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Factorial algebraic group actions and categorical quotients
Journal of Algebra. 2013. Vol. 387. P. 87-98.
Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of afinitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1502.02167.
According to the classical theorem, every irreducible algebraic variety endowed with a nontrivial rational action of a connected linear algebraic group is birationally isomorphic to a product of another algebraic variety and the s-dimensional projectice space with positive s. We show that the classical proof of this theorem actually works only in characteristic 0 and ...
Added: February 10, 2015
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2021. No. 2105.12861.
Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. ...
Added: May 28, 2021
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1503.08303.
For every pair (G, V ) where G is a connected simple
linear algebraic group and V is a simple algebraic G-module with
a free algebra of invariants, the number of irreducible components
of the nullcone of unstable vectors in V is found. ...
Added: March 31, 2015
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2019. No. 1901.07030.
This is an expanded version of the talk by the author at the conference Polynomial Rings and Affine Algebraic Geometry, February 12–16, 2018, Tokyo Metropolitan University, Tokyo, Japan. Considering a local version of the Zariski Cancellation Problem naturally leads to exploration of some classes of varieties of special kind and their equivariant versions. We discuss ...
Added: January 23, 2019
Arzhantsev I., Liendo A., Stasyuk T., Journal of Pure and Applied Algebra 2021 Vol. 225 No. 2 P. 106499
Let X be a normal variety endowed with an algebraic torus action. An additive group action alpha on X is called vertical if a general orbit of alpha is contained in the closure of an orbit of the torus action and the image of the torus normalizes the image of alpha in Aut(X). Our first result in this paper ...
Added: July 29, 2020
Р.С. Авдеев, Труды Московского математического общества 2011 Т. 72 № 1 С. 5-62
We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. ...
Added: February 25, 2014
Arzhantsev I., Bazhov I., Central European Journal of Mathematics 2013 Vol. 11 No. 10 P. 1713-1724
Let X be an affine toric variety. The total coordinates on X provide a canonical presentation !X -> X of X as a quotient of a vector space !X by a linear action of a quasitorus. We prove that the orbits of the connected component of the automorphism group Aut(X) on X coincide with the ...
Added: November 13, 2013
V. L. Popov, Mathematical notes 2018 Vol. 103 No. 5 P. 811-819
We prove that the family of all connected n-dimensional real Lie groups is uniformly Jordan for every n. This
implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some
transformation groups of complex spaces and Riemannian manifolds are Jordan. ...
Added: April 13, 2018
Arzhantsev I., Hausen J., Journal of Pure and Applied Algebra 2009 Vol. 213 No. 1 P. 154-172
We consider actions of reductive groups on a variety with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a ...
Added: July 10, 2014
Vladimir L. Popov, Transformation Groups 2014 Vol. 19 No. 2 P. 549-568
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: March 17, 2014
Попов В. Л., Известия РАН. Серия математическая 2019 Т. 84 № 4 С. 194-225
The rst group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...
Added: July 31, 2019
Р.С. Авдеев, Математические заметки 2013 Т. 94 № 1 С. 22-35
For an arbitrary connected solvable spherical subgroup H of a connected semisimple algebraic group G, we compute the group N_G(H), the normalizer of H in G. Thereby we complete a classification of all (not necessarily connected) solvable spherical subgroups in semisimple algebraic groups. ...
Added: February 25, 2014
Tokyo : American Mathematical Society, World Scientific, 2017
Preface
The workshop “Algebraic Varieties and Automorphism Groups” was held at the Research Institute of Mathematical Sciences (RIMS), Kyoto University during July 7-11, 2014. There were over eighty participants including twenty people from overseas Canada, France, Germany, India, Korea, Poland, Russia, Singapore, Switzerland, and USA.
Recently, there have been remarkable developments in algebraic geometry and related fields, ...
Added: July 12, 2017
Vladimir L. Popov, / Cornell University. Series math "arxiv.org". 2015. No. 1504.03867.
Exploring Bass' Triangulability Problem on unipotent algebraic subgroups of the affine Cremona groups, we prove a triangulability criterion, the existence of nontriangulable connected solvable affine algebraic subgroups of the Cremona groups, and stable triangulability of such subgroups; in particular, in the stable range we answer Bass' Triangulability Problem is the affirmative. To this end we ...
Added: April 16, 2015
V. L. Popov, Transformation Groups 2011 Vol. 16 No. 3 P. 827-856
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a
closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove
that in arbitrary G such a cross-section exists if and only if the ...
Added: March 16, 2013
Amerik E., Campana F., / Cornell University library. Series arxiv.org "algebraic geometry". 2013.
This is a note on Beauville's problem (solved by Greb, Lehn and Rollenske in the non-algebraic case and by Hwang and Weiss in general) whether a lagrangian torus on an irreducible holomorphic symplectic manifold is a fiber of a lagrangian fibration. We provide a different, very short solution in the non-algebraic case and make some ...
Added: April 9, 2013
Roman Avdeev, Transformation Groups 2021 Vol. 26 No. 2 P. 403-431
Given a connected reductive complex algebraic group G and a spherical subgroup H⊂G, the extended weight monoid Λˆ+G(G/H) encodes the G-module structures on spaces of regular sections of all G-linearized line bundles on G/H. Assuming that G is semisimple and simply connected and H is specified by a regular embedding in a parabolic subgroup P⊂G, ...
Added: September 9, 2021
V. L. Popov, Izvestiya: Mathematics, England 2019 Vol. 83 No. 4 P. 830-859
The first group of results of this paper concerns the compressibility of finite subgroups of the Cremona groups. The second concerns the embeddability of other groups in the Cremona groups and, conversely, the Cremona groups in
other groups. The third concerns the connectedness of the Cremona groups. ...
Added: September 29, 2019
Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13-25
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...
Added: November 5, 2022
Arzhantsev I., Zaitseva Y., Russian Mathematical Surveys 2022 Vol. 77 No. 4 P. 571-650
We survey recent results on open embeddings of the affine space C^n into a complete algebraic variety X such that the action of the vector group G_a^n on C^n by translations extends to an action of G_a^n on X. We begin with the Hassett–Tschinkel correspondence describing equivariant embeddings of \mathbb{C}^n into projective spaces and present its generalization for embeddings into projective hypersurfaces. Further sections deal with embeddings into flag ...
Added: February 26, 2023
V. L. Popov, Doklady Mathematics 2017 Vol. 96 No. 1 P. 312-314
For connected simple algebraic groups defined over an algebraically closed field of characteristic
zero, the classifications of irreducible algebraic representations of modalities 0, 1, and 2 are obtained. ...
Added: June 30, 2017
Abasheva A., / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2007.05773.
In this paper we study the geometry of the total space Y of a cotangent bundle to a Kähler manifold N where N is obtained as a Kähler reduction from Cn. Using the hyperkähler reduction we construct a hyperkähler metric on Y and prove that it coincides with the canonical Feix-Kaledin metric. This metric is in general non-complete. We show that the metric completion Y~ of ...
Added: July 21, 2020
Popov V., Известия РАН. Серия математическая 2022 Т. 86 № 5 С. 73-96
We explore to what extent the group variety of a connected algebraic group or the group manifold of a real Lie group determines its group structure. ...
Added: June 9, 2022
Roman Avdeev, Cupit-Foutou S., Advances in Mathematics 2018 Vol. 328 P. 1299-1352
Given a connected reductive algebraic group G and a finitely generated monoid Γ of dominant weights of G, in 2005 Alexeev and Brion constructed a moduli scheme M_Γ for multiplicity-free affine G-varieties with weight monoid Γ. This scheme is equipped with an action of an `adjoint torus' T_ad and has a distinguished T_ad-fixed point X_0. ...
Added: February 25, 2018