefolds
  15. Cubic complex
  16. ...and 19 more sections

Key Result

Theorem 1.1

For $X$ a cubic threefold and $Y$ an anticanonical smooth section, both of them general, the cubic $C$ is smooth.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.3
  • Definition 2.5
  • ...and 52 more