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## Torus actions of complexity one in non-general position

Cornell University
,
2019.
No. 1905.04761.

Ayzenberg A., Cherepanov V.

Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n;Z)=0). It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite simplicial complex L we construct an equivariantly formal manifold X2n such that X2n/Tn−1 is homotopy equivalent to Σ3L. The constructed manifold X2n is the total space of the projective line bundle over the permutohedral variety hence the action on X2n is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of the action in j-general position and prove that, for any simplicial complex M, there exists an equivariantly formal action of complexity one in j-general position such that its orbit space is homotopy equivalent to Σj+2M.

Keywords: toric varietyторическое многообразиеtorus actionдействие тораAlexander dualitycomplexity oneсложность одинorbit spaceequivariant formalitypermutohedronpermutohedral varietyprojective line bundleпространство орбитэквивариантная формальностьдвойственность Александерапермутоэдрпермутоэдрическое многообразиепроективизация векторного расслоения

Ayzenberg A., Masuda M., / Cornell University. Series arXiv "math". 2019.

Let a compact torus T=T^{n−1} act on a smooth compact manifold X=X^{2n} effectively, with nonempty finite set of fixed points, and suppose that stabilizers of all points are connected. If H^{odd}(X)=0 and the weights of tangent representation at each fixed point are in general position, we prove that the orbit space Q=X/T is a homology (n+1)-sphere. If, in addition, π_1(X)=0, then Q is homeomorphic to S^{n+1}. ...

Added: January 14, 2020

Ayzenberg A., / Cornell University. Series arXiv "math". 2019. No. 1903.03460.

For an action of a compact torus T on a smooth compact manifold~X with isolated fixed points the number 12dimX−dimT is called the complexity of the action. In this paper we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5 and S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3). Here the maximal tori of the corresponding Lie ...

Added: October 23, 2019

Ayzenberg A., Cherepanov V., Osaka Journal of Mathematics 2021 Vol. 58 No. 4 P. 839-853

Let the compact torus Tn1 act on a smooth compact manifold X2n eectively with nonempty nite set of xed points. We pose the question: what can be said
about the orbit space X2n{Tn1 if the action is cohomologically equivariantly formal
(which essentially means that HoddpX2n;Zq 0)? It happens that homology of the orbit
space can be arbitrary ...

Added: October 31, 2019

Ayzenberg A., Arnold Mathematical Journal 2020 P. 1-24

For an effective action of a compact torus T on a smooth compact manifold X with nonempty finite set of fixed points, the number 12dimX−dimT12dimX−dimT is called the complexity of the action. In this paper, we study certain examples of torus actions of complexity one and describe their orbit spaces. We prove that HP2/T3≅S5HP2/T3≅S5 and S6/T2≅S4S6/T2≅S4, for the homogeneous spaces HP2=Sp(3)/(Sp(2)×Sp(1))HP2=Sp(3)/(Sp(2)×Sp(1)) and S6=G2/SU(3)S6=G2/SU(3). Here, the maximal tori of ...

Added: November 19, 2020

Ayzenberg A., Algebraic and Geometric Topology 2020 Vol. 20 No. 6 P. 2957-2994

A periodic tridiagonal matrix is a tridiagonal matrix with additional two entries at the corners. We study the space $X_{n,\lambda}$ of Hermitian periodic tridiagonal $n\times n$-matrices with a fixed simple spectrum $\lambda$. Using the discretized S\edt{c}hr\"{o}dinger operator we describe all spectra $\lambda$ for which $X_{n,\lambda}$ is a topological manifold. The space $X_{n,\lambda}$ carries a natural effective action of a compact $(n-1)$-torus. ...

Added: January 14, 2020

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

Zaitseva Y., Математические заметки 2019 Т. 105 № 6 С. 824-838

В работе получено описание однородных локально нильпотентных дифференцирований алгебры регулярных функций некоторого класса триномиальных гиперповерхностей. Данный класс включает в себя все нефакториальные триномиальные гиперповерхности. ...

Added: September 19, 2019

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., / 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

Arzhantsev I., Hausen J., Mathematical Research Letters 2007 Vol. 14 No. 1 P. 129-136

Given a multigraded algebra A, it is a natural question whether or not for
two homogeneous components A_u and A_v, the product A_nuA_nv is the whole component
A_nu+nv for n big enough. We give combinatorial and geometric answers to this question. ...

Added: July 10, 2014

Arzhantsev I., Bragin S., Zaitseva Y., Communications in Contemporary Mathematics 2020 Vol. 22 No. 8 P. 1950064: 1

We study commutative associative polynomial operations A^n×A^n→A^n with unit on the affine space A^n over an algebraically closed field of characteristic zero. A classification of such operations is obtained up to dimension 3. Several series of operations are constructed in arbitrary dimension. Also we explore a connection between commutative algebraic monoids on affine spaces and additive actions on toric ...

Added: September 19, 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

Arzhantsev I., Ricerche di Matematica 2021

We show that an effective action of the one-dimensional torus G_m on a normal affine algebraic variety X can be extended to an effective action of a semi-direct product G_m⋌G_a with the same general orbit closures if and only if there is a divisor D on X that consists of G_m-fixed points. This result is applied to the study of orbits of the automorphism group Aut(X) on X. ...

Added: August 16, 2021

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

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

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

Cherepanov V., Математический сборник 2019

We consider effective actions of a compact torus Tn−1 on an even-dimensional smooth manifold M2n with isolated fixed points. We prove that under certain conditions on weights of tangent representations, the orbit space is a manifold with corners. Given that the action is Hamiltonian, the orbit space is homeomorphic to Sn+1∖(U1⊔…⊔Ul) where Sn+1 is the (n+1)--sphere and U1,…,Ul are open domains. We apply the results to ...

Added: October 28, 2020

Ayzenberg A., Gugnin D., / Cornell University. Series arXiv "math". 2019.

Let G1 and G2 be discrete subgroups of SO(3). The double quotients of the form X(G1,G2)=G1∖SO(3)/G2 were introduced in material science under the name misorientation spaces. In this paper we review several known results that allow to study topology of misorientation spaces. Neglecting the orbifold structure, all misorientation spaces are closed orientable topological 3-manifolds with finite fundamental groups. We compute ...

Added: January 14, 2020

Ayzenberg A., Бухштабер В. М., Математический сборник 2021

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space
$M_{\St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove this space to be a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe ...

Added: November 6, 2020

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

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

Ilten N. O., Lewis J., Victor Przyjalkowski, Journal of Algebra 2013 Vol. 374 P. 104-121

We show that every Picard rank one smooth Fano threefold has a weak Landau–Ginzburg model coming from a toric degeneration. The fibers of these Landau–Ginzburg models can be compactified to K3 surfaces with Picard lattice of rank 19. We also show that any smooth Fano variety of arbitrary dimension which is a complete intersection of ...

Added: July 2, 2013

Arzhantsev I., Acta Arithmetica 2018 Vol. 186 No. 1 P. 87-99

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface admits a Schwarz-Halphen curve if and only if the trinomial comes from a platonic triple. It is a generalization of Schwarz-Halphen's Theorem ...

Added: October 20, 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