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$Z$-critical connections and Bridgeland stability conditions

Ruadhaí Dervan, John Benjamin McCarthy, Lars Martin Sektnan

Abstract

We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations $Z$-critical connections, with $Z$ a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a $Z$-critical connection if and only if it is asymptotically $Z$-stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank.

$Z$-critical connections and Bridgeland stability conditions

Abstract

We associate geometric partial differential equations on holomorphic vector bundles to Bridgeland stability conditions. We call solutions to these equations -critical connections, with a central charge. Deformed Hermitian Yang--Mills connections are a special case. We explain how our equations arise naturally through infinite dimensional moment maps. Our main result shows that in the large volume limit, a sufficiently smooth holomorphic vector bundle admits a -critical connection if and only if it is asymptotically -stable. Even for the deformed Hermitian Yang--Mills equation, this provides the first examples of solutions in higher rank.

Paper Structure

This paper contains 22 sections, 60 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. Outlook
  3. Preliminaries
  4. Stability conditions
  5. An example
  6. $Z$-critical connections
  7. Moment maps, subsolutions and existence on surfaces
  8. Subsolutions
  9. Moment maps
  10. Existence on surfaces
  11. Self-adjointness
  12. Asymptotic $Z$-stability of $Z$-critical vector bundles
  13. Existence of $Z$-critical connections on $Z$-stable bundles
  14. The case when $E$ is slope stable
  15. The general case: finite-dimensional results
  16. ...and 7 more sections

Key Result

Theorem 1.0

A simple, sufficiently smooth holomorphic vector bundle admits a uniformly bounded family of $Z_k$-critical connections for all $k \gg 0$ if and only if it is asymptotically $Z$-stable with respect to subbundles.

Theorems & Definitions (147)

  • Theorem 1.0
  • Corollary 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 137 more