Computing persistent Stiefel–Whitney classes of line bundles

Let X be a Hopf manifold with non-Abelian fundamental group and E be a holomorphic vector bundle over X, with trivial pull-back to ℂn − {0}. The authors show that there exists a line bundle L over X such that E ⊗ L has a nowhere vanishing section. It is proved that in case dim(X) ≥ 3, π*(E) is trivial if and only if E is filtrable by vector bundles. With the structure theorem, the authors get the cohomology dimension of holomorphic bundle E over X with trivial pull-back and the vanishing of Chern class of E.

Let {Xt ↪→ Y | t ∈ M} be a family of compact complex submanifolds Xt of a complex manifold Y parametrised by a complex manifold M . In this paper we study vector bundles and linear connections on the moduli space M induced from holomorphic vector bundles on the space Y . The origin of this approach lies in the Ward transform [W] of holomorphic vector bundles E on Y which are trivial when restricted to each submanifoldXt of the family {Xt ↪→ Y | t ∈M}. We consider a more general class of holomorphic vector bundles E on twistor spaces Y which, when restricted to a submanifold Xt of the family, have the same integer h(Xt, E|Xt) for all t ∈M . With such a vector bundle E on Y there is an associated vector bundle E on M whose fibre at t ∈ M is isomorphic to H(Xt, E|Xt). It is shown that provided the cohomology groups H ( Xt, E|Xt ⊗N ∗ t ) and H ( Xt, E|Xt ⊗N ∗ t ) vanish (where Nt is the normal bundle of Xt ↪→ Y ), the vector bundle E induced on the parameter space M comes equipped with a distinguished linear connection ∇ satisfying some natural integrability conditions (cf. [L, E, B-E]). The curvature tensor of ∇ is represented, at each t ∈ M , by a cohomology class in H ( Xt, E|Xt ⊗ 2N∗ ) ⊗ ( H(Xt, E|Xt) )∗ . This theorem-construction gives a very simple way to estimate the curvature tensor of an induced connection directly from the original twistor data. As an application, we consider a pair X ↪→ Y consisting of a complex contact 3-fold (Y, L) and a Legendre submanifold X = CP such that the contact line bundle L restricts to X as O(3). Then the moduli space M of all holomorphic Legendre deformations of X inside Y is a 4-dimensional manifold which comes equipped with an induced 1-flat G3structure [Br2], and any such structure arises locally in this way. Suppose that there exists a line bundle E → Y such that L = E⊗3 (such an E always exists on a sufficiently small tubular neighbourhood of X in Y ). Since the normal bundle of X is isomorphic to JO(3) = C ⊗ O(2), the above cohomology conditions on E|Xt ⊗ N ∗ t are satisfied. Then the induced rank 2 vector bundle E comes equipped with a distinguished linear connection which, as easily follows from its integrability properties, is a “spinor” version of the torsion-free connection on M with holonomy in G3. In fact, any torsion-free connection with holonomy in G3 can be constructed, at least locally, in this way (cf. [Me1]). The point is that the way we prove the above mentioned theorem-construction gives a rather

Algebraic Vector Bundles

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

The object of this work is to define, by analogy with algebra, the Witt group and the graded Brauer group of a topological space X.A homomorphism is defined between them analogous to the generalized Hasse invariant.Upon evaluation, the Witt group is seen to be KO(X), the graded Brauer group 1+H1(X;ZZ) + H2(X; Z2) with truncated cup product multiplication, while the homomorphism is given by Stiefel-Whitney classes: I + m^ + h^.Introduction.The Hasse invariant of a quadratic form on a vector space over a field k is an element of the Brauer group of k.It can be generalized to a homomorphism from the Witt group of classes of quadratic forms to the graded Brauer group of classes of graded ^-algebras [3], [16].Curiously enough, in this guise it is quite analogous to the homomorphism given by the first two Stiefel-Whitney classes which maps real vector bundles over a space X to the group 1 + H\X; Z2) + H2(X;Z2), where multiplication is by truncated cup product.The purpose of this paper is to display this analogy.Delzant has defined Stiefel-Whitney classes of quadratic forms [4], [13] using the discriminant and Hasse invariant.It should add insight into his definition to see that the first two topological Stiefel-Whitney classes can also be defined in an algebraic way.A vector bundle L is known to be orientable for real L-theory if its first two Stiefel-Whitney classes vanish.In this case it can be read off from the diagram of Theorem 3.1 that the Clifford bundle of L is a bundle of graded endomorphism algebras of a graded module bundle, which is then used to construct the Thorn class of L. Donovan and Karoubi in studying this orientability question have done independently much of this same work, applying it further to define L-theory with local coefficients [6], [7].The connection between algebra and topology comes from considering bundles over X whose fibers are vector spaces with quadratic forms or graded algebras and subjecting them to the same analysis the fiber alone is accorded in algebra.This is described in some detail in 1.11 is devoted to the algebraic computations and 111 to a study of the topological problems.

A parameterization of the set of all stabilizing state feedback gains is given, and a geometric structure of the set of stable state feedback systems in the set of stable matrices is used. It is shown that it is natural to provide these sets with a vector bundle structure. Obtained results not only provide fundamental guidelines for designing state feedback gains using parameterization, but also a new approach to analyzing structures of linear stable systems. >

In this work we deal with vector bundles of rank two on a Fano manifold $X$ with $b_2=b_4=1$. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle. As consequences, we obtain the complete list of $\mathbb{P}^1$-bundles over $X$ that have a second $\mathbb{P}^1$-bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds and show the stability of indecomposable Fano bundles (with one exception on $\mathbb{P}^2$).

On the vector bundles over rationally connected varieties

A compactification of the moduli space of principal Higgs bundles over singular curves

The purpose of this paper is to carry out the abelianization program pro posed by Atiyah [1] and Hitchin [9] for the geometric quantization of SU(2) Wess-Zumino-Witten model. Let C be a Riemann surface of genus g. Let Mg be the moduli space of semi-stable rank 2 holomorphic vector bundles on C with trivial determinant. For a positive integer k, let T(Mg, Ck) be the space of holomorphic sections of the fc-th tensor product of the determinant line bundle C on Mg. An element of T(Mg, Ck) is called a rank 2 theta function of level k. The main result of our abelianization is to give an explicit representation of a base of T{Mg, Ck) as well as its transformation formula in terms of classical Riemann theta functions with automorphic form coefficients defined on the Prym variety P associated with a two-fold branched covering surface C of C. r(Mg,Ck) can be identified with the conformai block of level k of the SU(2) WZW model ([5], [15]). The abelianization procedure enables us to de duce the various known results about the conformai block in a uniform way. Firstly, we construct a projectively flat connection on the vector bundle over the Teichm?ller space with fibre T{Mg,Ck). Secondly, making use of our ex plicit representation of rank 2 theta functions we construct a Hermitian product on the vector bundle preserved by the connection. Also our explicit represen tation enables us to prove that Y(Mg,Ck) has the predicted dimension from the Quantum Clebsh-Gordan conditions. A natural connection on the said vector bundle for the SU(iV) WZW model was first constructed by Hitchin [11]. It will turn out that the connection constructed in this paper coincides with the Hitchin connection. Laszlo [16] showed that the Hitchin connection coincides with the con nection constructed by Tsuchiya, Ueno and Yamada [21] through the above identification. On the other hand Kirillov [13], [14] constructed a Hermitian product on the conformai block compatible with the Tsuchiya-Ueno-Yamada connection using the representation theory of affine Lie algebras together with the theory of hermitian modular tensor categories; cf. [22]. Laszlo's result

According to Horrocks (Quart. J. Math. Oxford (2) 17 (1966), 14–18), a vector bundle E on the projective n-space extends stably to the projective N-space, N > n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N > n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coandă and Trautmann (Commun. Algebra 34 (2006), 2485–2488). Combining this result with a theorem of Mohan Kumar et al. (Manuscripta Math. 112 (2003), 183–189), we get a new effective version of the Babylonian tower theorem for vector bundles on projective spaces.

The purpose of this paper is to study extendibility and stable extendibility of vector bundles over real projective spaces. We determine a necessary and sufficient condition that a vector bundle ζ over the real projective n-space RPn is extendible (or stably extendible) to RPm for every m>n in the case where ζ is the complexification of the tangent bundle of RPn and in the case where ζ is the normal bundle associated to an immersion of RPn in the Euclidean (n+k)-space Rn+k or its complexification, and give examples of the normal bundle which is extendible to RPN but is not stably extendible to RPN+1.

Spectrum of a Dynamical System and Applied Symbolic Dynamics

We investigate by embedding methods a class of nonholonomic and forced systems arising in particular from the rolling pneumatic tire. For these systems the existing embedding theorems of Takens-type are not applicable due to the nonholonomic constraints, the time-dependence of the forces, and the noncompactness of the phase space. To overcome these difficulties we interpret the given mechanical system as a vector field on a nontrivial fiber bundle with time-dependent Riemannian fiber metric. I. INTRODUCTION Our main goal is to interprete a nonautonomous and constrained system as a lifted system on a suitable fiber bundle with nontrivial bundle metric and to construct for this system a generic embedding. Let us recall for the precise characterization of the bundle structure some basic definitions (((5), (4)). A -fiber bundle is given by a surjective (i.e. a map onto) submersion , where and are -manifolds. The submersion is assumed to be locally trivial, i.e. there exists a manifold such that, for each point , there exists a neighborhood of and a -diffeomorphism with We say that is the bundle space, is the base space, is the projection, is a bundle chart and is the typical fiber. For any the set is the fiber over . A -vector bundle is a -fiber bundle whose fibers have a vector space structure.

We extend the Horrocks correspondence between vector bundles and cohomology modules on the projective plane to the product of two projective lines. We introduce a set of invariants for a vector bundle on the product of two projective lines, which includes the first cohomology module of the bundle, and prove that there is a one to one correspondence between these sets of invariants and isomorphism classes of vector bundles without line bundle summands.