Unobstructed symplectic packing by ellipsoids for tori and Hyperkähler manifolds (Q1652905)

Let \((M,J,\omega)\) be a closed connected Kähler manifold. The complex structure \(J\) is called Campana-simple if the union of all complex, proper subvarieties of \(M\) has measure zero in \(M\). (The only other case is when this union is all \(M\).) Even-dimensional tori equipped with Kähler symplectic forms and closed hyper-Kähler manifolds of maximal holonomy are examples of Campana-simple manifolds. Furthermore \(J\) is said to be approximated by Campana-simple complex structures if \(J\) can be approximated by a smooth family \(\{J_t\}_{t\in B^{2n}\subset\mathbb{C}^n}\) of complex structures with \(J_0=J\) and if there is some sequence \({t_i}\subset B^{2n}\) with limit 0 such that every \(\{J_{t_i}\}\) is Campana-simple. The main result of the present article asserts that if \(J\) can be approximated by Campana-simple complex structures then packing \(M\) by ellipsoids is unobstructed, i.e., any finite collection of pairwise disjoint closed ellipsoids in the standard symplectic \(\mathbb{R}^{2n}\) of total volume less than the symplectic volume of \(M\) can be symplectically embedded into \(M\). The proof of this flexibility result follows the sphere-packing results of \textit{D. McDuff} and \textit{L. Polterovich} [Invent. Math. 115, No. 3, 405--429 (1994; Zbl 0833.53028)] which study the structure of the symplectic cone in the cohomology of a blow-up of \(M\). The authors here consider instead the Kähler resolution (which is a smooth manifold) of the weighted blow-ups of \(M\) (which produce orbifolds). This result is a generalization of [\textit{J. Latschev} et al., Geom. Topol. 17, No. 5, 2813--2853 (2013; Zbl 1277.57024)].