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Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves

Soheyla Feyzbakhsh, Naoki Koseki, Zhiyu Liu, Nick Rekuski

TL;DR

The paper tackles the existence of Bridgeland stability conditions on Calabi–Yau threefolds by reducing the problem to a Brill–Noether-type inequality on curves. It develops a dimensional reduction framework—restricting stability questions from X to a surface S and then to a curve C ⊂ S—and introduces the Brill–Noether invariant BN_C to control the necessary Bogomolov–Gieseker-type bounds. By proving BN_C bounds in broad geometric settings (Del Pezzo and K3 surfaces, and more generally CY examples such as weighted complete intersections and Fano divisors), the authors establish BG-type inequalities near slope zero, which yield the desired Gamma-corrected stability conditions on X. The results apply to a wide class of CY threefolds, including quasi-smooth weighted CICYs and cyclic covers of Fano threefolds, thereby providing a unified pathway to Bridgeland stability with potential applications in enumerative geometry and birational geometry. This work integrates tilt-stability, wall-crossing, and Brill–Noether theory to extend the scope of stability conditions beyond previously known CY cases.

Abstract

Fix a polarised Calabi-Yau threefold $(X,H)$. We reduce a version of the Bayer-Macrì-Toda conjecture for $(X,H)$, which ensures the existence of Bridgeland stability conditions on $X$, to verifying a Brill-Noether-type inequality for curves on $X$. We then prove this inequality for a broad class of Calabi-Yau threefolds, including complete intersection Calabi-Yau threefolds in weighted projective spaces.

Stability conditions on Calabi-Yau threefolds via Brill-Noether theory of curves

TL;DR

The paper tackles the existence of Bridgeland stability conditions on Calabi–Yau threefolds by reducing the problem to a Brill–Noether-type inequality on curves. It develops a dimensional reduction framework—restricting stability questions from X to a surface S and then to a curve C ⊂ S—and introduces the Brill–Noether invariant BN_C to control the necessary Bogomolov–Gieseker-type bounds. By proving BN_C bounds in broad geometric settings (Del Pezzo and K3 surfaces, and more generally CY examples such as weighted complete intersections and Fano divisors), the authors establish BG-type inequalities near slope zero, which yield the desired Gamma-corrected stability conditions on X. The results apply to a wide class of CY threefolds, including quasi-smooth weighted CICYs and cyclic covers of Fano threefolds, thereby providing a unified pathway to Bridgeland stability with potential applications in enumerative geometry and birational geometry. This work integrates tilt-stability, wall-crossing, and Brill–Noether theory to extend the scope of stability conditions beyond previously known CY cases.

Abstract

Fix a polarised Calabi-Yau threefold . We reduce a version of the Bayer-Macrì-Toda conjecture for , which ensures the existence of Bridgeland stability conditions on , to verifying a Brill-Noether-type inequality for curves on . We then prove this inequality for a broad class of Calabi-Yau threefolds, including complete intersection Calabi-Yau threefolds in weighted projective spaces.

Paper Structure

This paper contains 20 sections, 31 theorems, 171 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Chern characters
  4. Slope-stability
  5. Tilt-stability
  6. Bridgeland stability conditions
  7. On Bayer--Macrì--Toda conjecture
  8. Step I: from $\operatorname{ch}_2$ to $\operatorname{ch}_3$
  9. Step II: from surfaces to threefolds
  10. Step III: from curves to surfaces
  11. Bounds on the Brill--Noether number
  12. Classical bounds
  13. Bounds on $\operatorname{BN}_C$ via wall-crossing
  14. Del Pezzo surfaces
  15. K3 surfaces
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that there exists $\epsilon>0$ such that BG3 holds. Then there exists a $1$-cycle $\Gamma$ with $\Gamma.H\geq 0$ such that conj:bmt holds for $(X, H)$. In particular, there exists a family of geometricHere, a stability condition $\sigma$ on $X$ is called geometric if the structure sheaf $\mat

Figures (1)

  • Figure : Figure. The blue graph represents the piecewise linear part of $f_{\epsilon}(x)$. The second line segment connects points $(\epsilon, -\frac{1}{2}\epsilon)$ and $(1, \frac{1}{2})$.

Theorems & Definitions (72)

  • Theorem 1.1: Theorem \ref{['thm:ch2']}
  • Definition 1.2
  • Theorem 1.3: Theorem \ref{['thm:main-criterion']}(a)
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 62 more