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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., Comment: 32 pages

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., 44 pages

Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details., 23 pages, LaTeX2e; minor changes

The short article [V.M. Buchstaber, E.Yu. Netay, $\mathbb{C}P(2)$-multiplicative Hirzebruch genera and elliptic cohomology, Russian Math. Surveys, 69:4 (2014), 757-759] states results on $\mathbb{C}P(2)$-multiplicative Hirzebruch genera. The aim of the following text is to give a proof of Theorem 3. This proof uses only the technique of functional differential equations., Comment: 4 pages

This paper is devoted to the one-dimensional heat equation and the non-linear ordinary differential equations associated to it. We consider homogeneous polynomial dynamical systems in the n-dimensional space, n = 0, 1, 2, .... For any such system our construction matches a non-linear ordinary differential equation. We describe the algorithm that brings the solution of such an equation to a solution of the heat equation. The classical fundamental solution of the heat equation corresponds to the case n=0 in terms of our construction. Solutions of the heat equation defined by the elliptic theta-function lead to the Chazy-3 equation and correspond to the case n=2. The group SL(2, C) acts on the space of solutions of the heat equation. We show this action for each n induces the action of SL(2, C) on the space of solutions of the corresponding ordinary differential equations. In the case n=2 this leads to the well-known action of this group on the space of solutions of the Chazy-3 equation. An explicit description of the family of ordinary differential equations arising in our approach is given., Comment: 20 pages

We consider the geometrical addition law on the elliptic curve in Tate coordinates. It corresponds to the general formal group law over the ring of polynomials with integer coefficients of the parametra of the curve. We study the structure of this law and the differential equation that determines its exponent. We describe a 5-parametric family of Hirzebruch genera with integer values on stably complex manifolds. We introduce the general Krichever genus, which is given by a generalized Baker-Akhiezer function. This function has many of the fundamental properties of the Baker-Akhiezer function, but unlike it, it is not meromorphic, because it can have two branch points in the parallelogram of periods., Comment: 39 pages

In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product $\times $ and a join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\mathcal{RP}$ has the structure of a graded Hopf algebra. It turns out that $\mathcal{P}$ has a natural Hopf comodule structure over $\mathcal{RP}$. Faces operators $d_k$ that send a polytope to the sum of all its $(n-k)$-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra $\mathcal{Z}$. This structure gives a ring homomorphism $\R\to\Qs\otimes\R$, where $\R$ is $\mathcal{P}$ or $\mathcal{RP}$. Composing this homomorphism with the characters $P^n\to\alpha^n$ of $\mathcal{P}$, $P^n\to\alpha^{n+1}$ of $\mathcal{RP}$, and with the counit we obtain the ring homomorphisms $f\colon\mathcal{P}\to\Qs[\alpha]$, $f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha]$, and $\F^*:\mathcal{RP}\to\Qs$, where $F$ is the Ehrenborg transformation. We describe the images of these homomorphisms in terms of functional equations, prove that these images are rings of polynomials over $\mathbb Q$, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism $f,\;f_{\mathcal{RP}}$, and $\F$ the images of two polytopes coincide if and only if they have equal flag $f$-vectors. Therefore algebraic structures on the images give the information about flag $f$-vectors of polytopes., Comment: 61 pages

Our primary aim is to develop a theory of equivariant genera for stably complex manifolds equipped with compatible actions of a torus T^k. In the case of omnioriented quasitoric manifolds, we present computations that depend only on their defining combinatorial data; these draw inspiration from analogous calculations in toric geometry, which seek to express arithmetic, elliptic, and associated genera of toric varieties in terms only of their fans. Our theory focuses on the universal toric genus \Phi, which was introduced independently by Krichever and Loeffler in 1974, albeit from radically different viewpoints. In fact \Phi is a version of tom Dieck's bundling transformation of 1970, defined on T^k-equivariant complex cobordism classes and taking values in the complex cobordism algebra of the classifying space. We proceed by combining the analytic, the formal group theoretic, and the homotopical approaches to genera, and refer to the index theoretic approach as a recurring source of insight and motivation. The resultant flexibility allows us to identify several distinct genera within our framework, and to introduce parametrised versions that apply to bundles equipped with a stably complex structure on the tangents along their fibres. In the presence of isolated fixed points, we obtain universal localisation formulae, whose applications include the identification of Krichever's generalised elliptic genus as universal amongst genera that are rigid on SU-manifolds. We follow the traditions of toric geometry by working with a variety of illustrative examples wherever possible. For background and prerequisites we attempt to reconcile the literature of east and west, which developed independently for several decades after the 1960s., Comment: 35 pages, LaTeX. In v2 references made to the index theoretical approach to genera; rigidity and multiplicativity results improved; acknowledgements added

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the complex, almost complex and stable complex structures which are invariant under the canonical left action of the maximal torus T^k on G/H. As it is known, on G/H we may have many such structures and the computations of their toric genus in terms of fixed points for the same torus action give the constraints on possible collections of weights for the corresponding representations of T^k in the tangent spaces at the fixed points, as well as on the signs at these points. In that context, the effectiveness is also approached due to an explicit description of the relations between the weights and signs for an arbitrary couple of such structures. Special attention is devoted to the structures which are invariant under the canonical action of the group G. Using classical results, we obtain an explicit description of the weights and signs in this case. We consequently obtain an expression for the cobordism classes of such structures in terms of coefficients of the formal group law in cobordisms, as well as in terms of Chern numbers in cohomology. These computations require no information on the cohomology ring of the manifold G/H, but, on their own, give important relations in this ring. As an application we provide an explicit formula for the cobordism classes and characteristic numbers of the flag manifolds U(n)/T^n, Grassmann manifolds G_{n,k}=U(n)/(U(k)\times U(n-k)) and some particular interesting examples., Comment: 30 pages; this is the extension of arXiv:0708.2054v2 [math.AT] with an application to some new problems

We prove that the integral cohomology algebra of the moment-angle complex Z_K, or of the corresponding coordinate subspace arrangement complement U(K), is isomorphic to the Tor-algebra of the face ring Z[K] of simplicial complex K., Comment: 3 pages

Let $\rho:(D^2)^m\to I^m$ be the orbit map for the diagonal action of the torus $T^m$ on the unit poly-disk $(D^2)^m$, $I^m=[0,1]^m$ is the unit cube. Let $C$ be a cubical subcomplex in $I^m$. The moment-angle complex $\ma(C)$ is a $T^m$-invariant bigraded cellular decomposition of the subset $\rho^{-1}(C)\subset(D^2)^m$ with cells corresponding to the faces of $C$. Different combinatorial problems concerning cubical complexes and related combinatorial objects can be treated by studying the equivariant topology of corresponding moment-angle complexes. Here we consider moment-angle complexes defined by canonical cubical subdivisions of simplicial complexes. We describe relations between the combinatorics of simplicial complexes and the bigraded cohomology of corresponding moment-angle complexes. In the case when the simplicial complex is a simplicial manifold the corresponding moment-angle complex has an orbit consisting of singular points. The complement of an invariant neighbourhood of this orbit is a manifold with boundary. The relative Poincare duality for this manifold implies the generalized Dehn-Sommerville equations for the number of faces of simplicial manifolds., Comment: 28 pages, LaTeX2e, extended version of the paper published in Russian Math. Surveys 55 (2000), no. 3