Torus Bundles Research Articles

The Borsuk–Ulam property for homotopy classes on bundles, parametrized braids groups and applications for surfaces bundles

Let M and N be fiber bundles over the same base B, where M is endowed with a free involution τ over B. A homotopy class δ∈[M,N]B (over B) is said to have the Borsuk–Ulam property with respect to τ if for every fiber-preserving map f:M→N over B which represents δ there exists a point x∈M such that f(τ(x))=f(x). In the cases that B is a K(π,1)-space and the fibers of the projections M→B and N→B are K(π,1) closed surfaces SM and SN, respectively, we show that the problem of decide if a homotopy class of a fiber-preserving map f:M→N over B has the Borsuk-Ulam property is equivalent of an algebraic problem involving the fundamental groups of M, the orbit space of M by τ and a type of generalized braid groups of N that we call parametrized braid groups. As an application, we determine the homotopy classes of fiber-preserving self maps over S1 that satisfy the Borsuk-Ulam property, with respect to all involutions τ over S1, for the torus bundles over S1 with M=N=MA and A=[1n01].

Extremal Kähler-Ricci solitons on Fano homogeneous toric bundles

In this short note, we prove that an extremal Kähler-Ricci soliton on a Fano homogeneous toric bundle is Kähler-Einstein.

A new approach to topological T-duality for principal torus bundles

In this paper, we introduce a new ‘Thom class’ formulation of topological T-duality for principal torus bundles. This definition is equivalent to the established one of Bunke, Rumpf, and Schick but has the virtue of removing the global assumptions on the H-flux required in the old definition. With the new definition, we provide easier and more transparent proofs of the classification of T-duals and generalize the local formulation of T-duality for circle bundles by Bunke, Schick, and Spitzweck to the torus case.

Toric principal bundles, Tits buildings and reduction of structure group

A toric principal G -bundle is a principal G -bundle over a toric variety together with a torus action commuting with the G -action. Recently, extending the Klyachko classification of toric vector bundles, Kaveh–Manon give a classification of toric principal G -bundles using piecewise linear maps to the (extended) Tits building of G . In this paper, we use the Kaveh–Manon classification to give a description of the (equivariant) automorphism group of a toric principal bundle as well as a simple criterion for (equivariant) reduction of structure group, recovering results of Dasgupta et al. Finally, motivated by the equivariant splitting problem for toric principal bundles, we introduce the notion of Helly’s number of a building and pose the problem of giving sharp upper bounds for Helly’s numbers of Tits buildings of semisimple algebraic groups G .

Algebra and geometry of irreducible toric vector bundles of rank n on [formula omitted]

We construct a presentation for the Cox ring of the projectivization PE of any rank n irreducible toric vector bundle on Pn. We use this presentation to show that PE always satisfies Fujita's freeness and ampleness conjectures.

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

On arithmetic quotients of the group SL2 over a quaternion division k-algebra

Abstract Given a totally real algebraic number field k of degree s, we consider locally symmetric spaces X G / Γ {X_{G}/\Gamma} associated with arithmetic subgroups Γ of the special linear algebraic k-group G = SL M 2 ⁢ ( D ) {G=\mathrm{SL}_{M_{2}(D)}} , attached to a quaternion division k-algebra D. The group G is k-simple, of k-rank one, and non-split over k. Using reduction theory, one can construct an open subset Y Γ ⊂ X G / Γ {Y_{\Gamma}\subset X_{G}/\Gamma} such that its closure Y ¯ Γ {\overline{Y}_{\Gamma}} is a compact manifold with boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} , and the inclusion Y ¯ Γ → X G / Γ {\overline{Y}_{\Gamma}\rightarrow X_{G}/\Gamma} is a homotopy equivalence. The connected components Y [ P ] {Y^{[P]}} of the boundary ∂ ⁡ Y ¯ Γ {\partial\overline{Y}_{\Gamma}} are in one-to-one correspondence with the finite set of Γ-conjugacy classes of minimal parabolic k-subgroups of G. We show that each boundary component carries the natural structure of a torus bundle. Firstly, if the quaternion division k-algebra D is totally definite, that is, D ramifies at all archimedean places of k, we prove that the basis of this bundle is homeomorphic to the torus T s - 1 {T^{s-1}} of dimension s - 1 {s-1} , has the compact fibre T 4 ⁢ s {T^{4s}} , and its structure group is SL 4 ⁢ s ⁢ ( ℤ ) {\mathrm{SL}_{4s}(\mathbb{Z})} . We determine the cohomology of Y [ P ] {Y^{[P]}} . Secondly, if the quaternion division k-algebra D is indefinite, thus, there exists at least one archimedean place v ∈ V k , ∞ {v\in V_{k,\infty}} at which D v {D_{v}} splits over ℝ {\mathbb{R}} , that is, D v ≅ M 2 ⁢ ( ℝ ) {D_{v}\cong M_{2}(\mathbb{R})} , the fibre is homeomorphic to T 4 ⁢ s {T^{4s}} , but the base space of the bundle is more complicated.

Flat F-theory and friends

We discuss F-theory backgrounds associated to flat torus bundles over Ricci-flat manifolds. In this setting the F-theory background can be understood as a IIB orientifold with a large radius limit described by a supersymmetric compactification of IIB supergravity on a smooth, Ricci flat, but in general non-spin geometry. When compactified on an additional circle these backgrounds are T-dual to IIA compactifications on smooth non-orientable manifolds with a Pin− structure.

Virasoro Constraints for Toric Bundles

Abstract We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle $E \to B$ if and only if it holds for the base B. The main steps are: (i) We establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action in terms of genus-zero invariants of the toric fiber and all-genus invariants of B, and (ii) we pass to the nonequivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

Convergence and analytic decomposition of quantum cohomology of toric bundles

We prove that the equivariant big quantum cohomology QHT⁎(E) of the total space E of a toric bundle E→B converges provided that the big quantum cohomology QH⁎(B) converges. The proof is based on Brown's mirror theorem for toric bundles [5]. It has been observed by Coates, Givental and Tseng that the quantum connection of E splits into copies of that of B[10]. Under the assumption that QH⁎(B) is convergent, we construct a decomposition of the quantum D-module of E into a direct sum of that of B, which is analytic with respect to parameters of QHT⁎(E). In particular, we obtain an analytic decomposition for the equivariant/non-equivariant big quantum cohomology of E.

Homology stability for asymptotic monopole moduli spaces

We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles . The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. Our results follow from a general homological stability result for configuration spaces equipped with non-local data.

Classification of torus bundles that bound rational homology circles

Classification of torus bundles that bound rational homology circles

Cohomology Rings of Toric Bundles and the Ring of Conditions

The celebrated BKK Theorem expresses the number of roots of a system of generic Laurent polynomials in terms of the mixed volume of the corresponding system of Newton polytopes. In Pukhlikov and Khovanskiĭ (Algebra i Analiz 4(4):188–216, 1992), Pukhlikov and the second author noticed that the cohomology ring of smooth projective toric varieties over C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {C}}$$\\end{document} can be computed via the BKK Theorem. This complemented the known descriptions of the cohomology ring of toric varieties, like the one in terms of Stanley–Reisner algebras. In Sankaran and Uma (Comment Math Helv 78(3):540–554, 2003), Sankaran and Uma generalized the “Stanley–Reisner description” to the case of toric bundles, i.e., equivariant compactifications of (not necessarily algebraic) torus principal bundles. We provide a description of the cohomology ring of toric bundles which is based on a generalization of the BKK Theorem, and thus extends the approach by Pukhlikov and the second author. Indeed, for every cohomology class of the base of the toric bundle, we obtain a BKK-type theorem. Furthermore, our proof relies on a description of graded-commutative algebras which satisfy Poincaré duality. From this computation of the cohomology ring of toric bundles, we obtain a description of the ring of conditions of horospherical homogeneous spaces as well as a version of Brion–Kazarnovskii theorem for them. We conclude the manuscript with a number of examples. In particular, we apply our results to toric bundles over a full flag variety G/B. The description that we get generalizes the corresponding description of the cohomology ring of toric varieties as well as the one of full flag varieties G/B previously obtained by Kaveh (J Lie Theory 21(2):263–283, 2011).

Solvability of a class of fully nonlinear elliptic equations on tori

We study the solvability of a class of fully nonlinear equations on the flat torus. The equations arise in the study of some Calabi–Yau type problems in torus bundles.

Cox rings of projectivized toric vector bundles and toric flag bundles

Work of González, Hering, Payne, and Süß shows that it is possible to find both examples and non-examples of Mori dream spaces among projectivized toric vector bundles. This result, and the combinatorial nature of the data of projectivized toric vector bundles make them an ideal test class for the question: what makes a variety a Mori dream space? In the present paper we consider this question with respect to natural algebraic operations on toric vector bundles.Suppose E is a toric vector bundle such that the projectivization PE is a Mori dream space, then when are the direct sum toric vector bundles P(E⊕E), P(E⊕E⊕E)… also Mori dream spaces? We give an answer to this question utilizing a relationship with the associated full flag bundle FL(E). We describe several classes of examples, and we compute a presentation for the Cox ring of the full flag bundle for the tangent bundle of projective space.

Extremal metrics on toric manifolds and homogeneous toric bundles

Abstract In this paper, we prove the Yau–Tian–Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles, thus for toric manifolds and homogeneous toric bundles, M admits an extremal metric in c 1 ⁢ ( L ) {c_{1}(L)} if and only if ( M , L ) {(M,L)} is relatively K ^ {\widehat{K}} -stable.

On cusps of hyperbolic once-punctured torus bundles over the circle

The geometry of certain canonical triangulation of once-punctured torus bundles over the circle is applied to the problem of computing their cusp tori. We are also concerned with the problem of finding the limit points of the set formed by such cusp tori, inside the moduli space of the torus. Our discussion generalizes examples which were elaborated by H. Helling (unpublished) and F. Guéritaud.

Extensions of toric line bundles

Extensions of toric line bundles

Type IIB parabolic (p, q)-strings from M2-branes with fluxes

We extend the work of Schwarz [1] to show that bound states of type IIB supersymmetric (p, q)-strings on a circle are associated with M2-branes irreducibly wrapped on T2, or equivalently with nontrivial worldvolume fluxes. Beyond this extension we consider the Hamiltonian of an M2-brane with C± fluxes formulated on a symplectic torus bundle with monodromy. In particular, we analyze the relevant case when the monodromy is parabolic. We show that the Hamiltonian is defined in terms of the coinvariant module. We also find that the mass operator is invariant under transformations between inequivalent coinvariants. These coinvariants classify the inequivalent classes of twisted torus bundles with nontrivial monodromy for a given flux. We obtain their associated (p, q)-strings via double dimensional reduction, which are invariant under a parabolic subgroup of SL(2, ℚ). This is the origin of the gauge symmetry of the associated gauged supergravity. These bound states could also be related to the parabolic Scherk-Schwarz reductions of type IIB string theory.

Structural stability for fibrewise Anosov diffeomorphisms on principal torus bundles

AbstractWe show that a fibre-preserving self-diffeomorphism which has hyperbolic splittings along the fibres on a compact principal torus bundle is topologically conjugate to a map that is linear in the fibres.