The geometry of foliations generated by some differentials of Abelian type is considered. The case where all fibers are closed is studied.
We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes characteristic classes of flags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences we avoid the need to use the methods of FFG, and moreover we are able to describe all the symmetric powers at once.
We investigated conformal foliations $(M,F)$ of codimension $q\geq 3$ and proved a criterion for them to be Riemannian. In particular, the application of this criterion allowed us to proof the existence of an attractor that is a minimal set for each non-Riemannian conformal foliation. Moreover, if foliated manifold is compact then non-Riemannian conformal foliation $(M,F)$ is $(Conf(S^q),S^q)$-foliation with finitely many minimal sets. They are all attractors, and each leaf of the foliation belongs to the basin of at least one of them. The specificity of the proper conformal foliations is indicated. Special attention is given to complete conformal foliations.
We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.
A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects
of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.
It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. For each such geometry $( N,\zeta)$ we prove the existence and uniqueness of the structure of a finite-dimensional Lie group in the group of all automorphisms $Aut (N},\zeta)$.
The applications to the orbifolds are considered.