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The Shafarevich conjecture for varieties with globally generated cotangent

Thomas Krämer, Marco Maculan

TL;DR

The paper proves the Shafarevich conjecture for smooth projective canonically polarized varieties with amply generated cotangent bundles under mild numerical hypotheses, extending finiteness results beyond subvarieties of abelian varieties. It combines intrinsic cotangent-bundle and Albanese-geometry analysis with extrinsic big monodromy criteria for families into abelian varieties, using a perverse-sheaf Tannaka framework to control monodromy representations. A key technical innovation is a finite constructible Hilbert-scheme cover that yields finiteness for subvarieties with good reduction and ample normal bundles, together with a systematic elimination of wedge powers, spin representations, and exceptional groups as potential Tannaka groups. The resulting approach yields finiteness results for families of such varieties over number fields and provides a robust toolkit that merges Hodge-theoretic, microlocal, and arithmetic methods to establish non-density of integral points in this broader class of moduli spaces.

Abstract

We prove the Shafarevich conjecture for varieties with globally generated cotangent bundle, subject to mild numerical conditions.

The Shafarevich conjecture for varieties with globally generated cotangent

TL;DR

The paper proves the Shafarevich conjecture for smooth projective canonically polarized varieties with amply generated cotangent bundles under mild numerical hypotheses, extending finiteness results beyond subvarieties of abelian varieties. It combines intrinsic cotangent-bundle and Albanese-geometry analysis with extrinsic big monodromy criteria for families into abelian varieties, using a perverse-sheaf Tannaka framework to control monodromy representations. A key technical innovation is a finite constructible Hilbert-scheme cover that yields finiteness for subvarieties with good reduction and ample normal bundles, together with a systematic elimination of wedge powers, spin representations, and exceptional groups as potential Tannaka groups. The resulting approach yields finiteness results for families of such varieties over number fields and provides a robust toolkit that merges Hodge-theoretic, microlocal, and arithmetic methods to establish non-density of integral points in this broader class of moduli spaces.

Abstract

We prove the Shafarevich conjecture for varieties with globally generated cotangent bundle, subject to mild numerical conditions.

Paper Structure

This paper contains 36 sections, 27 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Intrinsic results
  3. Numerical conditions
  4. Extrinsic result
  5. Big monodromy
  6. Big Tannaka groups
  7. Varieties with globally generated cotangent bundle
  8. Basic properties
  9. Models
  10. Normalization and stabilizer
  11. Proof of \ref{['IntroThmExtrinsic']}
  12. Finiteness for subvarieties of abelian varieties
  13. A constructible cover
  14. Proof of \ref{['Thm:FinitenessNormalization']}
  15. Perverse sheaves and characteristic cycles
  16. ...and 21 more sections

Key Result

Theorem A

Fix an integer $c \geqslant 1$. Then up to $K$-isomorphism there are only finitely many projective surfaces $Y$ over $K$ with amply generated cotangent bundle, good reduction, $c_2(Y) = c$ and $h^0(Y, \Omega^1_Y) \geqslant 6$.

Theorems & Definitions (49)

  • Definition
  • Example
  • Theorem A
  • Theorem B
  • Corollary
  • Definition
  • Theorem C
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 39 more