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Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic

Thibault Alexandre

TL;DR

The paper addresses how Siegel varieties in characteristic $p$ fail full hyperbolicity and can exhibit partial hyperbolicity controlled by positivity of vector bundles. It introduces $(\varphi,D)$-ampleness, a Frobenius-tTwisted positivity notion, and proves stability results for automorphic and Schur-constructed bundles, enabling a transfer of positivity through flag bundles and pushforwards. A key mechanism is plethysm of Schur functors in positive characteristic, which—under a prime-size bound—admits filtrations by Schur functors, allowing one to deduce $(\varphi,D)$-ampleness for many automorphic bundles. The main results show that for $p \ge g^2+3g+1$, exterior powers of $\Omega^1_{\mathrm{Sh}^{\mathrm{tor}}}(\log D_{\mathrm{red}})$ become $(\varphi,D)$-ample in appropriate degrees, implying subvarieties of codimension up to $g-1$ are of log general type, with an exceptional locus of codimension at least $g$. Collectively, these findings reveal a pseudo-hyperbolic structure in positive characteristic and offer a framework to study hyperbolicity via Frobenius-twisted positivity and plethysm alongside automorphic geometry.

Abstract

We study hyperbolicity properties of the moduli space of polarized abelian varieties (also known as the Siegel modular variety) in characteristic $p$. Our method uses the plethysm operation for Schur functors as a key ingredient and requires a new positivity notion for vector bundles in characteristic $p$ called $(\varphi,D)$-ampleness. Generalizing what was known for the Hodge line bundle, we also show that many automorphic vector bundles on the Siegel modular variety are $(\varphi,D)$-ample.

Positivity, plethysm and hyperbolicity of Siegel varieties in positive characteristic

TL;DR

The paper addresses how Siegel varieties in characteristic fail full hyperbolicity and can exhibit partial hyperbolicity controlled by positivity of vector bundles. It introduces -ampleness, a Frobenius-tTwisted positivity notion, and proves stability results for automorphic and Schur-constructed bundles, enabling a transfer of positivity through flag bundles and pushforwards. A key mechanism is plethysm of Schur functors in positive characteristic, which—under a prime-size bound—admits filtrations by Schur functors, allowing one to deduce -ampleness for many automorphic bundles. The main results show that for , exterior powers of become -ample in appropriate degrees, implying subvarieties of codimension up to are of log general type, with an exceptional locus of codimension at least . Collectively, these findings reveal a pseudo-hyperbolic structure in positive characteristic and offer a framework to study hyperbolicity via Frobenius-twisted positivity and plethysm alongside automorphic geometry.

Abstract

We study hyperbolicity properties of the moduli space of polarized abelian varieties (also known as the Siegel modular variety) in characteristic . Our method uses the plethysm operation for Schur functors as a key ingredient and requires a new positivity notion for vector bundles in characteristic called -ampleness. Generalizing what was known for the Hodge line bundle, we also show that many automorphic vector bundles on the Siegel modular variety are -ample.
Paper Structure (36 sections, 77 theorems, 161 equations, 1 figure, 1 table)

This paper contains 36 sections, 77 theorems, 161 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. General picture
  3. History and motivation
  4. Our main result
  5. A new positivity notion for vector bundles in characteristic $p$
  6. Positivity of automorphic vector bundles on the Siegel variety
  7. Schur functor and the plethysm operation
  8. Plethysm and hyperbolicity
  9. Organization of the paper
  10. Representations of algebraic groups
  11. Plethysm for Schur functors in positive characteristic
  12. Over the complex numbers
  13. Strict polynomial functors
  14. A plethysm in positive characteristic under additional hypothesis
  15. Positive vector bundles
  16. ...and 21 more sections

Key Result

Theorem 1

Consider a geometrically integral smooth projective curve $C$ over a number field $K/\mathop{\mathrm{\mathbb{Q}}}\nolimits$. The following three assertions are equivalent.

Figures (1)

  • Figure 7.1: $(\varphi,D)$-ampleness of automorphic bundles $\nabla(\lambda)$ when $g = 2$.

Theorems & Definitions (183)

  • Theorem : MR718935
  • Definition
  • Remark
  • Definition
  • Remark
  • Proposition : Proposition \ref{['th_plethysm']}
  • Proposition 2.1: MR2015057
  • Definition 2.2: MR2015057
  • Proposition 2.3: MR2015057
  • Definition 2.4: MR2015057
  • ...and 173 more