Complexifications of Morse functions and the directed Donaldson-Fukaya category
type: Item
instance of: scholarly article
title: Complexifications of Morse functions and the directed Donaldson-Fukaya category
zbMATH Open document ID: 1211.53100
author: Joseph Johns
published in: The Journal of Symplectic Geometry
publication date: 2011-01-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/0808.0262
full work available at URL: https://projecteuclid.org/euclid.jsg/1291387952
review text: Floer theory of symplectic Lefschetz fibrations derived from a Morse function on a four manifold is studied. Let \(f: N\to\mathbb{R}\) be a real analytic Morse function. Then, in favorable circumstances, the complexification \(f_{\mathbb{C}}\) of \(f\) can be regarded as a symplectic Lefschetz fibration [cf. \textit{B. Hall} and \textit{W. Kirwin}, Adapted complex structure and the geodesic flow, \url{arXiv:0811.3083}, to appear in Math. Ann.]. For smooth \(f\) (not necessary real analytic), taking a Riemannian metric \(g\) such that \((f, g)\) is Morse-Smale, and constructing an exact symplectic manifold \(M\), \(\dim M= 2\dim N- 2\), with some exact Lagrangian spheres \(V_1,\dots, V_m\subset M\), one for each critical point of \(f\), for a given handle decomposition of \(N\), we can derive a symplectic Lefschetz fibration from \(f\) [cf. \textit{P. Seidel}, Fukaya categories and Picard-Lefschetz theory. Zürich Lectures in Advanced Mathematics. Zürich: European Mathematical Society (EMS). vi, 326 p. (2008; Zbl 1159.53001); hereafter referred to as [1]]. Assuming \(N\) is compact and \(f\) is a self-indexing Morse function with either two, three, or four critical values: \(\{0, n\}\), \(\{0, n,2n\}\) or \(\{0,n,n+ 1, 2n+ 1\}\), the author carried out this procedure in [The Picard-Lefschetz theory of complexified Morse functions, \url{arXiv:0906.1218}, to appear in Geometry and Topology, hereafter referred to as [2]]. In 82), Let \(\pi: E\to\mathbb{C}\) be the corresponding symplectic Lefschetz fibration, then 1. \(N\) embeds in \(E\) as an exact Lagrangean submanifold; 2. All the critiacal points of \(\pi\) lie in \(N\) and \(\text{Crit}(T)= \text{Crit}(f)\); 3. \(\pi|N= f: N\to\mathbb{R}\) (referred Theorem A in this paper). Adopting a version of plumbing [\textit{R. E. Gompf} and \textit{A. I. Stipsicz}, 4-manifolds and Kirby calculus, Graduate Studies in Mathematics. 20. Providence, RI: American Mathematical Society (AMS). xv, 558 p. (1999; Zbl 0933.57020)], expl…
reviewed by: Akira Asada
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 5839422
zbMATH Keywords: Morse function
zbMATH Keywords: symplectic Lefschetz fibration
zbMATH Keywords: vanishing cycle
zbMATH Keywords: Morse-Butt-Lagrangian Floer homology
zbMATH Keywords: directed Donaldson-Fukaya category
zbMATH Keywords: flow category
MaRDI profile type: Publication
arXiv ID: 0808.0262
DOI: 10.4310/JSG.2010.V8.N4.A3
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description: scientific article