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Examples of real stable bundles on K3 surfaces

Dino Festi, Daniel Platt, Ragini Singhal, Yuuji Tanaka

TL;DR

The paper systematically constructs explicit stable vector bundles on K3 surfaces equipped with both a holomorphic and an anti-holomorphic involution by combining monad/bundle machinery with carefully chosen branched double covers that yield small Picard groups. Stability is established using the Generalised Hoppe Criterion, which the authors verify through targeted arithmetic checks on the Picard lattice aided by Magma/Macaulay2 computations. The work provides a collection of concrete examples—including pulls from P^2 and P^1×P^1 and several monad-based rank-2 and rank-3 bundles—as well as a real-structure framework, and demonstrates the feasibility of generating further instances via computational approaches. The results have implications for constructing G2- and Spin(7)-instantons, as the bundles are designed to lift to higher-dimensional manifolds built from X, with explicit data and code available. Overall, the paper offers a practical toolkit for producing invariant stable bundles on low-Picard K3 surfaces and illustrates how computational methods can drive explicit geometric constructions in this setting.

Abstract

Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-Sá Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$ with Picard rank $2$ using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.

Examples of real stable bundles on K3 surfaces

TL;DR

The paper systematically constructs explicit stable vector bundles on K3 surfaces equipped with both a holomorphic and an anti-holomorphic involution by combining monad/bundle machinery with carefully chosen branched double covers that yield small Picard groups. Stability is established using the Generalised Hoppe Criterion, which the authors verify through targeted arithmetic checks on the Picard lattice aided by Magma/Macaulay2 computations. The work provides a collection of concrete examples—including pulls from P^2 and P^1×P^1 and several monad-based rank-2 and rank-3 bundles—as well as a real-structure framework, and demonstrates the feasibility of generating further instances via computational approaches. The results have implications for constructing G2- and Spin(7)-instantons, as the bundles are designed to lift to higher-dimensional manifolds built from X, with explicit data and code available. Overall, the paper offers a practical toolkit for producing invariant stable bundles on low-Picard K3 surfaces and illustrates how computational methods can drive explicit geometric constructions in this setting.

Abstract

Motivated by gauge theory on manifolds with exceptional holonomy, we construct examples of stable bundles on K3 surfaces that are invariant under two involutions: one is holomorphic; and the other is anti-holomorphic. These bundles are obtained via the monad construction, and stability is examined using the Generalised Hoppe Criterion of Jardim-Menet-Prata-Sá Earp, which requires verifying an arithmetic condition for elements in the Picard group of the surfaces. We establish this by using computer aid in two critical steps: first, we construct K3 surfaces with small Picard group-one branched double cover of with Picard rank using a new method which may be of independent interest; and second, we verify the arithmetic condition for carefully chosen elements of the Picard group, which provides a systematic approach for constructing further examples.

Paper Structure

This paper contains 17 sections, 21 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Background
  3. K3 surfaces
  4. Moduli of semistable sheaves
  5. Monad bundles
  6. Stable bundles on covering spaces
  7. K3 surfaces with small Picard group
  8. A branched double cover of $\mathbb{P}^2$
  9. A branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$
  10. Stable bundles on K3 surfaces with small Picard group
  11. Examples of bundles on the branched double cover of $\mathbb{P}^2$
  12. The pullback of the cotangent bundle of $\mathbb{P}^2$
  13. Another rank $2$ example
  14. Examples of bundles on the branched double cover of $\mathbb{P}^1 \times \mathbb{P}^1$
  15. A rank $2$ example
  16. ...and 2 more sections

Key Result

Theorem 1.1

If $E \rightarrow X$ is a stable bundle that is infinitesimally rigid and admits a holomorphic involution $\hat{\iota}: E \rightarrow E$ covering $\iota$ and an anti-holomorphic involution $\hat{\sigma}: E \rightarrow E$ covering $\sigma$, then there exists a bundle $E' \rightarrow N^7$ constructed

Theorems & Definitions (48)

  • Theorem 1.1: Platt2024
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Theorem 6.1.6 in Huybrechts2010
  • Theorem 2.6: Generalised Hoppe Criterion, Theorem 3 in Jardim2017
  • Proposition 2.7
  • proof
  • ...and 38 more