Centered complexity one Hamiltonian torus actions (Q2750925)

Let a torus \(T\) act effectively on a symplectic manifold \((M,\omega)\) by symplectic transformations with a moment map \(\Phi:M\to {\mathbf t}^*\), where \({\mathbf t}\) is the Lie algebra of \(T\). The dimension of the torus is at most half the dimension of the manifold. The difference \(k=\frac 12 \dim M-\dim T\) is called the complexity. The examples where the complexity is one are compact symplectic 4-manifolds with Hamiltonian circle actions, which were classified by the first author [Periodic Hamiltonian flows on four-dimensional manifolds. Mem. Am. Math. Soc. 672 (1999; Zbl 0982.70011)], see also \textit{K. Ahara} and \textit{A. Hattori} [J. Fac. Sci., Univ. Tokyo, Sect. I A 38, 251-298 (1991; Zbl 0749.53018)], \textit{M. Audin} [Lect. Notes Math. 1416, 1-25 (1990; Zbl 0699.58031); The topology of torus actions on symplectic manifolds, Progress in Mathematics, 93. Basel Birkhäuser (1991; Zbl 0726.57029)] and \textit{D. A. Timashev} [Russ. Math. Surv. 51, No. 3, 567-568 (1996); translation from Usp. Mat. Nauk 51, No. 3, 213-214 (1996; Zbl 0892.14020); Izv. Math. 61, No. 2, 363-397 (1997); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 61, No. 2, 127-162 (1997; Zbl 0911.14022)]. NEWLINENEWLINENEWLINEThis is the first in a series of papers in which the authors study complexity one torus actions in arbitrary dimension. A proper Hamiltonian \(T\)-manifold is a connected symplectic manifold \((M,\omega)\) together with an effective action of \(T\), an open convex subset \(U \subset {\mathbf t}^*\), and a proper moment map \(\Phi:M\to U\). NEWLINENEWLINENEWLINETheorem 1 (Local Uniqueness). Let \((M,\omega, \Phi, U)\) and \((M',\omega', \Phi', U)\) be complexity one spaces. Assume that their Duistermaat-Heckman measures and that their genus and isotropydata over \(\alpha \in U\) are the same. Then there exists a neighborhood of \(\alpha\) over which the spaces are isomorphic. NEWLINENEWLINENEWLINEA proper Hamiltonian \(T\)-manifold \((M,\omega, \Phi, U)\) is said to be centered about a point \(\alpha \in U\) if \(\alpha\) is contained in the closure of the moment image of every orbit type stratum. NEWLINENEWLINENEWLINETheorem 2 (Centered Uniqueness). Let \((M,\omega, \Phi, U)\) and\((M',\omega', \Phi', U)\) be complexity one spaces that are centered about \(\alpha \in U\). Assume that their Duistermaat-Heckman measures are the same and that their genus and isotropy data over \(\alpha\) are the same. Then the spaces are isomorphic. NEWLINENEWLINENEWLINETheorem 3. Let \(M\) be the Grassmannian \(Gr^{+}(2,{\mathbb R}^5)\) or \(\text{Gr}^{+}(2,{\mathbb R}^6)\). There exists an equivariant symplectic embedding of a disjoint union of two open symplectic balls with linear actions and with equal radii into \(M\) such that the complement of the image has zerovolume. A fortiori, each of these Grassmannians can be fully packed by two equal symplectic balls.