A classification of toric, folded-symplectic manifolds

folded-symplectic;toric;Delzant;origami manifolds;classification;completely integrable system

Hockensmith, Daniel Lawrence

关键词: folded-symplectic;    toric;    Delzant;    origami manifolds;    classification;    completely integrable system;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/88015/HOCKENSMITH-DISSERTATION-2015.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

Given a $G$-toric, folded-symplectic manifold with co-orientable folding hypersurface, we show that its orbit space is naturally a manifold with corners $W$ equipped with a smooth map $\psi: W \to \frak{g}^*$, where $\frak{g}^*$ is the dual of the Lie algebra of the torus, $G$.The map $\psi$ has fold singularities at points in the image of the folding hypersurface under the quotient map and it is a unimodular local embedding away from these points.Thus, to every $G$-toric, folded-symplectic manifold we can associate its orbit space data $\psi:W \to \fg^*$, a unimodular map with folds.We fix a unimodular map with folds $\psi:W \to \fg^*$ and show that isomorphism classes of $G$-toric, folded-symplectic manifolds whose orbit space data is $\psi:W \to \fg^*$ are in bijection with $H^2(W; \mathbb{Z}_G\times \R)$, where $\mathbb{Z}_G= \ker(\exp:\frak{g} \to G)$ is the integral lattice of $G$.Thus, there is a pair of characteristic classes associated to every $G$-toric, folded-symplectic manifold.This result generalizes a classical theorem of Delzant as well as the classification of toric, origami manifolds, due to Cannas da Silva, Guillemin, and Pires, in the case where the folding hypersurface is co-orientable.We spend a significant amount of time discussing the fundamentals of equivariant and non-equivariant folded-symplectic geometry.In particular, we characterize folded-symplectic forms in terms of their induced map from the sheaf of vector fields into a distinguished sheaf of one-forms, we relate the existence of an orientation on the folding hypersurface of a fold-form to the intrinsic derivative of the contraction mapping from the tangent bundle to the cotangent bundle, and we show that $G$-toric, folded-symplectic manifolds are stratified by $K$-toric, folded-symplectic submanifolds, where $K$ varies over the subtori of $G$ and the action is principal on each stratum.We show how these structures give rise to the rigid orbit space structure of a toric, folded-symplectic manifold used in the classification.We also give a robust description of folded-symplectic reduction, which we use to construct local models of toric, folded-symplectic manifolds.