Nearby Special Lagrangians

TL;DR

This work addresses the local rigidity of closed embedded special Lagrangians $Q$ in Calabi–Yau manifolds by examining closed irreducibly immersed nearby special Lagrangians with unobstructed Floer cohomology inside Weinstein neighbourhoods. It develops a comprehensive Fukaya-category framework, including Novikov fields, gapped curved $A_ abla$-categories, wrapped and exact variants, and Yoneda theory, to translate geometric proximity into algebraic data tied to $oldsymbol{ extpi}_1Q$ representations. The main results show that, under favorable group-theoretic conditions on $oldsymbol{ extpi}_1Q$ (abelian, virtually solvable, or having no non-abelian free subgroups), any nearby closed irreducibly immersed special Lagrangian with unobstructed HF is unbranched, and in the abelian case it is a $C^1$-perturbation of $Q$; stronger conclusions hold when $oldsymbol{ extpi}_1Q$ is finite. The methods combine split-generation, McLean-type deformation theory, finite covers, Lie–Kolchin/Tits-alternative arguments, and Viterbo-type restriction to control representations arising from $HF^*(T_q^*Q, oldsymbol{b})$, yielding a robust local classification and rigidity phenomenon. The paper also constructs branched and obstructed examples to demonstrate sharpness of hypotheses and develops Tsai–Wang techniques for C^0 nearby special Lagrangians, clarifying the landscape of possible nearby Lagrangian geometries and their Floer-theoretic properties.

Abstract

Let $X$ be a Calabi--Yau manifold and $Q\subset X$ a closed connected embedded special Lagrangian; closed Lagrangians mean compact Lagrangian submanifolds without boundary. We prove that if the fundamental group $π_1Q$ is abelian then there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian with unobstructed Floer cohomology is $C^1$ close to $Q.$ We prove also that if $π_1Q$ is virtually solvable then for every positive integer $R$ there exists a Weinstein neighbourhood of $Q\subset X$ in which every closed irreducibly immersed special Lagrangian of degree $\le R$ and with unobstructed Floer cohomology is unbranched; that is, the projection $L\to Q$ is a covering map. We prove a stronger statement when $π_1Q$ is finite and a weaker statement when $π_1Q$ has no non-abelian free subgroups. The $π_1Q$ conditions, the Floer cohomology condition and the special Lagrangian condition are all essential as we show by counterexamples.

Figures (2)

• Figure 2.1: Moduli spaces of discs for the duality map
• Figure 2.2: a pseudo-holomorphic disc for $(q,\hat{x},q)$

Theorems & Definitions (221)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Proposition 2.4: Artin--Schreier
• proof