Quasitoric totally normally split representatives in unitary cobordism ring

Solomadin, Grigory

doi:10.48550/arxiv.1704.07403

Cited by 2 publications

(8 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recall that the bounded flag variety BF n is a non-singular projective toric variety of dimension n (see [6], [25]). Choose a basis e 0 , .…”

Section: Definitionsmentioning

confidence: 99%

“…In [25], the dualisation of β i ⊗β ′ j on the toric variety BF i ×BF j was given by explicit equation as a smooth complex hypersurface (divisor) in BF i × BF j . We denote such a hypersurface by R i,j and call it the Ray's hypersurface.…”

Section: Introductionmentioning

confidence: 99%

“…The importance of this question is due to the problem of representatives of the complex bordism ring Ω U * in the family of quasitoric TTS-and TNS-manifolds. This problem was completely solved in [25], by constructing a family of toric TTS-and TNS-manifolds whose bordism classes multiplicatively generate the ring Ω U * . However, this construction is rather involved.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

Solomadin¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we present a necessary condition for a non-singular projective variety with an effective algebraic torus action of positive complexity with isolated fixed points to be a toric variety. The method is based on the monodromy map on the weight hypergraph associated with the torus action. The introduced notion of a weight hypergraph generalises the GKM-graph in the case of a torus action satisfying all axioms from the definition of a GKM-manifold, except 2-linear independence of weights at fixed points. We apply this method to the generalised Buchstaber-Ray varieties BR i , j ⊂ BF i × P j , i , j 0, as well as the Ray's varieties R i , j ⊂ BF i × BF j and give a full list of toric varieties among them. The importance of the varieties R i , j is due to the problem of representatives in the unitary bordism ring in the family of quasitoric TTS-and TNS-manifolds. Also in this paper the automorphism group Aut H i , j is computed for the Milnor hypersurface H i , j ⊂ P i × P j . As a corollary, any effective action of (S 1 ) k on the Milnor hypersurface H i , j preserving the natural complex structure has k max {i , j }. * (0 i j ). However, BR i,j are not TNS-manifolds, generally speaking. In [7], the diamond sum operation was introduced, in order to replace a disjoint union of two quasitoric manifolds (in sense of [13]) by a complex bordant quasitoric manifold. As a corollary ([7]), in any class x ∈ Ω U 2n a quasitoric representative was constructed from the Buchstaber-Ray varieties, n > 1. It remained unknown whether R i,j are toric varieties. The importance of this question is due to the problem of representatives of the complex bordism ring Ω U * in the family of quasitoric TTS-and TNS-manifolds. This problem was completely solved in [25], by constructing a family of toric TTS-and TNS-manifolds whose bordism classes multiplicatively generate the ring Ω U * . However, this construction is rather involved. On the other hand, if R i,j would be toric varieties, then they form a family of toric TTS-and TNS-generators of Ω U * , which is simpler to describe. This provides a motivation to study torus actions on the aforementioned varieties.The Picard group Pic(BF i × BF j ) is isomorhic to H 2 (BF i × BF j ; Z) (see [21, p.127, §15.9]). It implies that the first Chern class c 1 (β i ⊗ β ′ j ) does not belong to the cone of effective classes (spanned by maximal torus-invariant effective divisors of the toric variety), hence it is not represented by any non-singular closed subvariety of codimension 1 in X (see [9]). This justifies our choice of the linear bundle β i ⊗ β ′ j on BF i × BF j . The choice of the particular variety R i,j among all possible dualisations of β i ⊗ β ′ j on BF i × BF j is due to its relation to the well-known MilnorThe publication has been prepared with the support of the "RUDN University Program 5-100" and by the grant of "Young mathematics of Russia" foundation.

show abstract

“…Recall that the bounded flag variety BF n is a non-singular projective toric variety of dimension n (see [6], [25]). Choose a basis e 0 , .…”

Section: Definitionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

Solomadin¹

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…TTS-and TNS-manifolds appeared in the works of Arthan and Bullet [1], Ochanine and Schwartz [12], Ray [16] and [18] related to a representation of a given complex cobordism class with a manifold from a prescribed family. A naturally arising problem here is to study TTS/TNS-manifolds in well-known families of manifolds, for example, quasitoric manifolds (see [8], [9]).…”

Section: Introductionmentioning

confidence: 99%

“…Let us remind some facts about quasitoric TNS-manifolds (see [18]). The complex projective space CP n is a TNS-manifold iff n < 2.…”

Section: Introductionmentioning

confidence: 99%

Quasitoric Totally Normally Split Manifolds

Solomadin

2018

Proc. Steklov Inst. Math.

View full text Add to dashboard Cite

A smooth stably complex manifold is called a totally tangentially/normally split manifold (TTS/TNSmanifold, for short, resp.) if the respective complex tangential/normal vector bundle is stably isomorphic to a Whitney sum of complex linear bundles, resp. In this paper we construct manifolds M s.t. any complex vector bundle over M is stably equivalent to a Whitney sum of complex linear bundles. A quasitoric manifold shares this property iff it is a TNS-manifold. We establish a new criterion of the TNS-property for a quasitoric manifold M via non-semidefiniteness of certain higher-degree forms in the respective cohomology ring of M . In the family of quasitoric manifolds, this generalises the theorem of J. Lannes about the signature of a simply connected stably complex TNS 4-manifold. We apply our criterion to show the flag property of the moment polytope for a nonsingular toric projective TNS-manifold of complex dimension 3.

show abstract

Quasitoric totally normally split representatives in unitary cobordism ring

Cited by 2 publications

References 0 publications

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

Cohomology rings and algebraic torus actions on hypersurfaces in the product of projective spaces and bounded flag varieties

Quasitoric Totally Normally Split Manifolds

Contact Info

Product

Resources

About