Reductions of locally conformal symplectic structures and de Rham cohomology tangent to a foliation

The Basic de Rham Complex of a Singular Foliation

A singular foliation $\mathcal F$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal F$, and that of the basic differential forms on $M$. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when $\mathcal F$ is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of $M$, and, as a special case of the l…

叶wise de Rham cohomology of generic Reeb foliations

Leafwise de Rham cohomology of generic Reeb foliations

本文证明了对于通用Reeb foliation，其叶wise de Rham cohomology的H^0为实数域，而H^1在三维情况下为无限维。

Foliations associated to regular Poisson structures

A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.

数值平坦叶状结构与全纯泊松几何

Numerically flat foliations and holomorphic Poisson geometry

本文研究了紧Kähler流形上具有数值平坦切丛的光滑全纯叶状结构，证明了其诱导切丛分解等性质，并由此导出了射影流形上秩为2的Poisson结构的全局Weinstein分裂定理及零点存在性的新Hodge理论条件。

Poisson structures on tangent bundles

The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be p-related (p is the projection of TM onto M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fibe…

叶状结构的射流与 $b^k$-代数胚

Jets of foliations and $b^k$-algebroids

本文引入并分类了 $b^k$-型奇异叶状结构，揭示其与形式邻域中射流及和乐表示的关系，并研究了阶延拓的示性类阻碍、超曲面情形下的代数胚上同调以及代数胚辛形式与诱导辛变分的联系。