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Generic special Lagrangian moduli spaces of a non-Kähler Calabi--Yau threefold

Benjamin Friedman

TL;DR

The paper develops a perturbed special Lagrangian theory for non-Kähler Calabi–Yau threefolds by introducing PSL submanifolds governed by the system $f^*\omega + \mathrm{d}^{\dagger}_{f^*\omega}\rho = 0$ and $f^*\mathrm{Im}\,\Omega = 0$ with $[\rho]=0$, then proves a Sard–Smale transversality result showing that for a comeagre set of Hermitian metrics $\omega$ the PSL moduli space is 0-dimensional (consisting of isolated submanifolds). The analysis builds a local universal moduli space as a Banach manifold, analyzes infinitesimal PSL deformations via a split linearization $\mathcal{L}=(\mathrm{D}s)_{P_0}$ with a Fredholm block $\mathcal{L}_1$, and uses transversality and the Taubes trick to extend to smooth parameters, culminating in a global result for generic $\omega$. This work contrasts the non-Kähler setting with McLean’s Kähler result and suggests potential applications to counting isolated PSL submanifolds and constructing invariants in non-Kähler Calabi–Yau geometries, with implications for SYZ-type structures. The methods combine geometric analysis of totally real submanifolds, elliptic regularity, and infinite-dimensional transversality to establish the discrete nature of PSL moduli under metric perturbations. Overall, it provides a robust framework for understanding PSL moduli in non-Kähler Calabi–Yau threefolds and sets the stage for potential invariants and geometric applications.

Abstract

Given a (possibly non-Kähler) Calabi--Yau threefold $(X,Ω)$, we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of $(X,ω,Ω)$, where $ω$ is a Hermitian metric on $X$. The equations defining this class of submanifolds reduce to the usual SL equations when $ω$ is a Kähler metric. Using the Sard--Smale technique, we prove the existence of a comeagre set of Hermitian metrics $ω$ on $X$ such that the moduli space of perturbed SL submanifolds in $(X,ω,Ω)$ consists of isolated points.

Generic special Lagrangian moduli spaces of a non-Kähler Calabi--Yau threefold

TL;DR

The paper develops a perturbed special Lagrangian theory for non-Kähler Calabi–Yau threefolds by introducing PSL submanifolds governed by the system and with , then proves a Sard–Smale transversality result showing that for a comeagre set of Hermitian metrics the PSL moduli space is 0-dimensional (consisting of isolated submanifolds). The analysis builds a local universal moduli space as a Banach manifold, analyzes infinitesimal PSL deformations via a split linearization with a Fredholm block , and uses transversality and the Taubes trick to extend to smooth parameters, culminating in a global result for generic . This work contrasts the non-Kähler setting with McLean’s Kähler result and suggests potential applications to counting isolated PSL submanifolds and constructing invariants in non-Kähler Calabi–Yau geometries, with implications for SYZ-type structures. The methods combine geometric analysis of totally real submanifolds, elliptic regularity, and infinite-dimensional transversality to establish the discrete nature of PSL moduli under metric perturbations. Overall, it provides a robust framework for understanding PSL moduli in non-Kähler Calabi–Yau threefolds and sets the stage for potential invariants and geometric applications.

Abstract

Given a (possibly non-Kähler) Calabi--Yau threefold , we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of , where is a Hermitian metric on . The equations defining this class of submanifolds reduce to the usual SL equations when is a Kähler metric. Using the Sard--Smale technique, we prove the existence of a comeagre set of Hermitian metrics on such that the moduli space of perturbed SL submanifolds in consists of isolated points.
Paper Structure (18 sections, 18 theorems, 122 equations, 2 figures)

This paper contains 18 sections, 18 theorems, 122 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation and main results
  3. Acknowledgements
  4. PSL submanifolds
  5. The SL and PSL systems
  6. Totally real geometry
  7. Minimality and regularity of PSL submanifolds
  8. Structure of the PSL moduli space
  9. The parameter space and the moduli space
  10. Reduction to the local picture
  11. Infinitesimal PSL deformations
  12. The local universal moduli space is a Banach manifold
  13. Transversality
  14. Smooth parameters and the Taubes Trick
  15. Returning to the global picture
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $(X,\Omega)$ be a compact Calabi--Yau threefold, and let $L^3$ be a fixed compact manifold of dimension $3$. Then within the space $\mathrm{Herm}(X)$ of Hermitian metrics on $X$ with the $C^{\infty}$ topology, there is a comeagre set $\mathrm{Herm}_{\mathrm{reg}}(X)$ such that for all $\omega \i

Figures (2)

  • Figure 1: Poof! A small perturbation of the complex structure causes the family of special Lagrangian circles with phase $e^{i\theta}$ in the Calabi--Yau torus to disappear completely.
  • Figure 2: On the other hand, after a small perturbation of the metric, some closed geodesics will survive, but these are isolated for generic perturbations.

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • Theorem 3.1: Local version of Theorem \ref{['thm-main']}, $C^{\ell,a}$ parameters
  • Lemma 3.2
  • ...and 20 more