The paper concerns a -rigidity result for the characteristic foliations in symplectic geometry. A symplectic homeomorphism (in the sense of Eliashberg-Gromov) which preserves a smooth hypersurface also preserves its characteristic foliation.
The note is about a connection between Seshadri constants and packing constants and presents another proof of Lazarsfeld's result from [Math. Res. Lett. 3 (1996), 439-447].
We classify the -dimensional compact nilmanifolds that admit abelian complex structures, and for any such complex structure we describe the space of symplectic forms which are compatible with .
Some relations between normal complex surface singularities and symplectic fillings of the links of the singularities are discussed. For a certain class of singularities of general type, which are called hypersurface K3 singularities in this paper, an inequality for numerical invariants of any minimal symplectic fillings of the links of the singularities is derived. This inequality can be regarded as a symplectic/contact analog of the 11/8-conjecture in 4-dimensional topology.
On a symplectic manifold, there is a natural elliptic complex replacing the de Rham complex. It can be coupled to a vector bundle with connection and, when the curvature of this connection is constrained to be a multiple of the symplectic form, we find a new complex. In particular, on complex projective space with its Fubini–Study form and connection, we can build a series of differential complexes akin to the Bernstein–Gelfand–Gelfand complexes from parabolic differential geometry.
Let M be a smooth manifold of dimension m>0, and denote by the canonical Nijenhuis tensor on TM. Let Π be a Poisson bivector on M and the complete lift of Π on TM. In a previous paper, we have shown that is a Poisson-Nijenhuis manifold. Recently, the higher order tangent lifts of Poisson manifolds from M to have been studied and some properties were given. Furthermore, the canonical Nijenhuis tensors on are described by A. Cabras and I. Kolář [Arch. Math. (Brno) 38 (2002), 243-257],...
We describe all canonical 2-forms Λ(ω) on the r-th order tangent bundle TrM = Jr0 (R;M) of a symplectic manifold (M, ω). As a corollary we deduce that all canonical symplectic structures Λ(ω) on TrM over a symplectic manifold (M, ω) are of the form Λ(ω) = Σrk=0 αkω(k) for all real numbers αk with αr ≠ 0, where ω(k) is the (k)-lift (in the sense of A. Morimoto) of ω to TrM.
Let and be compact symplectic manifolds (resp. symplectic manifolds) of dimension 2n > 2. Fix 0 < s < n (resp. 0 < k ≤ n) and assume that a diffeomorphism Φ : X → Y maps all 2s-dimensional symplectic submanifolds of X to symplectic submanifolds of Y (resp. all isotropic k-dimensional tori of X to isotropic tori of Y). We prove that in both cases Φ is a conformal symplectomorphism, i.e., there is a constant c ≠0 such that .
A cluster ensemble is a pair of positive spaces (i.e. varieties equipped with positive atlases), coming with an action of a symmetry group . The space is closely related to the spectrum of a cluster algebra [12]. The two spaces are related by a morphism . The space is equipped with a closed -form, possibly degenerate, and the space has a Poisson structure. The map is compatible with these structures. The dilogarithm together with its motivic and quantum avatars plays a central role...
On cotangent bundles the Liouville field, the Liouville 1-form and the canonical symplectic structure d exist. In this paper interactions between these objects and -tensor fields on cotangent bundles are studied. Properties of the connections induced by the above structures are investigated.