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Rational curves on foliated varieties
P. 1–22.
Bogomolov F. A., McQuillan M.
This article represents a study of ample subbundles of the tangent sheaf of a variety in a formal neighbourhood of a curve. With the added hypothesis of integrability it is best possible. A particular corollary is Mori’s cone theorem for foliations by curves.
Publication based on the results of:
In book
Springer, 2015.
Lvovsky S., / Series arXiv "math". 2024.
Suppose that F is a smooth and connected complex surface (not necessarily compact) containing a smooth rational curve with positive self-intersection. We prove that if there exists a non-constant meromorphic function on F, then the field of meromorphic functions on F is isomorphic to the field of rational functions in one or two variables over ℂ. ...
Added: December 3, 2024
Lvovsky S., / Series arXiv "math". 2023.
We prove that for any d>0 there exists an embedding of the Riemann sphere ℙ1in a smooth complex surface, with self-intersection d, such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along C has transcendence degree 2 over ℂ. We give two different constructions of such neighborhoods, either ...
Added: December 3, 2023
Amerik E., Verbitsky M., , in: Rationality of Varieties.: Birkhäuser, 2021. P. 75–96.
Added: April 6, 2022
Zhukova N., Applied Mathematics and Nonlinear Sciences 2020 Vol. 5 No. 2 P. 279–292
The purpose of this article is to review the author's results on the existence and
structure of minimal sets and attractors of conformal foliations of codimension $q,$ ${q\geq 3.}$
Results on strong transversal equivalence of conformal foliations are also presented.
Connections with works of other authors are indicated. Examples of conformal foliations with
exceptional, exotic and regular minimal sets ...
Added: December 30, 2019
Ilyashenko Y., Shilin I., , in: Contemporary MathematicsVol. 692: Modern Theory of Dynamical Systems: A Tribute to Dmitry Victorovich Anosov.: Providence, Rhode Island: American Mathematical Society, 2017. P. 155–175.
Different notions of attractors and relations between them are considered. The major new result claims that Lyapunov unstable Milnor attractors are topologically generic in a space of diffeomorphisms of any manifold of dimension greater than one. This result is due to the second author. A sketch of the proof is given. New robust properties of ...
Added: April 10, 2018
Zhukova N., Sheina K., Журнал Средневолжского математического общества 2016 Т. 18 № 2 С. 30–40
We find necessary and sufficient conditions for a foliation of codimension $q$ on $n$-dimensional manifold with transverse linear connection to admit a transverse invariant pseudo-Riemannian metric of a given signature which is parallel with the respect to the indicated connection. In particular, we obtain a criterion for a foliation with transverse linear connection to be ...
Added: June 7, 2016
Dolgonosova A., Zhukova N., Журнал Средневолжского математического общества 2015 Т. 17 № 4 С. 14–23
We prove the equivalence of three different approaches to the definition of completeness of a foliation with transverse linear connection. It is shown that for the transverse a ne foliations
(M, F) of codimension q, q > 1, each of the mentioned above conditions are equivalent to
ful llment of the following two conditions: 1) there exists an Ehresmann ...
Added: March 12, 2016
Springer, 2015.
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. ...
Added: November 24, 2015
Amerik E., Verbitsky M., International Mathematics Research Notices 2015 Vol. 2015 No. 23 P. 13009–13045
Let M be an irreducible holomorphically symplectic manifold. We show that all faces of the Kähler cone of M are hyperplanes Hi orthogonal to certain homology classes, called monodromy birationally minimal (MBM) classes. Moreover, the Kähler cone is a connected component of a complement of the positive cone to the union of all Hi. We ...
Added: October 28, 2015
Verbitsky M., Pure and Applied Mathematics Quarterly 2014 Vol. 10 No. 2 P. 325–354
Let S be a smooth rational curve on a complex manifold M. It is called ample if its normal bundle is positive: NS=⨁O(i_k),i_k<0. We assume that M is covered by smooth holomorphic deformations of S. The basic example of such a manifold is a twistor space of a hyperkähler or a 4–dimensional anti-selfdual Riemannian manifold ...
Added: January 23, 2015
Boston: Birkhäuser, 2013.
This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the ...
Added: February 14, 2013