Generic singularities of symplectic and quasi-symplectic immersions (Q2845627)

This paper is devoted to the classification of first occurring singularities of immersed symplectic or quasi-submanifolds of a symplectic manifold, i.e. classification of the tuples NEWLINE\[NEWLINE \left(\mathbb{R}^{2n},\omega,S_1^k\cup S_2^k\right)_0,\tag{1} NEWLINE\]NEWLINE where \(\omega\) is a symplectic form on \(\mathbb{R}^{2n}\) and \(S_1^k\), \(S_2^k\) are \(k\)-dimensional symplectic or quasi-symplectic submanifolds of \(\left(\mathbb{R}^{2n},\omega\right)\) whose intersection contains \(0\in\mathbb{R}^{2n}\). The notation \(( )_0\) means that objects in the parenthesis are germs at \(0\in\mathbb{R}^{2n}\). A tuple (1) is equivalent to a tuple of the same form given by \(\widetilde{\omega}\), \(\widetilde{S}_1^k\), \(\widetilde{S}_2^k\) if there exists a local diffeomorphism of \(\mathbb{R}^{2n}\) which brings \(\widetilde{\omega}\) to \(\omega\) and \(S_1^k\cup S_2^k\) to \(\widetilde{S}_1^k\cup \widetilde{S}_2^k\). For any \(k<2n\) the authors construct a complete system of invariants in the problem of classifying singularities of immersed \(k\)-dimensional submanifolds of a symplectic \(2n\)-manifold at a generic double point. They work in a fixed category which is smooth or real-analytic and restrict themselves to generic germs of (1) which means that their results concern a certain open and dense set in the space of such germs.