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## Rigid geometries and their automorphism groups on leaf spaces of foliations

Cornell University
,
2018.
No. 1704.04220.

We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifold. We separate out a special category F_0 of leaf manifolds containing the orbifold category as a complete subcategory. Objects of F_0 may be non-Hausdorff unlike orbifolds. The topology of some objects of F_0 do not satisfies the separation axiom T_0. It is shown that for every object N of F_0 a rigid geometry on N admits a desingularization. Moreover, for every such N we prove the existence and the uniqueness of a finite-dimensional Lie group structure on the group of all automorphisms of the rigid geometry on N.

Publication based on the results of:

Zhukova N., В кн. : Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции. : Каз. : Издательство Казанского университета, 2017. С. 48-51.

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. ...

Added: April 1, 2018

Zhukova N., Moscow Mathematical Journal 2018

We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...

Added: April 2, 2018

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Zhukova N.I., K. I. Sheina, / Cornell University. Series math "arxiv.org". 2015. No. 1410.1144.

We get sufficient conditions for the full basic automorphism group of a complete
Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category
of Cartan foliations. In particular, we obtain sufficient conditions for this group
to be discrete. Emphasize that the transverse Cartan geometry may be noneffective.
Some estimates of the dimension of this group depending ...

Added: November 10, 2014

Zhukova N., , in : The Conference NOMA-2017. Book of Abstracts. : Nizhny Novgorod : Nizhny Novgorod State University, 2017. P. 67-68.

We present a new method of investigation of G-structures on orbifolds. This method is founded on the consideration of a G-structure on an n-dimensional orbifold as the corresponding transversal structure of an associated foliation. For a given orbifold, there are different associated foliations. We construct and apply a compact associated foliation (M,F) on a compact ...

Added: April 14, 2018

Nikolay Konovalov, / Cornell University. Series "Working papers by Cornell University". 2022. No. 2202.07507.

Let $V_{n,d}$ be the variety of equations for hypersurfaces of degree $d$ in $\mathbb{P}^n(\mathbb{C})$ with singularities not worse than simple nodes. We prove that the orbit map $G'=SL_{n+1}(\mathbb{C}) \to V_{n,d}$, $g\mapsto g\cdot s_0$, $s_0\in V_{n,d}$ is surjective on the rational cohomology if $n>1$, $d\geq 3$, and $(n,d)\neq (2,3)$. As a result, the Leray-Serre spectral sequence ...

Added: September 12, 2022

N. I. Zhukova, Transformation Groups 2017

We prove an analog of the Lichnerowicz conjecture for compact and noncompact
Riemannian orbifolds. In particular, we prove that any compact Riemannian
orbifold of dimension n >2 with an essential connected Lie group of conformal
transformations is conformally equivalent to the canonical Riemannian orbifold which is the
quotient space of the standard n-dimensional sphere by a finite isometry group ...

Added: April 4, 2017

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2013. No. 1307.5522.

This is an expanded version of my talk at the workshop ``Groups of Automorphisms in Birational and Affine Geometry'', October 29–November 3, 2012, Levico Terme, Italy. The first section is focused on Jordan groups in abstract setting, the second on that in the settings of automorphisms groups and groups of birational self-maps of algebraic varieties. ...

Added: July 21, 2013

Avilov A., Sbornik Mathematics 2016 Vol. 307 No. 3 P. 315-330

We prove that any G-del Pezzo threefold of degree 4, except for a one-parameter family and four distinguished cases, can be equivariantly reconstructed to the projective space ℙ3, a quadric Q ⊂ ℙ4 , a G-conic bundle or a del Pezzo fibration. We also show that one of these four distinguished varieties is birationally rigid ...

Added: July 6, 2016

Bagaev A. V., Zhukova N., Journal of Geometry and Physics 2019 Vol. 142 P. 80-91

S.S. Chern conjectured that the Euler characteristic of every closed affine
manifold has to vanish. We present an analog of this conjecture stating that
the Euler-Satake characteristic of any compact affine orbifold is equal to zero.
We prove that Chern's conjecture is equivalent to its analog for
the Euler-Satake characteristic of compact affine orbifolds, and
orbifolds may be ineffective. This ...

Added: April 26, 2019

Prokhorov Y., Shramov K., / Cornell University. Series arXiv "math". 2018.

We prove that automorphism groups of Inoue and primary Kodaira surfaces are Jordan. ...

Added: June 8, 2019

Zhukova N., Математический сборник 2012 Т. 203 № 3 С. 79-106

Доказано, что любое полное конформное слоение (M,F) коразмерности q> 2 является либо римановым, либо (Conf(S^q),S^q)-слоением. Если (M,F) не является римановым слоением, то оно имеет глобальный аттрактор, представляющий собой либо нетривиальное минимальное множество, либо один замкнутый слой или объединение двух замкнутых слоев. При этом компактность многообразия M не предполагается. В частности, каждое собственное полное конформное не риманово ...

Added: September 28, 2014

Zhukova N. I., Proceedings of the Steklov Institute of Mathematics 2012 Vol. 278 No. 1 P. 94-105

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable ...

Added: October 19, 2014

Zhukova N., Журнал Средневолжского математического общества 2017 Т. 19 № 4 С. 33-44

For any smooth orbifold $\mathcal N$ is constructed a foliated model, which is a foliation
with an Ehresmann, the leaf space of which is the same as $\mathcal N$. We investigate
the relationship relationship between some properties of orbifold and its foliated model.
The article discusses the application to Cartan orbifolds, that is orbifolds endowed with Cartan geometry. ...

Added: February 20, 2018

Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 600-614

The second part of the paper is devoted to enumeration of r-regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to ...

Added: September 21, 2018

А.Ю. Долгоносова .., Н.И. Жукова, Труды Математического центра им. Н.И. Лобачевского 2013 Т. 47 С. 43-46

Different equivalent approaches to the notion of a foliation with transverse linear connection are
represented. ...

Added: October 18, 2014

Zhukova N., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 395-407

It is shown that the structural theory of Molino for Riemannian foliations on compact
manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with
Ehresmann connection. There are no restrictions on the codimension of the foliation
and the dimension of the foliated manifold.
For a Riemannian foliation $(M, F)$ with Ehresmann connection
it is proved that the closure of ...

Added: December 27, 2019

N. I. Zhukova, Труды Математического института им. В.А. Стеклова РАН 2012 Т. 278 С. 102-113

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable ...

Added: September 28, 2014

Gusein-Zade S., Алгебра и анализ 2021 Т. 33 № 3 С. 73-84

Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...

Added: May 2, 2021

Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2014. No. 1401.0278.

We explore orbits, rational invariant functions, and quotients of the natural actions of connected, not necessarily finite dimensional subgroups of the automorphism groups of irreducible algebraic varieties. The applications of the results obtained are given. ...

Added: January 3, 2014

Shramov K., Przyjalkowski V., Proceedings of the Steklov Institute of Mathematics 2019 Vol. 307 P. 198-209

We show that smooth well-formed weighted complete intersections have finite automorphism groups, with several obvious exceptions. ...

Added: August 12, 2020

Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723-3731

We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...

Added: August 18, 2020

Vladimir L. Popov, Springer Proceedings in Mathematics & Statistics 2014 Vol. 79 P. 185-213

This is an expanded version of my talk at the workshop
``Groups of Automorphisms in Birational and Affine Geometry'',
October 29–November 3, 2012, Levico Terme, Italy.
The first section is focused on Jordan groups in abstract setting,
the second on that in the settings of automorphisms groups and
groups of birational self-maps of algebraic varieties.
The appendix is an expanded version ...

Added: April 28, 2014