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## SU-bordism: structure results and geometric representatives

Russian Mathematical Surveys. 2019. Vol. 74. No. 3. P. 461-524.

The first part of this survey gives a modernised exposition of the structure of the special unitary bordism ring, by combining the classical geometric methods of Conner-Floyd, Wall, and Stong with the Adams-Novikov spectral sequence and formal group law techniques that emerged after the fundamental 1967 paper of Novikov. In the second part toric topology is used to describe geometric representatives in SU-bordism classes, including toric, quasi-toric, and Calabi-Yau manifolds.

Limonchenko I., Lu Z., Panov T., Proceedings of the Steklov Institute of Mathematics 2018 Vol. 302 P. 270-278

Batyrev constructed a family of Calabi-Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi-Yau manifolds whose SU-bordism classes generate the special unitary bordism ring with 2 inverted. We also describe explicit Calabi-Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions. ...

Added: October 29, 2021

Khovanskii A., Limonchenko I., Monin L., Filomat 2022 Vol. 36 No. 19 P. 6513-6537

The classical BKK theorem computes the intersection number of divisors on toric variety in terms of volumes of corresponding polytopes. It was observed by Pukhlikov and the first author that the BKK theorem leads to a presentation of the cohomology ring of toric variety as a quotient of the ring of differential operators with constant ...

Added: June 1, 2023

Limonchenko I., Лю Ж., Панов Т. Е., Труды Математического института им. В.А. Стеклова РАН 2018 Т. 302 С. 287-295

V. V. Batyrev constructed a family of Calabi–Yau hypersurfaces dual to the first Chern class in toric Fano varieties. Using this construction, we introduce a family of Calabi–Yau manifolds whose SU-bordism classes generate the special unitary bordism ring. We also describe explicit Calabi–Yau representatives for multiplicative generators of the SU-bordism ring in low dimensions. ...

Added: September 25, 2019

Galkin S., / Cornell University. Series math "arxiv.org". 2014. No. 1404.7388.

Consider a Laurent polynomial with real positive coefficients such that the origin is strictly inside its Newton polytope. Then it is strongly convex as a function of real positive argument. So it has a distinguished Morse critical point --- the unique critical point with real positive coordinates. As a consequence we obtain a positive answer ...

Added: May 4, 2014

Dzhunusov S., Zaitseva Y., Forum Mathematicum 2021 Vol. 33 No. 1 P. 177-191

We classify commutative algebraic monoid structures on normal affine surfaces over an algebraically closed field of characteristic zero. The answer is given in two languages: comultiplications and Cox coordinates. The result follows from a more general classification of commutative monoid structures of rank 0, n-1 or n on a normal affine variety of dimension n. ...

Added: January 15, 2021

Shafarevich A., Indagationes Mathematicae 2023 Vol. 34 No. 1 P. 42-53

Let G_a be the additive group of the field of complex numbers ℂ. We say that an irreducible algebraic variety X of dimension n admits an additive action if there is a regular action of the group G_a^n =G_a×…×G_a (n times) on X with an open orbit. In 2017 Baohua Fu and Jun-Muk Hwang introduced a class of Euler-symmetric varieties. They gave a classification ...

Added: February 6, 2023

I. Arzhantsev, Kaliman S., M. Zaidenberg, Advances in Mathematics 2024 Vol. 437 Article 109449

It was shown by Kaliman and Zaidenberg (2023) 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 article we extend this to smooth projective spherical varieties (in particular, toric varieties) successively ...

Added: December 17, 2023

Arzhantsev I., Indagationes Mathematicae 2023 Vol. 34 No. 4 P. 812-819

We prove that every non-degenerate toric variety, every homogeneous space of a connected linear algebraic group without non-constant invertible regular functions, and every variety covered by affine spaces admit a surjective morphism from an affine space. ...

Added: May 24, 2023

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

Denis V. Osipov, Bulletin of the Korean Mathematical Society 2017 Vol. 54 No. 5 P. 1699-1718

We gave a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic K-theory and depends on the canonical Z-torsor of a locally linearly compact k-vector space. Analogs of certain auxiliary results for ...

Added: January 26, 2018

Leonid Monin, Smirnov E., Seminaire Lotharingien de Combinatoire 2023 Vol. 89B Article 76

In 1992, Pukhlikov and Khovanskii provided a description of the cohomology ring of toric variety as a quotient of the ring of differential operators on spaces of virtual polytopes. Later Kaveh generalized this construction to the case of cohomology rings for full flag varieties.
In this paper we extend Pukhlikov--Khovanskii type presentation to the case of K-theory ...

Added: October 26, 2023

Ayzenberg A., Masuda M., Park S. et al., Proceedings of the Steklov Institute of Mathematics 2015 Vol. 288 No. 1 P. 10-28

We construct quasitoric manifolds of dimension 6 and higher which are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given and the statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metric properties. ...

Added: September 24, 2015

Shakhmatov K., Математические заметки 2021 Т. 109 № 6 С. 929-937

An open translation-equivariant embedding of the affine space A^n into a complete nonprojective algebraic variety X is constructed for any n >= 3. The main tool is the theory of toric varieties. In the case n = 3, the orbit structure of the obtained action on the variety X is described. ...

Added: June 6, 2021

Arzhantsev I., Zaidenberg M., International Mathematics Research Notices 2022 Vol. 2022 No. 11 P. 8162-8195

Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U_1,…,U_s of Aut(X), which are normalized by the torus acting on X, we show that the group G generated by U_1,…,U_s verifies the following alternative of Tits type: either G is a unipotent algebraic group or it contains a non-abelian free subgroup. We deduce that if G is 2-transitive on a G-orbit in X, then G contains a non-abelian ...

Added: January 31, 2021

Limonchenko I., Монин Л. В., Хованский А. Г., Труды Математического института им. В.А. Стеклова РАН 2022 Т. 317 С. 1-27

We develop a theory of volume polynomials of generalized virtual polytopes based
on the study of topology of affine subspace arrangements in a real Euclidean space. We apply
this theory to obtain a topological version of the Bernstein–Kushnirenko theorem as well
as Stanley–Reisner and Pukhlikov–Khovanskii type descriptions for the cohomology rings of
generalized quasitoric manifolds. ...

Added: October 28, 2022

Белев С. А., Tyurin N. A., Теоретическая и математическая физика 2013 Т. 175 № 2 С. 147-158

We prove the existence of a rank-one pseudotoric structure on an arbitrary smooth toric symplectic manifold. As a consequence, we propose a method for constructing Chekanov-type nonstandard Lagrangian tori on arbitrary toric manifolds. ...

Added: February 18, 2013

Arzhantsev I., Алгебра и анализ 2022 Т. 34 № 2 С. 1-55

В работе дан обзор результатов последних лет о кратной транзитивности действий групп автоморфизмов аффинных алгебраических многообразий. Рассматривается свойство бесконечной транзитивности действия группы специальных автоморфизмов и эквивалентное ему свойство гибкости многообразия. Данные свойства имеют важные алгебраические и геометрические следствия, и при этом они выполнены для широких классов многообразий. Отдельно изучаются случаи, когда бесконечная транзитивность имеет место ...

Added: March 14, 2022

Bilich B., / Cornell University. Series math "arxiv.org". 2021. No. 2106.04884.

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by ...

Added: June 13, 2021

Galkin S., Belmans P., Mukhopadhyay S., Experimental Mathematics 2019

We give the first examples of smooth Fano and Calabi–Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have ...

Added: October 4, 2019

Arzhantsev I., St Petersburg Mathematical Journal 2023 Vol. 34 No. 2 P. 143-178

We survey recent results on multiple transitivity for automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of ...

Added: March 30, 2023

Arzhantsev I., Perepechko A., / Bulletin des sciences mathématiques. Series 22-00305 "BULSCI-D". 2023.

We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups ...

Added: October 6, 2023

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

Kotenkova P., Beitrage zur Algebra und Geometrie 2014 Vol. 55 No. 2 P. 621-634

Let X be a normal affine algebraic variety with regular action of a torus T and T ⊂ T be a subtorus. We prove that each root of X with respect to T can be obtained by restriction of some root of X with respect to T. This allows to get an elementary proof of ...

Added: September 17, 2015

Ayzenberg A., Труды Математического института им. В.А. Стеклова РАН 2018 Т. 302 С. 23-40

We consider an effective action of a compact (n-1)-torus on a smooth 2n-manifold with isolated xed points. We prove that under certain conditions the orbit space is a closed topological manifold. In particular, this holds for certain torus actions with disconnected stabilizers. There is a ltration of the orbit manifold by orbit dimensions. The subset ...

Added: October 15, 2018