Bott Towers Research Articles

Strong cohomological rigidity of Hirzebruch surface bundles in Bott towers

We show the strong cohomological rigidity of Hirzebruch surface bundles over Bott manifolds. As a corollary, we have that the strong cohomological rigidity conjecture is true for Bott manifolds of dimension 8.

Maximal log Fano manifolds are generalized Bott towers

We prove that maximal log Fano manifolds are generalized Bott towers. As an application, we prove that in each dimension, there is a unique maximal snc Fano variety satisfying Friedman's d-semistability condition.

Seshadri constants on Bott towers

For a positive integer n, let Xn→Xn−1→…→X2→X1→X0 be a Bott tower of height n, and let L be a nef line bundle on Xn. We compute Seshadri constants ε(Xn,L,x) of L at any point x∈Xn under some conditions.

Seshadri constants of equivariant vector bundles on toric varieties

We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.

Stability of Equivariant Vector Bundles over Toric Varieties

We give a complete answer to the question of (semi)stability of tangent bundles on any nonsingular complex projective toric variety with Picard number 2 by using combinatorial criterion of (semi)stability of an equivariant sheaf. We also give a complete answer to the question of (semi)stability of tangent bundles on all toric Fano 4-folds with Picard number \leq 3 which are classified by Batyrev [1]. We construct a collection of equivariant indecomposable rank 2 vector bundles on Bott towers and pseudo-symmetric toric Fano varieties. Furthermore, in case of Bott towers, we show the existence of an equivariant stable rank 2 vector bundle with certain Chern classes with respect to a suitable polarization.

Toric vector bundles on Bott tower

In this paper, using Klyachko's classification theorem we study positivity and semi-stability of toric vector bundles on a class of nonsingular projective toric varieties, known as Bott towers. In particular, we give a criterion of s-jet ampleness of line bundles and characterize nef and big line bundles on Bott towers using Bott numbers. We obtain a criterion for the ampleness of discriminant zero semi-stable toric vector bundles on nonsingular projective toric varieties. We also describe toric subbundles of toric vector bundles on nonsingular toric varieties.

On Mori cone of Bott towers

A Bott tower of height r is a sequence of projective bundlesXr⟶πrXr−1⟶πr−1⋯⟶π2X1=P1⟶π1X0={pt}, where Xi=P(OXi−1⊕Li−1) for a line bundle Li−1 over Xi−1 for all 1≤i≤r and P(−) denotes the projectivization. These are smooth projective toric varieties and we refer to the top object Xr also as a Bott tower. In this article, we study the Mori cone and numerically effective (nef) cone of Bott towers, and we classify Fano, weak Fano and log Fano Bott towers. We prove some vanishing theorems for the cohomology of tangent bundle of Bott towers.

ON A GENERALIZATION OF HIRZEBRUCH'S THEOREM TO BOTT TOWERS

The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds Bn(�1,...,�n−1,�n) and Bn(�1,...,�n−1,�') are isomorphic to each other, as Bott towers if and only if bothn � � ' mod 2 and � 2 = (� ') 2 hold in the cohomology ring of Bn−1(�1,...,�n−1) over integer coefficients. This result will complete a circle of ideas initiated in (11) by Ishida. We also give some partial affirmative remarks toward the assertion that under certain c our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.

Classification of Bott towers by matrix

A criterion for the classification of Bott towers is presented, i.e., two Bott towers B*(A) and B*(A′) are isomorphic if and only if the matrices A and A′ are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B*(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.

Filtered cohomological rigidity of bott towers

A Bott tower is an iterated CP-bundle over a point, where each CP-bundle is the projectivization of a rank 2 decomposable complex vector bundle. For a Bott tower, the filtered cohomology is naturally defined. We show that isomorphism classes of Bott towers are distinguished by their filtered cohomology rings. We even show that any filtered cohomology ring isomorphism between two Bott towers is induced by an isomorphism of the Bott towers.

Topological classification of generalized Bott towers

If B is a toric manifold and E is a Whitney sum of complex line bundles over B, then the projectivization P(E) of E is again a toric manifold. Starting with B as a point and repeating this construction, we obtain a sequence of complex projective bundles which we call a generalized Bott tower. We prove that if the top manifold in the tower has the same cohomology ring as a product of complex projective spaces, then every fibration in the tower is trivial so that the top manifold is diffeomorphic to the product of complex projective spaces. This gives supporting evidence to what we call the cohomological rigidity problem for toric manifolds, Are toric manifolds diffeomorphic (or homeomorphic) if their cohomology rings are isomorphic? We provide two more results which support the cohomological rigidity problem.

Semifree circle actions, Bott towers and quasitoric manifolds

A Bott tower is the total space of a tower of fibre bundles with base and fibres . Every Bott tower of height n is a smooth projective toric variety whose moment polytope is combinatorially equivalent to an n-cube. A circle action is semifree if it is free on the complement to the fixed points. We show that a quasitoric manifold over a combinatorial n-cube admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is a Bott tower. Then we show that every Bott tower obtained in this way is topologically trivial, that is, homeomorphic to a product of 2-spheres. This extends a recent result of Il'inskiĭ, who showed that a smooth compact toric variety admitting a semifree action of a 1-dimensional subtorus with isolated fixed points is homeomorphic to a product of 2-spheres, and makes a further step towards our understanding of Hattori's problem of semifree circle actions. Finally, we show that if the cohomology ring of a quasitoric manifold is isomorphic to that of a product of 2-spheres, then the manifold is homeomorphic to this product. In the case of Bott towers the homeomorphism is actually a diffeomorphism. Bibliography: 18 titles.

K-théorie équivariante des tours de Bott. Application à la structure multiplicative de la K-théorie équivariante des variétés de drapeaux

(Equivariant K-theory of Bott towers. Application to the multiplicative structure of the equivariant K-theory of flag varieties) We construct a basis of the equivariant K-theory of Bott towers, and we describe precisely the multiplicative structure of these algebras. We deduce similar results for Bott-Samelson varieties. Thanks to the link between flag varieties and Bott-Samelson varieties, we give a method to compute the structure constants of the equivariant K-theory of flag varieties in the basis constructed by Kostant and Kumar in [18]

Bott Towers, Crosspolytopes and Torus Actions

We study the geometry and topology of Bott towers in the context of toric geometry. We show that any kth stage of a Bott tower is a smooth projective toric variety associated to a fan arising from a crosspolytope; conversely, we prove that any toric variety associated to a fan obtained from a crosspolytope actually gives rise to a Bott tower. The former leads us to a description of the tangent bundle of the kth stage of the tower, considered as a complex manifold, which splits into a sum of complex line bundles. Applying Danilov–Jurkiewicz theorem, we compute the cohomology ring of any kth stage, and by way of construction, we provide all the monomial identities defining the related affine toric varieties.

Homotopy Decompositions and K-Theory of Bott Towers

We describe Bott towers as sequences of toric manifolds $M^k$, and identify the omniorientations which correspond to their original construction as complex varieties. We show that the suspension of M^k is homotopy equivalent to a wedge of Thom complexes, and display its complex K-theory as an algebra over the coefficient ring. We extend the results to KO-theory for several families of examples, and compute the effects of the realification homomorphism; these calculations breathe geometric life into Bahri and Bendersky's analysis of the Adams Spectral Sequence. By way of application we consider the enumeration of stably complex structures on M^k, obtaining estimates for those which arise from omniorientations and those which are almost complex. We conclude with observations on the role of Bott towers in complex cobordism theory.

Bott towers, complete integrability, and the extended character of representations

Bott towers, complete integrability, and the extended character of representations