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Higher discriminants of vector bundles and Schur functors

Alessandro D'Andrea, Enrico Fatighenti, Claudio Onorati

TL;DR

The paper develops a representation-theoretic framework to compute the first three logarithmic Chern characters of Schur functors applied to a vector bundle, linking these invariants to traces of quadratic and cubic Casimir elements via the universal enveloping algebra. By establishing explicit polynomials $\dot{\delta}_r^{(2)}(\alpha)$ and $\dot{\delta}_r^{(3)}(\alpha)$ that govern the action of Casimir elements on Schur modules, the authors derive closed formulas for discriminants $\Delta_2$ and $\Delta_3$ of Schur powers, and, through a splitting principle, translate these into concrete Chern-character expressions for Schur bundles. They then apply these results to geometry, showing that Schur functors of locally free sheaves preserve (poly)stability and modularity on hyper-Kähler manifolds, yielding a powerful method to construct infinite families of slope polystable modular bundles. The work connects algebraic representation theory, Casimir actions, and complex-geometry stability notions, with explicit formulas useful for calculations and for exploring Bogomolov-type inequalities and modularity in higher dimensions.

Abstract

We prove some closed formulas for the logarithmic Chern character of a locally free sheaf. The argument used is representation-theoretic and we connect these formulas with the actions of some Casimir elements of $\mathfrak{sl}_r$. As an application, we give a recipe to construct slope polystable modular bundles on hyper-Kähler manifolds from old ones.

Higher discriminants of vector bundles and Schur functors

TL;DR

The paper develops a representation-theoretic framework to compute the first three logarithmic Chern characters of Schur functors applied to a vector bundle, linking these invariants to traces of quadratic and cubic Casimir elements via the universal enveloping algebra. By establishing explicit polynomials and that govern the action of Casimir elements on Schur modules, the authors derive closed formulas for discriminants and of Schur powers, and, through a splitting principle, translate these into concrete Chern-character expressions for Schur bundles. They then apply these results to geometry, showing that Schur functors of locally free sheaves preserve (poly)stability and modularity on hyper-Kähler manifolds, yielding a powerful method to construct infinite families of slope polystable modular bundles. The work connects algebraic representation theory, Casimir actions, and complex-geometry stability notions, with explicit formulas useful for calculations and for exploring Bogomolov-type inequalities and modularity in higher dimensions.

Abstract

We prove some closed formulas for the logarithmic Chern character of a locally free sheaf. The argument used is representation-theoretic and we connect these formulas with the actions of some Casimir elements of . As an application, we give a recipe to construct slope polystable modular bundles on hyper-Kähler manifolds from old ones.

Paper Structure

This paper contains 18 sections, 20 theorems, 98 equations.

Table of Contents

  1. Review of polynomial representations of $\mathop{\mathrm{GL}}\nolimits_r$
  2. The Grothendieck group of representations
  3. Chern character of a representation
  4. Discriminants
  5. Slope
  6. Logarithmic Chern character
  7. What about higher discriminants?
  8. Casimir actions
  9. Universal enveloping algebra and Casimir elements
  10. Proof of Theorem \ref{['thm:casimiri']}
  11. Geometric applications
  12. Schur functors of locally free sheaves
  13. Modular bundles on hyper-Kähler manifolds
  14. Discriminants for low-rank bundles
  15. Explicit formulas
  16. ...and 3 more sections

Key Result

Theorem A

Let $X$ be a compact Kähler manifold or a proper smooth complex variety, and let $E$ be a vector bundle of rank $r$ on $X$. Denote by $\mathsf{\mathbf{S}}^\alpha E$ the Schur functor associated to a partition $\alpha=(\alpha_1,\dots,\alpha_r)$, and denote by $r_\alpha$ its rank. Then where $\dot\delta_r^{(2)}(\alpha)$ and $\dot\delta_r^{(3)}(\alpha)$ are polynomials in $\alpha=(\alpha_1, \ldots,

Theorems & Definitions (65)

  • Theorem A: Theorem \ref{['thm:fibrati']}
  • Theorem B: Theorem \ref{['thm:modular']}
  • Example 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6: Svrtan
  • Remark 1.7
  • Proposition 2.1
  • ...and 55 more