Chern-Dold character in complex cobordisms and theta divisors

Buchstaber, V. M.; Veselov, A. P.

doi:10.48550/arxiv.2007.05782

Cited by 3 publications

(4 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, G 6,2 is of particular importance in algebraic geometry being one of the six Severi varieties, [21]. For n = 6 we start with the manifold together with the point S = (1 : 1) 6 . At this point any of these two subvarieties intersect.…”

Section: The Main Resultsmentioning

confidence: 99%

“…It is well known fact that M(0, 5) is the del Pezzo surface of degree 5, while algebro-geometric characterization of the space M(0, n) for n > 5 is an open well known problem. In the paper [6] it has been shown that the problem of algebro-topological characterization of these spaces is related to well known problems of complex cobordisms theory.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Buchstaber,

Terzić

2021

Preprint

View full text Add to dashboard Cite

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 wellknown paper Kapranov has proved that the Deligne-Mumford compactification M(0, n) of n-pointed curves of genus zero can be realized as the Chow quotient G n,2 //(C * ) 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 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 F n . Together with Keel's description of M(0, n), this construction enabled us to obtain an explicit diffeomorphism between F n and M(0, n). Thus, we showed that the space G n,2 //(C * ) n can be realized as our universal space of parameters F n . In this way, we give description of the structure in G n,2 //(C * ) n , that is M(0, n) in terms of the CW-complex of the admissible polytopes for G n,2 and their spaces of parameters. 1

show abstract

Section: The Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Buchstaber,

Terzić

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The cohomology rings of the spaces M(0, n) were described in [22] (see also [21]). Complex cobordism classes of the spaces M(0, n) are important for applications of the Chern character to complex cobordism theory; see [6] and [4].…”

mentioning

confidence: 99%

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

Buchstaber,

Terzic

2023

Sb. Math.

View full text Add to dashboard Cite

The complex Grassmann manifolds $G_{n,k}$ appear as one of the fundamental objects in developing an interaction between algebraic geometry and algebraic topology. The case $k=2$ is of special interest on its own as the manifolds $G_{n,2}$ have several remarkable properties which distinguish them from the $G_{n,k}$ for $k>2$. In our paper we obtain results which, essentially using the specifics of the Grassmann manifolds $G_{n,2}$, develop connections between algebraic geometry and equivariant topology. They are related to well-known problems of the canonical action of the algebraic torus $(\mathbb{C}^{\ast})^n$ on $G_{n,2}$ and the induced action of the compact torus $T^n\subset(\mathbb{C}^{\ast})^n$. Kapranov proved that the Deligne-Mumford-Grothendieck-Knudsen compactification $\overline{\mathcal{M}}(0,n)$ of the space of $n$-pointed rational stable curves can be realized as the Chow quotient $G_{n,2}//(\mathbb{C}^{\ast})^{n}$. In recent papers of the authors a constructive description of the orbit space $G_{n,2}/T^n$ was obtained. In deducing this result the notions of the complex of admissible polytopes and the universal space of parameters $\mathcal{F}_{n}$ for the $T^n$-action on $G_{n,2}$ were of essential use. Using the techniques of wonderful compactification, in this paper an explicit construction of the space $\mathcal{F}_{n}$ is presented. In combination with Keel's description of $\overline{\mathcal{M}}(0,n)$, this construction enabled one to obtain an explicit diffeomorphism between $\mathcal{F}_{n}$ and $\overline{\mathcal{M}}(0,n)$. In this way, we give a description of $G_{n,2}//(\mathbb{C}^{\ast})^{n}$ as the space $\mathcal{F}_{n}$ with a structure described in terms of admissible polytopes $P_\sigma$ and spaces $F_\sigma$. Bibliography: 32 titles.

show abstract

“…We are as yet unsure of the relationship between the quantum deformation of cobordism-theory obtained by incorporating our invariants into the product and the deformation recently introduced by Buchstaber-Veselov in[1].…”

mentioning

confidence: 99%

Multiplicative Quantum Cobordism Theory

Huq-Kuruvilla

2021

Preprint

View full text Add to dashboard Cite

We prove a twisting theorem for nodal classes in permutation-equivariant quantum K-theory, and combine it with existing theorems of Givental [5] to obtain a twisting a theorem for general characteristic classes of the virtual tangent bundle. Using this result, we develop complex cobordismvalued Gromov-Witten invariants defined via K-theory, and relate those invariants to K-theoretic ones via the quantization of suitable symplectic transformations. This procedure is a K-theoretic analogue of the quantum cobordism theory developed by Givental and Coates in [3]. Using the universality of cobordism theory, we give an example of these results in the context of "Hirzebruch K-theory", which is the cohomology theory determined by the Hirzebruch χ−y-genus.

show abstract

Chern-Dold character in complex cobordisms and theta divisors

Cited by 3 publications

References 10 publications

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

Multiplicative Quantum Cobordism Theory

Contact Info

Product

Resources

About