arXiv subjects

Victor M. Buchstaber

Publications and source records attributed to Victor M. Buchstaber.

At least 19 recordsLinked to original sources

arXiv2026

A solution to the quantum Yang--Baxter equation associated with a universal two-valued algebraic group

We construct spectral-parameter solutions of the quantum Yang--Baxter equation from the associativity condition of the universal symmetric $2$-algebraic two-valued group, whose multiplication law depends on four coefficients $k_2,k_4,k_6,k_8$ subject to the single relation $4k_8-k_4^2+k_2k_6=0$ -- which coincides with the three-dimensional WDVV equation for Dubrovin's normalised potential. Every three-dimensional Frobenius manifold with a cyclic vector field carries such a spectral two-valued group, whose finite branch points are the eigenvalues of the multiplication operator. For an arbitrary unital algebra we show that a two-parameter spectral $R$-operator built from multiplication and unit has Yang--Baxter defect proportional to an explicit tensor built from the associator; for tangent Frobenius algebras this yields a regular, unitary $9\times9$ $R$-matrix satisfying the quantum Yang--Baxter equation exactly at WDVV points, with spectral parameter the Abel coordinate of a bielliptic curve whose elliptic quotient is the spectral curve, together with an associated integrable chain. The $A_3$ Frobenius manifold is worked out explicitly.

nlin.SI↗

arXiv2026

Smooth manifolds in $G_{n,2}$ and $\mathbb{C} P^{N}$ defined by symplectic reductions of $T^n$-action

We study symplectic reductions arising from the canonical action of the maximal compact torus $T^n$ on the complex Grassmann manifold $G_{n,2}$ as well as those arising from the $T^n$-action on the complex projective space $\C P^{N}$, $N=\binom{n}{2}-1$ for which the Plücker embedding $G_{n,2}\to \C P^{N}$ is $T^n$-equivariant. We investigate the topology of regular level sets of the moment maps and the corresponding symplectic reductions. For $n=4$ we show that the regular level sets of the moment maps do not depend on a regular value. We prove this set to be homeomorphic to $S^3\times T^2$ in the case $G_{4,2}$, while in the case $\C P^5$ it is homeomorphic to $ S^5\times T^2$. We relate our constructions to moduli spaces of weighted pointed stable genus zero curves. The Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$ is proved to arise as a symplectic reduction of $G_{n,2}$ by the canonical $T^n$-action in precisely the cases $n=4,5$. In the case $n\geq 5$, there is well known the Losev-Manin compactification different from Deligne-Mumford and it appears to be this symplectic reduction only in the case $n=5$. In this way we show that for $n=5$, a symplectic reduction depends on a regular value of the moment map.

math.AT↗

arXiv2026

Algebraic topology of the Lagrange inversion

The Lagrange inversion formula for power series is one of the classical formulas from analysis and combinatorics. A nice geometric interpretation of this formula in terms of the Stasheff polytopes was discovered by Loday. We show that it also admits a natural topological interpretation in terms of the Chern numbers of the complex projective space. The proof is based on our earlier work on the Chern-Dold character in complex cobordism theory and leads to a new derivation of the Lagrange inversion formula. We provide a similar interpretation of the multiplicative inversion formula in terms of Chern numbers of the smooth theta divisors. In this relation we introduce a new formal group defined by the Catalan numbers and explain the topological meaning of the corresponding Hirzebruch genus. Finally, we discuss a related general problem of when all Chern numbers of an algebraic variety are divisible by its Euler characteristic.

math.AG↗

arXiv2026

Cohomology of symplectic $T^n$ - reductions and compactifications of $\mathcal{M}_{0, n}$

A symplectic $T^n$ - reduction on a complex Grassmann manifold $G_{n,2}$ for the canonical action of the maximal compact torus depends on the $S_n$ - orbit of a maximal chamber in a hypersimplex $Δ_{n,2}$. The chamber decomposition of $Δ_{n,2}$ is defined by the admissible polytopes, which can be realized as matroids. In our previous work we described this chamber decomposition by means of the hyperplane arrangement. It is important to note that to any chamber it corresponds the compact space, which is a smooth compact manifold for a chamber of maximal dimension. In this paper we obtain explicit description of the cohomology rings of the symplectic $T^n$ - reductions on $G_{n,2}$ for the standard moment map in terms of the chamber decomposition of $Δ_{n,2}$ and the well known results of Kirwan and Goldin. We recently introduced the Hassett category whose objects are special compactifications of the space $\mathcal{M}_{0, n}$. The initial object in this category is the Deligne-Mumford compactification $\overline{\mathcal{M}}_{0, n}$. In this paper we describe the cohomology rings of various compactifications for $\mathcal{M}_{0, n}$, which correspond to the objects of the Hassett category. Our results show that even in the case $n=5$ there are objects in the Hassett category with different cohomology rings.

math.AT↗

arXiv2025

Sigma function associated with a hyperelliptic curve with two points at infinity

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel-Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve. In our previous paper, a solution to the KP equation was constructed in terms of the Baker function. This paper is devoted to the properties of the Baker functions. In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions. We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically. We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.

math.AG↗

arXiv2025

Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation

In this paper, a theory of hyperelliptic functions based on multidimensional sigma functions is developed and explicit formulas for hyperelliptic solutions to the Kadomtsev-Petviashvili equations KP-I and KP-II are obtained. The long-standing problem of describing the dependence of these solutions on the variation of the coefficients of the defining equation of a hyperelliptic curve, which are integrals of the equations, is solved.

math-ph↗

arXiv2024

Todd polynomials and Hirzebruch numbers

In 1956 Hirzebruch found an explicit formula for the denominators of the Todd polynomials, which was proved later in his joint work with Atiyah. We present a new formula for the Todd polynomials in terms of the ``forgotten symmetric functions", which follows from our previous work on complex cobordisms. In particular, this leads to a simpler proof of the Hirzebruch formula and provides new interpretations for the Hirzebruch numbers.

math.AT↗

arXiv2024

Moduli space of weighted pointed stable curves and toric topology of Grassmann manifolds

We relate the theory of moduli spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ of stable weighted curves of genus $0$ to the equivariant topology of complex Grassmann manifolds $G_{n,2}$, with the canonical action of the compact torus $T^n$. We prove that all spaces $\overline{\mathcal{M}}_{0,\mathcal{A}}$ can be isomorphically or up to birational morphisms embedded in $G_{n,2}/T^n$ . The crucial role for proving this result play the chamber decomposition of the hypersimplex $Δ_{n,2}$ which corresponds to $(\mathbb{C} ^{\ast})^{n}$-stratification of $G_{n,2}$ and the spaces of parameters over the chambers, which are subspaces in $G_{n,2}/T^n$. We show that the points of these moduli spaces $\overline{\mathcal{M}}_{0, \mathcal{A}}$ have the geometric realization as the points of the spaces of parameters over the chambers. We single out the characteristic categories among such moduli spaces. The morphisms in these categories correspond to the natural projections between the universal space of parameters and the spaces of parameters over the chambers. As a corollary, we obtain the realization of the orbit space $G_{n,2}/T^n$ as an universal object for the introduced categories. As one of our main results we describe the structure of the canonical projection from the Deligne-Mumford compactification to the Losev-Manin compactification of $\mathcal{M}_{0,n}$, using the embedding of $\mathcal{M}_{0, n}\subset \bar{L}_{0, n, 2}$ in $(\mathbb{C} P^{1})^{N}$, $N=\binom{n-2}{2}$, the action of the algebraic torus $(\mathbb{C} ^{\ast})^{n-3}$ on $(\mathbb{C} P^{1})^{N}$ for which $\bar{L}_{0, n, 2}$ is invariant, and the realization of the Losev-Manin compactification as the corresponding permutohedral toric variety.

math.AG↗

arXiv2023

Classification of involutive commutative two-valued groups

A complete classification of finitely generated involutive commutative two-valued groups is obtained. Three series of such two-valued groups are constructed: principal, unipotent and special, and it is shown that any finitely generated involutive commutative two-valued group is isomorphic to a two-valued group belonging to one of these series. A number of classification results are obtained for topological involutive commutative two-valued groups in the Hausdorff and locally compact cases. The classification of algebraic involutive two-valued groups in the one-dimensional case is also discussed.

math.GR↗

arXiv2022

Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions

The article is devoted to the classical problems about the relationships between elliptic functions and hyperelliptic functions of genus 2. It contains new results, as well as a derivation from them of well-known results on these issues. Our research was motivated by applications to the theory of equations and dynamical systems integrable in hyperelliptic functions of genus 2. We consider a hyperelliptic curve $V$ of genus 2 which admits a morphism of degree 2 to an elliptic curve. Then there exist two elliptic curves $E_i$, $i=1,2$, and morphisms of degree 2 from $V$ to $E_i$. We construct hyperelliptic functions associated with $V$ from the Weierstrass elliptic functions associated with $E_i$ and describe them in terms of the fundamental hyperelliptic functions defined by the logarithmic derivatives of the two-dimensional sigma functions. We show that the restrictions of hyperelliptic functions associated with $V$ to the appropriate subspaces in $\mathbb{C}^2$ are elliptic functions and describe them in terms of the Weierstrass elliptic functions associated with $E_i$. Further, we express the hyperelliptic functions associated with $V$ on $\mathbb{C}^2$ in terms of the Weierstrass elliptic functions associated with $E_i$. We derive these results by describing the homomorphisms between the Jacobian varieties of the curves $V$ and $E_i$ induced by the morphisms from $V$ to $E_i$ explicitly.

math.AG↗

arXiv2021

The orbit spaces $G_{n,2}/T^n$ and the Chow quotients $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ of the Grassmann manifolds $G_{n,2}$

The focus of our paper is on the complex Grassmann manifolds $G_{n,2}$ which appear as one of the fundamental objects in developing the interaction between algebraic geometry and algebraic topology. In his well-known paper Kapranov has proved that the Deligne-Mumford compactification $\overline{\mathcal{M}}(0,n)$ of $n$-pointed curves of genus zero can be realized as the Chow quotient $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$. In our recent papers, the constructive description of the orbit space $G_{n,2}/T^n$ has been obtained. In getting this result our notions of the CW-complex of the admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for $T^n$-action on $G_{n,2}$ were of essential use. Using technique of the wonderful compactification, in this paper it is given an explicit construction of the space $\mathcal{F}_{n}$. Together with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled us to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. Thus, we showed that the space $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$ can be realized as our universal space of parameters $\mathcal{F}_{n}$. In this way, we give description of the structure in $G_{n,2}\!/\!/(\mathbb{C} ^{\ast})^{n}$, that is $\overline{\mathcal{M}}(0,n)$ in terms of the CW-complex of the admissible polytopes for $G_{n,2}$ and their spaces of parameters.

math.AG↗

arXiv2020

A resolution of singularities for the orbit spaces $G_{n,2}/T^n$

The problem of the description of the orbit space $X_{n} = G_{n,2}/T^n$ for the standard action of the torus $T^n$ on a complex Grassmann manifold $G_{n,2}$ is widely known and it appears in diversity of mathematical questions. A point $x\in X_{n}$ is said to be a critical point if the stabilizer of its corresponding orbit is nontrivial. In this paper, the notion of singular points of $X_n$ is introduced which opened the new approach to this problem. It is showed that for $n>4$ the set of critical points $\text{Crit}X_n$ belongs to our set of singular points $\text{Sing}X_{n}$, while the case $n=4$ is somewhat special for which $\text{Sing}X_4\subset \text{Crit}X_4$, but there are critical points which are not singular. The central result of this paper is the construction of the smooth manifold $U_n$ with corners, $\dim U_n = \dim X_n$ and an explicit description of the projection $p_{n} : U_{n}\to X_{n}$ which in the defined sense resolve all singular points of the space $X_n$. Thus, we obtain the description of the orbit space $G_{n,2}/T^n$ combinatorial structure. Moreover, the $T^n$-action on $G_{n,2}$ is a seminal example of complexity $(n-3)$ - action. Our results demonstrate the method for general description of orbit spaces for torus actions of positive complexity.

math.AT↗

arXiv2020

Analytical and number-theoretical properties of the two-dimensional sigma function

This survey is devoted to the classical and modern problems related to the entire function ${σ({\bf u};λ)}$, defined by a family of nonsingular algebraic curves of genus $2$, where ${\bf u} = (u_1,u_3)$ and $λ= (λ_4, λ_6,λ_8,λ_{10})$. It is an analogue of the Weierstrass sigma function $σ(u;g_2,g_3)$ of a family of elliptic curves. Logarithmic derivatives of order 2 and higher of the function ${σ({\bf u};λ)}$ generate fields of hyperelliptic functions of ${\bf u} = (u_1,u_3)$ on the Jacobians of curves with a fixed parameter vector $λ$. We consider three Hurwitz series $σ({\bf u};λ)=\sum_{m,n\ge 0}a_{m,n}(λ)\frac{u_1^mu_3^n}{m!n!}$, $σ({\bf u};λ) = \sum_{k\ge 0}ξ_k(u_1;λ)\frac{u_3^k}{k!}$ and $σ({\bf u};λ) = \sum_{k\ge 0}μ_k(u_3;λ)\frac{u_1^k}{k!}$. The survey is devoted to the number-theoretic properties of the functions $a_{m,n}(λ)$, $ξ_k(u_1;λ)$ and $μ_k(u_3;λ)$. It includes the latest results, which proofs use the fundamental fact that the function ${σ({\bf u};λ)}$ is determined by the system of four heat equations in a nonholonomic frame of six-dimensional space.

math.AG↗

arXiv2019

Toric topology of the complex Grassmann manifolds

The family of the complex Grassmann manifolds $G_{n,k}$ with a canonical action of the torus $T^n=\mathbb{T}^{n}$ and the analogue of the moment map $μ: G_{n,k}\to Δ_{n,k}$ for the hypersimplex $Δ_{n,k}$, is well known. In this paper we study the structure of the orbit space $G_{n,k}/T^n$ by developing the methods of toric geometry and toric topology. We use a subdivision of $G_{n,k}$ into the strata $W_σ$ and determine all regular and singular points of the moment map $μ$, introduce the notion of the admissible polytopes $P_σ$ such that $μ(W_σ) = \stackrel{\circ}{P_σ}$ and the notion of the spaces of parameters $F_σ$, which together describe $W_σ/T^{n}$ as the product $\stackrel{\circ}{P_σ} \times F_σ$. To find the appropriate topology for the set $\cup _σ \stackrel{\circ}{P_σ} \times F_σ$ we introduce the notions of the universal space of parameters $\tilde{\mathcal{F}}$ and the virtual spaces of parameters $\tilde{F}_σ\subset \tilde{\mathcal{F}}$ such that there exist the projections $\tilde{F}_σ\to F_σ$. Hence, we propose a method for the description of the orbit space $G_{n,k}/T^n$. Earlier we proved that the orbit space $G_{4,2}/T^4$, defined by the canonical $T^4$-action of complexity $1$, is homeomorphic to $\partial Δ_{4,2}\ast \mathbb{C} P^1$. We prove here that the orbit space $G_{5,2}/T^5$, defined by the canonical $T^5$-action of complexity $2$, is homotopy equivalent to the space obtained by attaching the disc $D^8$ to the space $Σ^{4}\mathbb{R} P^2$ by the generator of the group $π_{7}(Σ^{4}\mathbb{R} P^2)=\mathbb{Z} _{4}$. In particular, $(G_{5,2}/G_{4,2})/T^5$ is homotopy equivalent to $\partial Δ_{5,2}\ast \mathbb{C} P^2$. The methods and the results of this paper are fundaments for our theory of $(2l,q)$-manifolds.

math.AT↗

arXiv2019

The foundations of $(2n,k)$-manifolds

In the focus of our paper is a system of axioms that serves as a basis for introducing structural data for $(2n,k)$-manifolds $M^{2n}$, where $M^{2n}$ is a smooth, compact $2n$-dimensional manifold with a smooth effective action of the $k$-dimensional torus $T^k$. In terms of these data a construction of the model space $\mathfrak{E}$ with an action of the torus $T^k$ is given, such that there exists a $T^k$-equivariant homeomorphism $\mathfrak{E}\to M^{2n}$. This homeomorphism induces a homeomorphism $\mathfrak{E}/T^k\to M^{2n}/T^k$. The number $d=n-k$ is called the complexity of an $(2n,k)$-manifold. Our theory comprises toric geometry and toric topology, where $d=0$. It is shown that the class of homogeneous spaces $G/H$ of compact Lie groups, where rk$G=$rk$H$, contains $(2n,k)$-manifolds that have non zero complexity. The results are demonstrated on the complex Grassmann manifolds $G_{k+1,q}$ with an effective action of the torus $T^k$.

math.AT↗

arXiv2017

On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{λ,μ,n}$ is a family of differential operators of order two. They depend on complex parameters $λ$, $μ$, $n$. Its restriction to real parameter domain $λ+μ^2>0$ is a linearization of the family of nonlinear equations on two-torus modeling the Josephson effect in superconductivity. We show that for every $b,n\in\mathbb C$ satisfying a certain "non-resonance condition" and every $λ,μ\in\mathbb C$, $μ\neq0$ there exists an entire function $f_{\pm}:\mathbb C\to\mathbb C$ (unique up to constant factor) such that $z^{-b}\mathcal L(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb C$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those $λ$, $μ$, $n$, $b$ for which the monodromy of the Heun equation has eigenvalue $e^{2πi b}$. It also describes those $λ$, $μ$, $n$ for which the monodromy is parabolic: has multiple eigenvalue. We consider the rotation number $ρ$ of the dynamical system on two-torus as a function of parameters restricted to a surface $λ+μ^2=const$. The phase-lock areas are its level sets having non-empty interiors. For general families of dynamical systems the problem to describe the boundaries of the phase-lock areas is known to be very complicated. Here we include the results in this direction obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). The result on parabolic monodromy implies the description of the union of their boundaries by an explicit functional equation. For every $θ\notin\mathbb Z$ we get a description of the set $\{ρ\equiv\pmθ(mod2\mathbb Z)\}$.

math.DS↗

arXiv2016

Finite sets of operations sufficient to construct any fullerene from $C_{20}$

We study the well-known problem of combinatorial classification of fullerenes. By a (mathematical) fullerene we mean a convex simple three dimensional polytope with all facets pentagons and hexagons. We analyse approaches of construction of arbitrary fullerene from the dodecahedron (a fullerene $C_{20}$). A growth operation is a combinatorial operation that substitutes the patch with more facets and the same boundary for the patch on the surface of a simple polytope to produce a new simple polytope. It is known that an infinite set of different growth operations transforming fullerenes into fullerenes is needed to construct any fullerene from the dodecahedron. We prove that if we allow a polytope to contain one exceptional facet, which is a quadrangle or a heptagon, then a finite set of growth operation is sufficient. We analyze pairs of objects: a finite set of operations, and a family of acceptable polytopes containing fullerenes such that any polytope of the family can be obtained from the dodecahedron by a sequence of operations from the corresponding set. We describe explicitly three such pairs. First two pairs contain seven operations, and the last -- eleven operations. Each of these operations corresponds to a finite set of growth operations and is a composition of edge- and two edges-truncations.

math.CO↗

arXiv2016

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.

math.DS↗