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Bridgeland stability conditions on some higher-dimensional Calabi--Yau manifolds and generalized Kummer varieties

Yiran Cheng

TL;DR

This work advances Bridgeland stability theory into higher dimensions by developing a toolkit to construct stability conditions via product constructions, equivariant methods, and restriction techniques. The authors establish restriction maps that preserve geometric stability and the support property under isotrivial fiber conditions, and they extend stability to products with curves using Liu’s framework. By combining these methods with Bondal–Orlov/BKR-type equivalences for quotient and orbifold settings, they produce new stability conditions on a diverse array of hyper-Kähler and Calabi–Yau manifolds, including generalized Kummer varieties, Enriques-type CYs, bielliptic constructions, and Cynk–Hulek CYs in arbitrary dimensions. These results substantially broaden the known landscape of higher-dimensional stability conditions and open avenues for exploring moduli, Donaldson–Thomas theory, and derived-category geometry in HK/CY settings. The techniques also provide a flexible pathway for lifting stability conditions through finite covers and group actions, with potential applications to broader classes of varieties where explicit Bogomolov–Gieseker-type inequalities are unavailable.

Abstract

We construct Bridgeland stability conditions on the the following hyper-Kähler or strict Calabi--Yau manifolds: - Generalized Kummer varieties associated to an abelian surface that is isogenous to a product of elliptic curves. - Universal covers of Hilbert schemes of some Enriques surfaces; this provides examples of stability conditions on strict Calabi--Yau manifolds in each even dimension. - Albanese fibers of some finite étale covering of Hilbert schemes of some bielliptic surfaces; this provides examples of stability conditions on strict Calabi-Yau manifolds in each odd dimension. - Cynk--Hulek Calabi--Yau manifolds with an automorphism of order $2$ or $3$.

Bridgeland stability conditions on some higher-dimensional Calabi--Yau manifolds and generalized Kummer varieties

TL;DR

This work advances Bridgeland stability theory into higher dimensions by developing a toolkit to construct stability conditions via product constructions, equivariant methods, and restriction techniques. The authors establish restriction maps that preserve geometric stability and the support property under isotrivial fiber conditions, and they extend stability to products with curves using Liu’s framework. By combining these methods with Bondal–Orlov/BKR-type equivalences for quotient and orbifold settings, they produce new stability conditions on a diverse array of hyper-Kähler and Calabi–Yau manifolds, including generalized Kummer varieties, Enriques-type CYs, bielliptic constructions, and Cynk–Hulek CYs in arbitrary dimensions. These results substantially broaden the known landscape of higher-dimensional stability conditions and open avenues for exploring moduli, Donaldson–Thomas theory, and derived-category geometry in HK/CY settings. The techniques also provide a flexible pathway for lifting stability conditions through finite covers and group actions, with potential applications to broader classes of varieties where explicit Bogomolov–Gieseker-type inequalities are unavailable.

Abstract

We construct Bridgeland stability conditions on the the following hyper-Kähler or strict Calabi--Yau manifolds: - Generalized Kummer varieties associated to an abelian surface that is isogenous to a product of elliptic curves. - Universal covers of Hilbert schemes of some Enriques surfaces; this provides examples of stability conditions on strict Calabi--Yau manifolds in each even dimension. - Albanese fibers of some finite étale covering of Hilbert schemes of some bielliptic surfaces; this provides examples of stability conditions on strict Calabi-Yau manifolds in each odd dimension. - Cynk--Hulek Calabi--Yau manifolds with an automorphism of order or .
Paper Structure (25 sections, 52 theorems, 152 equations)

This paper contains 25 sections, 52 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Main applications
  3. Restricting stability conditions
  4. Organization of the paper
  5. Acknowledgements
  6. Preliminaries
  7. $t$-structures and stability conditions
  8. Equivariant categories
  9. Inducing stability conditions
  10. Constructions on products
  11. Local t-structures over base
  12. Constant $t$-structures on products
  13. Open heart property
  14. Stability conditions on products with curves
  15. Restricting stability conditions
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $S$ be an abelian surface which is isogenous to a product of elliptic curves, i.e., $S \simeq (C_1 \times C_2)/G$, where $C_i$ ($i=1,2$) are elliptic curves and $G$ is a finite group acting freely on $C_1 \times C_2$. Consider the generalized Kummer variety $K_n(S)$ associated to the abelian sur where $\alpha_n$ is the Albanese map for the Hilbert scheme $S^{[n]}$. Then, there exist stability

Theorems & Definitions (121)

  • Theorem 1.1: Generalized Kummer varieties
  • Corollary 1.2
  • Theorem 1.3: Even dimensional Calabi--Yau manifolds arising from Enriques surfaces
  • Theorem 1.4: Odd dimensional Calabi--Yau manifolds arising from bielliptic surfaces
  • Theorem 1.5: Cynk--Hulek Calabi--Yau manifolds
  • Remark 1.6
  • Theorem 1.7: Theorem \ref{['thm:main']}
  • Corollary 1.8: Corollary \ref{['pushforward of stable obj g>1']}
  • Definition 2.1
  • Remark 2.2
  • ...and 111 more