Discrete and continuous symmetries in monotone Floer theory
type: Item
instance of: scholarly article
title: Discrete and continuous symmetries in monotone Floer theory
zbMATH Open document ID: 1479.53087
author: Jack Smith
published in: Selecta Mathematica. New Series
publication date: 2020-08-07T00:00:00Z
full work available at URL: https://arxiv.org/abs/1703.05343
review text: Let \(L\) be a closed, connected, monotone Lagrangian submanifold of a compact symplectic manifold \((X,\omega)\). The author develops two new techniques for computing the Floer cohomology \(H^\ast(L,L)\) when \(L\) has certain symmetries inside \(X\). The techniques are applied to a free \(\operatorname{PSU}(N-1)\) orbit \(L\) in \((\mathbb{CP}^{N-2})^N\), where \(N\geq 3\) and \((\mathbb{CP}^{N-2},\omega_{FS})\) is equipped with the Fubini-Study form. The author's first technique involves the (length-zero) closed-open string map \[ \mathcal{CO}^0:QH^\ast(X;R) \rightarrow HF^\ast(L,L;R) \] from the quantum cohomology to the Lagrangian Floer cohomology, which is a unital \(R\)-algebra homomorphism. The Lagrangians considered are \(K\)-homogeneous, i.e., \(L\) is a Lagrangian orbit of the action of a compact connected Lie group \(K\) on a compact Kähler manifold \(X\) by holomorphic automorphisms. The following theorem is the main result in the paper concerning the closed-open string map, where \(N_L^+\) is the minimal Chern number for rational curves in \(X\) that are holomorphic. Throughout the paper the author assumes that if the characteristic of the coefficient ring \(R\) is not 2, then \(L\) is orientable and equipped with a relative spin structure. \textbf{Theorem 3} (Theorem 3.5.3) Let \(Z \subset X\) be a setwise \(K\)-invariant analytic subvariety of complex codimension \(\leq N_X^+\), disjoint from \(L\) and Poincaré dual to a class \(\alpha \in QH^\ast(X;R)\). Then \[ \mathcal{CO}^0(\alpha) = \lambda \cdot 1_L, \] where \(\lambda\) is a count of axial discs passing through \(Z\). If \(L\) is equipped with the standard spin structure (see Definition 3.1.3) then all of these discs contribute with positive sign. The author's second technique concerns the Oh spectral sequence \(E_\ast\). If \(L\) is a closed Lagrangian submanifold of a compact symplectic manifold \((X,\omega)\), then there is an action of the group Symp\((X,L)\) of symplectomorphisms that pre…
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 7229487
zbMATH Keywords: Floer cohomology
zbMATH Keywords: symmetry
zbMATH Keywords: pearl complex
zbMATH Keywords: length-zero closed-open string map
zbMATH Keywords: Oh spectral sequence
reviewed by: David E. Hurtubise
MaRDI profile type: Publication
OpenAlex ID: W3039169248
arXiv ID: 1703.05343
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DOI: 10.1007/S00029-020-00575-5
description: scientific article