Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6
type: Item
instance of: scholarly article
title: Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6
zbMATH Open document ID: 1415.57018
author: Pablo Suárez-Serrato
author: J. Torres Orozco
author: Ramón Vera
published in: Annals of Global Analysis and Geometry
publication date: 2019-05-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/1609.06768
review text: If $X$ is a smooth closed 4-manifold, then a broken Lefschetz fibration is a smooth map $f:X\to S^2$ that is a submersion outside the singularity set. The allowed singularities are of the following types: (i)\, indefinite fold singularities, also called broken singularities, contained in the smooth embedded one-dimensional submanifold $\Gamma\subset X\backslash C$, which are locally modeled by the real charts $\mathbb R^4\to\mathbb R^2$, $(t,x_1,x_2,x_3)\mapsto(t,-x_1^2+x_2^2+x_3^2)$, (ii)\,Lefschetz singularities: finitely many points $C=\{p_1,\dots,p_k\}\subset X$, which are locally modeled by complex charts $\mathbb C^2\to\mathbb C$, $(z_1,z_2)\mapsto z_1^2+z_2^2$. A purely wrinkled fibration is a submersion $f$ on a closed 4-manifold $X$ to a closed surface having two types of singularities: (i)\,indefinite fold singularities, (ii)\,cusps, finitely many points contained in the set $B=\{p_1,\dots,p_k\}\subset X$, which are locally modeled by real charts $\mathbb R^4\to\mathbb R^2$, $(t,x_1,x_2,x_3)\mapsto(t,x_1^3+3tx_2^2-x_3^2)$. If $M$, $X$ are smooth manifolds of dimensions $2n$ and $2n-2$, respectively, then a generalized broken Lefschetz fibration is a submersion $f:M\to X$ with two types of singularities: (i)\,indefinite fold singularities, locally modeled by $\mathbb R^{2n}\to\mathbb R^{2n-2}$, $(t_1,\dots,t_{2n-3},,x_1,x_2,x_3)\mapsto(t_1,\dots,t_{2n-3},-x_1^2+x_2^2+x_3^2)$, (ii)\,Lefschetz singularities locally modeled by complex charts $\mathbb C^n\to\mathbb C^{n-1}$, $(z_1,\dots,z_n)\mapsto (z_1,\dots,z_{n-2},z_{n-1}^2+z_n^2)$. If $M$ is a smooth 6-manifold, then a generalized wrinkled fibration $f:M\to X$ is a submersion to a smooth closed 4-manifold $X$ with the following four indefinite singularities each locally modeled by real charts $\mathbb R^6\to\mathbb R^4$: (i)\,folds $x=(t_1,t_2,t_3,x_1,x_2,x_3)\mapsto(t_1,t_2,t_3,-x_1^2+x_2^2+x_3^2)$, (ii)\,cusps $x\mapsto(t_1,t_2,t_3,x_1^3-3t_1x_1+x_2^2-x_3^2)$, (iii)\,swallowtails $x\mapsto(t_1,t_2,t_3,x_1…
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D17
zbMATH DE Number: 7061063
zbMATH Keywords: singular Poisson
zbMATH Keywords: near-symplectic
zbMATH Keywords: broken Lefschetz fibrations
zbMATH Keywords: wrinkled
zbMATH Keywords: singularity theory
zbMATH Keywords: stable maps
zbMATH Keywords: fold
zbMATH Keywords: cusp
zbMATH Keywords: swallowtail
zbMATH Keywords: butterfly
MaRDI profile type: Publication
OpenAlex ID: W3106044734
arXiv ID: 1609.06768
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DOI: 10.1007/S10455-019-09651-2
description: scientific article