Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Nakai-Moishezon criteria and the toric Thomas-Yau conjecture

Jacopo Stoppa

TL;DR

The paper develops a toric analogue of the Thomas–Yau stability conjecture by connecting Lagrangian sections in Calabi–Yau toric fibrations to Bridgeland stability through a chain: SYZ transform to the Fukaya–Seidel category, dHYM/Nakai–Moishezon positivity, and toric Gamma-theoretic central charges. A central technical engine is the Gamma theorem, which transports K-theory data to period integrals of the LG potential, allowing the authors to translate stability into phase inequalities for periods with respect to $e^{-W(\omega_0)}\Omega_0$. Under this framework, they prove that for toric weak Fano manifolds (and certain general toric cases), a Lagrangian section $\mathcal{L}$ is Hamiltonian isotopic to a supercritical special SYZ representative iff the relevant phase inequalities hold for all toric subvarieties, with explicit statements in the weak del Pezzo setting. They also establish Bridgeland-stability–dHYM equivalences in dimension two (and discuss higher-rank/2D obstructions) and provide concrete unstable-case phenomena via weak dHYM solutions and minimal slope data, all compatible with Joyce–Li–Collins–Yau visions. Collectively, the results tie geometric stability of Lagrangian sections to algebraic stability notions through toric mirror symmetry, yielding a robust toric Thomas–Yau picture with explicit, computable criteria.

Abstract

We consider a class of Lagrangian sections $L$ contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: $L$ is Hamiltonian isotopic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. We use the SYZ transform, the toric gamma theorem, and toric homological mirror symmetry in order to reduce the statement to one about supercritical deformed Hermitian Yang-Mills connections, known as the Nakai-Moishezon criterion. As an application, we prove that, on the mirror of a toric weak del Pezzo surface, if $L$ defines a Bridgeland stable object in the Fukaya-Seidel category in a natural sense, then it is Hamiltonian isotopic to a special Lagrangian section in the class. The converse also holds for the mirror of the projective plane blown-up at one or two points, and always holds assuming a conjecture of Arcara and Miles. When $L$ is Bridgeland unstable, we obtain a morphism from $L$ to a weak solution of the special Lagrangian equation with phase angle satisfying a minimality condition. These results are consistent with general conjectures due to Joyce. We discuss some generalisations, including a weaker analogue of our main result for general projective toric manifolds, and a similar obstruction, related to Lagrangian multi-sections, in a special case.

Nakai-Moishezon criteria and the toric Thomas-Yau conjecture

TL;DR

The paper develops a toric analogue of the Thomas–Yau stability conjecture by connecting Lagrangian sections in Calabi–Yau toric fibrations to Bridgeland stability through a chain: SYZ transform to the Fukaya–Seidel category, dHYM/Nakai–Moishezon positivity, and toric Gamma-theoretic central charges. A central technical engine is the Gamma theorem, which transports K-theory data to period integrals of the LG potential, allowing the authors to translate stability into phase inequalities for periods with respect to . Under this framework, they prove that for toric weak Fano manifolds (and certain general toric cases), a Lagrangian section is Hamiltonian isotopic to a supercritical special SYZ representative iff the relevant phase inequalities hold for all toric subvarieties, with explicit statements in the weak del Pezzo setting. They also establish Bridgeland-stability–dHYM equivalences in dimension two (and discuss higher-rank/2D obstructions) and provide concrete unstable-case phenomena via weak dHYM solutions and minimal slope data, all compatible with Joyce–Li–Collins–Yau visions. Collectively, the results tie geometric stability of Lagrangian sections to algebraic stability notions through toric mirror symmetry, yielding a robust toric Thomas–Yau picture with explicit, computable criteria.

Abstract

We consider a class of Lagrangian sections contained in certain Calabi-Yau Lagrangian fibrations (mirrors of toric weak Fano manifolds). We prove that a form of the Thomas-Yau conjecture holds in this case: is Hamiltonian isotopic to a special Lagrangian section in this class if and only if a stability condition holds, in the sense of a slope inequality on objects in a set of exact triangles in the Fukaya-Seidel category. This agrees with general proposals by Li. We use the SYZ transform, the toric gamma theorem, and toric homological mirror symmetry in order to reduce the statement to one about supercritical deformed Hermitian Yang-Mills connections, known as the Nakai-Moishezon criterion. As an application, we prove that, on the mirror of a toric weak del Pezzo surface, if defines a Bridgeland stable object in the Fukaya-Seidel category in a natural sense, then it is Hamiltonian isotopic to a special Lagrangian section in the class. The converse also holds for the mirror of the projective plane blown-up at one or two points, and always holds assuming a conjecture of Arcara and Miles. When is Bridgeland unstable, we obtain a morphism from to a weak solution of the special Lagrangian equation with phase angle satisfying a minimality condition. These results are consistent with general conjectures due to Joyce. We discuss some generalisations, including a weaker analogue of our main result for general projective toric manifolds, and a similar obstruction, related to Lagrangian multi-sections, in a special case.
Paper Structure (36 sections, 16 theorems, 253 equations)

This paper contains 36 sections, 16 theorems, 253 equations.

Table of Contents

  1. Introduction
  2. Toric special Lagrangian sections and stability
  3. Relation to Bridgeland stability on surfaces
  4. The unstable case
  5. Some generalisations of Theorem \ref{['MainThm']}
  6. Background and main result
  7. dHYM connections and Nakai-Moishezon criteria
  8. Toric weak Fano manifolds
  9. Lower phase
  10. Higher rank
  11. General toric manifolds
  12. dHYM positivity
  13. Supercritical phase
  14. Phase inequality
  15. Higher rank
  16. ...and 21 more sections

Key Result

Theorem 1.2

Let $X$ be a toric weak Fano manifold with a fixed torus-invariant Kähler form $\omega_0$ and a holomorphic line bundle $L$, with ample inverse $L^{\vee}$. Fix $k > 0$ sufficiently large. Let $\mathcal{L} = \mathcal{L}(L^{\otimes k}, h) \subset \mathcal{U}_{q_k}$ be a SYZ Lagrangian section in the m with morphisms such that $\mathcal{L}$ is Hamiltonian isotopic to a supercritical special SYZ Lagr

Theorems & Definitions (48)

  • Remark 1.1
  • Theorem 1.2: Corollary \ref{['FSCor']}
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 38 more