We study the transversely metaplectic structure and the transversely symplectic Dirac operator on a transversely symplectic foliation. Moreover, we give the Weitzenbock type formula for transversely symplectic Dirac operators and we estimate the lower bound of the eigenvalues of the transversely symplectic Dirac operator on special spinors space on transverse Kahler foliations.
Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a $C^\infty$ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphism group of a transversely holomorphic foliation or a riemannian foliation is a strong ILH Lie goup in the sense of Omori.
We develop the deformation theory of symplectic foliations, i.e. regular foliations equipped with a leafwise symplectic form. The main result of this paper is that each symplectic foliation has an attached $L_\infty$-algebra controlling its deformation problem. Indeed, viewing symplectic foliations as regular Poisson structures, we establish a one-to-one correspondence between the small deformations of a given symplectic foliation and the Maurer-Cartan elements of the associated $L_\infty$-algebra. Using this, we show that infinitesimal deformations of symplectic foliations can be obstructed. …
We consider the deformation theory of two kinds of geometric objects: foliations on one hand, pre-symplectic forms on the other. For each of them, we prove that the geometric notion of equivalence given by isotopies agrees with the algebraic notion of gauge equivalence obtained from the $L_{\infty}$-algebras governing these deformation problems.
The group of diffeomorphisms commuting with an elliptic operator on a manifold is a compact Lie group under Compact-Open topology. In foliation theory, pseudogroup is introduced by Sacksteder. The pseudogroup of local transformations commuting with a basic differential operator possesses the equicontinuity and the quasi-analyticity when conditions on operator are given. These properties serve to construct a transverse metric on the normal bundle under a good condition on operator. For this, the Average Method is applied as in the construction of basic connection on foliated bundles.
Consider the Hamiltonian action of a torus on a transversely symplectic foliation that is also Riemannian. When the transverse hard Lefschetz property is satisfied, we establish a foliated version of the Kirwan injectivity theorem, and use it to study Hamiltonian torus actions on transversely Kähler foliations. Among other things, we prove a foliated analogue of the Carrell--Liberman theorem. As an application, this confirms a conjecture raised by Battaglia--Zaffran on the basic Hodge numbers of symplectic toric quasifolds. Our methods also allow us to present a symplectic approach to the calc…
An affirmative answer to a conjecture for Metoki class
在“The {G}el'fand-{K}alinin-{F}uks class and characteristic classes of transversatile symplectic foliations”arXiv：0910.3414中，Kotschick和Morita证明了$\ds\HGF{7}{2}{}{8}$中的Gel'fand-Kalinin-Fuks类被分解为某个叶上同调类$η$和一个transversatile辛类$ω$的乘积$η\wedge ω$。我们表明，同样的公式适用于Metoki类，这是一个非平凡的元素在$\ds \HGF{9}{2}{}{14}$。这个结果由Kotschick和Morita证明，他们研究了Kontsevich的辛叶理的特征类。我们的证明依赖于Groebner基理论使用计算机计算。
We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantization-commutes-with-reduction theorem in this context. Examples related to symplectic toric quasi-folds, suspensions of isometric actions of discrete groups, and K-contact manifolds are discussed.
