Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6
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 $x\mapsto(t_1,t_2,t_3,x_1^4+t_1\cdot x_1^2+t_2x_1+x_2^2-x_3^2)$, (iv)\,butterflies $x\mapsto(t_1,t_2,t_3,x_1^5+t_1\cdot x_1^3+t_2x_1^2+t_3x_1+x_2^2-x_3^2)$. A Poisson bracket, or a Poisson structure, on a smooth manifold $M$ is a bilinear operation $\{\cdot,\cdot\}$ on the set $C^\infty(M)$ of real valued smooth functions on $M$ that satisfies (i)\,$(C^\infty(M),\{\cdot,\cdot\})$ is a Lie algebra, (ii)\,$\{gh,k\}=g\{h,k\}+h\{g,k\}$ for any $g$, $h$, $k\in C^\infty(M)$. A manifold $M$ with a Poisson bracket is called a Poisson manifold. If $(M,\omega)$ is a symplectic manifold, then the symplectic form $\omega$ produces a Poisson structure $\{g,h\}=\omega(X_g,X_h)$. \par In this paper, the authors consider Poisson and near-symplectic structures on generalized wrinkled fibrations in dimension 6. They prove that if $M$ and $X$ are closed oriented smooth manifolds with dimensions $\dim(M)=2n$, $\dim(X)=2n-2$ and $f:M\to X$ is a generalized broken Lefschetz fibration, then there is a complete singular Poisson structure of rank 2 whose associated bivector vanishes on the singularity set of $f$. Also, they show that if $M$ is a closed oriented 6-manifold, $(X,\omega)$ is a closed symplectic 4-manifold, $f:M\to X$ is a generalized wrinkled fibration, a 3-submanifold $Z$ of $M$ is the singularity set of $f$, a 3-submanifold of $M$, there there is a class $\alpha\in H^2(M)$ such that it pairs positively with every component of every fiber, and $\alpha\vert_{Z}=[{\omega_X}\vert_{Z}]$, then there exists a near-symplectic form $\omega$ on $M$ with singular locus $Z$ such that it restricts to a symplectic form on the smooth fibers of the fibration.
scientific article