On local invariants of singular symplectic forms

TL;DR

This paper characterizes local invariants of singular symplectic forms with stable Martinet hypersurfaces, providing complete invariants in various categories and conditions for equivalence classes, especially in dimension four.

Contribution

It introduces a complete set of local invariants for singular symplectic forms with stable Martinet hypersurfaces across different categories and dimensions, including new conditions for form equivalence.

Findings

Complete invariants in complex, real, and smooth categories.

Conditions to determine the kernel of bla^{n-1} using invariants.

Sufficient conditions for classifying singular symplectic forms in dimension four.

Abstract

We find a complete set of local invariants of singular symplectic forms with the structurally stable Martinet hypersurface on a $2n$-dimensional manifold. In the $\mathbb C$-analytic category this set consists of the Martinet hypersurface $\Sigma _2$, the restriction of the singular symplectic form $\omega$ to $T\Sigma_2$ and the kernel of $\omega^{n-1}$ at the point $p\in \Sigma_2$. In the $\mathbb R$-analytic and smooth categories this set contains one more invariant: the canonical orientation of $\Sigma_2$. We find the conditions to determine the kernel of $\omega^{n-1}$ at $p$ by the other invariants. In dimension $4$ we find sufficient conditions to determine the equivalence class of a singular symplectic form-germ with the structurally smooth Martinet hypersurface by the Martinet hypersurface and the restriction of the singular symplectic form to it. We also study the singular symplectic forms with singular Martinet hypersurfaces. We prove that the equivalence class of such singular symplectic form-germ is determined by the Martinet hypersurface, the canonical orientation of its regular part and the restriction of the singular symplectic form to its regular part if the Martinet hypersurface is a quasi-homogeneous hypersurface with an isolated singularity.