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Unobstructed symplectic packing for tori and hyper-Kähler manifolds. (English) Zbl 1353.53055

J. Topol. Anal. 8, No. 4, 589-626 (2016); erratum ibid. 11, No. 1, 249-250 (2019).
An almost complex structure \(J\) and a differential \(2\)-form \(\omega\) on a smooth manifold \(M\) are said to be compatible if \(\omega(\cdot,J\cdot)\) is a \(J\)-invariant Riemannian metric on \(M\). Any closed differential \(2\)-form \(\omega\) compatible with an almost complex structure \(J\) is automatically symplectic. The compatibility between a complex structure \(J\) and a symplectic form \(\omega\) means that \(\omega(\cdot,J\cdot)+i\omega(\cdot,\cdot)\) is a Kähler metric on \(M\). A symplectic form \(\omega\) is called Kähler, if it is compatible with some complex structure on \(M\), and a complex structure is said to be of Kähler type if it is compatible with some symplectic form. A hyper-Kähler manifold \(M\) is a manifold equipped with three complex structures \(I_i\) satisfying the quaternionic relations and three symplectic forms \(\omega_i\) compatible, respectively, with \(I_i\), so that the three Riemannian metrics \(\omega_i(\cdot,I_i\cdot)\) coincide.
Such a collection of complex structures and symplectic forms on \(M\) is called a hyper-Kähler structure and is denoted by \(\mathfrak h=\{I_i,\omega_i\}\), \(i=1,2,3\). A symplectic form is hyper-Kähler and a complex structure is of hyper-Kähler type, if each of them appears in some hyper-Kähler structure. In particular, any hyper-Kähler symplectic form is Kähler and any complex structure of hyper-Kähler type is also of Kähler type. Two hyper-Kähler forms are hyper-Kähler deformation equivalent if they can be connected by a smooth path of hyper-Kähler forms. A hyper-Kähler manifold \((M,\mathfrak h)\) is called irreducible holomorphically symplectic if \(\pi_1(M)=0\) and \(\dim_{\mathbb C}H^{2,0}_I(M,\mathbb C)=1\), where \(I\) is any of the three complex structures appearing in \(\mathfrak h\) and \(H^{2,0}_I(M,\mathbb C)\) is the \((2,0)\)-part in the Hodge decomposition of \(H^{2}(M,\mathbb C)\) defined by \(I\). The irreducible holomorphically symplectic hyper-Kähler manifold is called hyper-Kähler manifold of maximum holonomy. If \((M,\omega)\), \(\dim_{\mathbb R}M=2n\), is a closed connected symplectic manifold with the symplectic volume \(\mathrm{Vol}\), then the symplectic packings of \((M,\omega)\) by balls are said to be unobstructed, if any finite collection of pairwise disjoint closed round balls in the standard symplectic \(\mathbb R^{2n}\) of total volume less than \(\mathrm{Vol}(M,\omega)\) has an open neighborhood that can be symplectically embedded into \((M,\omega)\). If \(\nu(M,\omega,V)=\frac{\sup_\alpha\mathrm{Vol}(V,\alpha\eta)}{\mathrm{Vol}(M,\omega)}\) for all \(\alpha\) such that \((V,\alpha\eta)\) admits a symplectic embedding into \((M,\omega)\), then it is said that \((M,\omega)\) is fully packed by \(k\) equal copies of \((V,\eta)\) if \(\nu(M,\omega,W)=1\), where \(W\) is a disjoint union of \(k\) equal copies of \((V,\eta)\). A complex structure \(J\) on a closed, connected manifold \(M\), \(\dim_{\mathbb C}M>l\), is called Campana simple, if the union \(\mathfrak U\) of all complex subvarieties \(Z\subset M\) satisfying \(0<\dim_{\mathbb C}Z<\dim_{\mathbb C}M\) has measure zero.
The symplectic packing problem is one of the major problems of symplectic topology. In this paper the authors use several strong results from complex geometry in order to prove the flexibility of symplectic packings by balls for all even-dimensional tori equipped with Kähler symplectic forms as well as for certain hyper-Kähler manifolds. They show that if \(M\) is a torus \(T^{2n}=\mathbb R^{2n}/\mathbb Z^{2n}\) with a Kähler form \(\omega\), or an irreducible holomorphically symplectic hyper-Kähler manifold with a hyper-Kähler symplectic form \(\omega\), then the symplectic packings of \((M,\omega)\) by balls are unobstructed.
If \((U,\eta)\), \(\dim_{\mathbb R}M=2n\), is an open, possibly disconnected, symplectic manifold, and \(V\subset U\), \(\dim_{\mathbb R}M=2n\), is a compact, possibly disconnected, submanifold of \(U\) with piecewise smooth boundary, then by a symplectic embedding of \((V,\eta)\) in \((M,\omega)\) is meant a symplectic embedding of an open neighborhood of \(V\) in \((U,\eta)\) to \((M,\omega)\). The authors prove that if \(H^2(V,\mathbb R)=0\) and \(M\), \(\dim_{\mathbb R}M=2n>4\), is either an oriented torus \(T^{2n}\) or, respectively, a closed connected oriented manifold admitting irreducible holomorphically symplectic hyper-Kähler structures, and if \(\omega_1\), \(\omega_2\) are either Kähler forms on \(T^{2n}\) or, respectively, hyper-Kähler forms on \(M\) such that \(\int_M\omega_1^n=\int_M\omega_2^n>0\) and that the cohomology classes \([\omega_1]\), \([\omega_2]\) are not proportional to rational ones, and in the hyper-Kähler case \(\omega_1\), \(\omega_2\) are hyper-Kähler deformation equivalent, then \(\nu(M,\omega_1,V)=\nu(M,\omega_2,V)\). Also, the authors show that for any Campana simple Kähler manifold, as well as for any manifold which is a limit of Campana simple manifolds in a smooth deformation, the symplectic packings by balls are unobstructed.

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53D05 Symplectic manifolds (general theory)
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References:

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