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Exceptional Tannaka groups only arise from cubic threefolds

Thomas Krämer, Christian Lehn, Marco Maculan

TL;DR

This work classifies exceptional Tannaka groups arising from subvarieties of abelian varieties, showing that in the range relevant to arithmetic finiteness and big monodromy results, the only exceptional case is the Fano surface of lines on a smooth cubic threefold, with Tannaka group $E_6$. The authors upgrade the Tannakian framework from perverse sheaves to complex Hodge modules, proving a comparison that aligns the Hodge decomposition with a cocharacter of the perverse Tannaka group, thereby imposing strong Hodge-theoretic restrictions. They derive precise Hodge-number and Euler-characteristic constraints, and through a detailed analysis of the difference morphism and Fano geometry, identify $E_6$-types with the classical cubic-threefold Fano surface; they also rule out $E_7$ under the same hypotheses using Kashiwara's estimates and a Larsen-type argument. The results strengthen prior finiteness theorems and provide a robust method to constrain Tannaka groups via Hodge-theoretic data, with potential applications to monodromy and arithmetic geometry. Overall, the paper demonstrates that exceptional Tannaka groups for smooth subvarieties of abelian varieties essentially arise only from cubic threefold geometry, solidifying a sharp dichotomy between classical and exceptional cases.

Abstract

We show that under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.

Exceptional Tannaka groups only arise from cubic threefolds

TL;DR

This work classifies exceptional Tannaka groups arising from subvarieties of abelian varieties, showing that in the range relevant to arithmetic finiteness and big monodromy results, the only exceptional case is the Fano surface of lines on a smooth cubic threefold, with Tannaka group . The authors upgrade the Tannakian framework from perverse sheaves to complex Hodge modules, proving a comparison that aligns the Hodge decomposition with a cocharacter of the perverse Tannaka group, thereby imposing strong Hodge-theoretic restrictions. They derive precise Hodge-number and Euler-characteristic constraints, and through a detailed analysis of the difference morphism and Fano geometry, identify -types with the classical cubic-threefold Fano surface; they also rule out under the same hypotheses using Kashiwara's estimates and a Larsen-type argument. The results strengthen prior finiteness theorems and provide a robust method to constrain Tannaka groups via Hodge-theoretic data, with potential applications to monodromy and arithmetic geometry. Overall, the paper demonstrates that exceptional Tannaka groups for smooth subvarieties of abelian varieties essentially arise only from cubic threefold geometry, solidifying a sharp dichotomy between classical and exceptional cases.

Abstract

We show that under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.
Paper Structure (45 sections, 52 theorems, 235 equations, 1 figure)

This paper contains 45 sections, 52 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tannaka groups of subvarieties
  3. Exceptional Tannaka groups
  4. The Hodge filtration
  5. No country for $E_7$
  6. Big Monodromy
  7. Tannaka groups of Hodge modules
  8. Complex Hodge modules
  9. Hodge structure on cohomology
  10. Tannaka categories
  11. An exact sequence of Tannaka groups
  12. Character twists
  13. The Hodge cocharacter
  14. The Hodge decomposition
  15. Hodge level of intersection complexes
  16. ...and 30 more sections

Key Result

Theorem A

Let $X \subset A$ be a smooth irreducible subvariety with ample normal bundle and dimension $< g/2$. Then the following are equivalent:

Figures (1)

  • Figure 1: The two possibilities for $p_v \colon S \to X$ in \ref{['prop:ProjectionHirzebruchSurface']}: The first line shows the case $q(v) \in C$, the second the case $q(v) \not \in C$.

Theorems & Definitions (102)

  • Example
  • Theorem A
  • Theorem B
  • Example
  • Theorem C
  • Theorem D
  • Theorem E
  • Example 2.1
  • Example 2.2
  • Corollary 2.3
  • ...and 92 more