96 research outputs found
Flexible varieties and automorphism groups
Given an affine algebraic variety X of dimension at least 2, we let SAut (X)
denote the special automorphism group of X i.e., the subgroup of the full
automorphism group Aut (X) generated by all one-parameter unipotent subgroups.
We show that if SAut (X) is transitive on the smooth locus of X then it is
infinitely transitive on this locus. In turn, the transitivity is equivalent to
the flexibility of X. The latter means that for every smooth point x of X the
tangent space at x is spanned by the velocity vectors of one-parameter
unipotent subgroups of Aut (X). We provide also different variations and
applications.Comment: Final version; to appear in Duke Math.
Varieties covered by affine spaces and their cones
It was shown in arXiv:2303.02036 that the affine cones over flag manifolds
and rational smooth projective surfaces are elliptic in the sense of Gromov.
The latter remains true after successive blowups of points on these varieties.
In the present note we extend this to smooth projective spherical varieties (in
particular, toric varieties) successively blown up along linear subvarieties.
The same also holds, more generally, for projective varieties covered by affine
spaces.Comment: 10 page
Counting and computing regions of -decomposition: algebro-geometric approach
New methods for -decomposition analysis are presented. They are based on
topology of real algebraic varieties and computational real algebraic geometry.
The estimate of number of root invariant regions for polynomial parametric
families of polynomial and matrices is given. For the case of two parametric
family more sharp estimate is proven. Theoretic results are supported by
various numerical simulations that show higher precision of presented methods
with respect to traditional ones. The presented methods are inherently global
and could be applied for studying -decomposition for the space of parameters
as a whole instead of some prescribed regions. For symbolic computations the
Maple v.14 software and its package RegularChains are used.Comment: 16 pages, 8 figure
Flexibility properties in Complex Analysis and Affine Algebraic Geometry
In the last decades affine algebraic varieties and Stein manifolds with big
(infinite-dimensional) automorphism groups have been intensively studied.
Several notions expressing that the automorphisms group is big have been
proposed. All of them imply that the manifold in question is an
Oka-Forstneri\v{c} manifold. This important notion has also recently merged
from the intensive studies around the homotopy principle in Complex Analysis.
This homotopy principle, which goes back to the 1930's, has had an enormous
impact on the development of the area of Several Complex Variables and the
number of its applications is constantly growing. In this overview article we
present 3 classes of properties: 1. density property, 2. flexibility 3.
Oka-Forstneri\v{c}. For each class we give the relevant definitions, its most
significant features and explain the known implications between all these
properties. Many difficult mathematical problems could be solved by applying
the developed theory, we indicate some of the most spectacular ones.Comment: thanks added, minor correction
Birational geometry of hypersurfaces in products of projective spaces
We study the birational properties of hypersurfaces in products of projective spaces. In the case of hypersurfaces in Pm x Pn, we describe their nef, movable and e ective cones and determine when they are Mori dream spaces. Using this, we give new simple examples of non-Mori dream spaces and analogues of Mumford's example of a strictly nef line bundle which is not ample.This is the author accepted manuscript. The final version is available from Springer via http://dx.doi.org/10.1007/s00209-015-1415-
Geodesic flows on Riemannian g.o. spaces
We prove the integrability of geodesic flows on the Riemannian g.o. spaces of
compact Lie groups, as well as on a related class of Riemannian homogeneous
spaces having an additional principal bundle structure.Comment: 12 pages, minor corrections, final versio
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Flexible varieties and automorphism groups
Duke Math. J. 162, no. 4 (2013), 767-823 Given an affine algebraic variety X of dimension at least 2, we let SAut (X)
denote the special automorphism group of X i.e., the subgroup of the full
automorphism group Aut (X) generated by all one-parameter unipotent subgroups.
We show that if SAut (X) is transitive on the smooth locus of X then it is
infinitely transitive on this locus. In turn, the transitivity is equivalent to
the flexibility of X. The latter means that for every smooth point x of X the
tangent space at x is spanned by the velocity vectors of one-parameter
unipotent subgroups of Aut (X). We provide also different variations and
applications
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