Recognition of objects through symplectic capacities (Q2168649)

The article is concerned with the following question in symplectic geometry from [\textit{K. Cieliebak} et al., Math. Sci. Res. Inst. Publ. 54, 1--44 (2007; Zbl 1143.53341)]: if two symplectic manifolds \((M, \omega_M)\) and \((N, \omega_N)\) are not isomorphic to each other, then is there any symplectic capacity \(c\) such that \(c(M, \omega_M) \neq c(N, \omega_N)\)? This is called a \textit{recognization problem}. This problem is investigated by the authors in a general framework. Consider a category called \((m,k)\)-form category where its object-set consists of the pairs \((M^m, \omega)\) where \(\omega \in \Omega^k(M)\), that is, a differential \(k\)-form on an \(m\)-dimensional manifold \(M\), and its hom-set consists of smooth embeddings, up to conjugation of isomorphisms, that pullback the paired forms. For instance, the category of all symplectic manifolds is a subcategory of the \((2n, 2)\)-form category. One can generalize the concept of symplectic capacities to a certain family of real-valued functions, still called capacities, on a \((m,k)\)-form category. The main result of this article proves that if a \((kn,k)\)-form category \( \mathcal C\) (where \(n \geq 2\)) has all its objects \((M, \omega)\) satisfying the following topological conditions: (1) \(M\) is compact, connected and simply connected; (2) \(\omega\) is exact and \(\omega^k\) is a volume form; (3) there exists a connected component \(A\) of \(\partial M\) such that \(\int_A \alpha \wedge (\omega|_A)^{n-1} <0\) where \(\alpha\) is any primitive of \(\omega|_A\), then capacities on \(\mathcal C\) can recognize objects. This article also provides examples to confirm that neither the compactness condition in (1) above nor the integration condition (3) above can be dropped.