Quaternionic Bundles Research Articles

Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero

The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.

Quaternionic algebraic cycles and reality

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group G a l ( C / R ) Gal({\mathbb C} / {\mathbb R}) . Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymour’s quaternionic K K -theory, and the other one classifies an equivariant cohomology theory Z ∗ ( − ) {\mathfrak Z}^*(-) which is a natural recipient of characteristic classes K H ∗ ( X ) → Z ∗ ( X ) KH^*(X) \to {\mathfrak Z}^*(X) for quaternionic bundles over Real spaces X X .

Quaternionic Bundles on Algebraic Spheres

Quaternionic Bundles on Algebraic Spheres

Group actions on quasi-symplectic manifolds

The purpose of this paper is to prove a vanishing theorem for characteristic numbers of quasi-symplectic manifolds, which admit a differentiable structure preserving action of the circle group S ~. Here a dosed differentiable manifold is called a quasi-symplectic manifold when the tangent bundle is a direct sum of quaternionic tensor products. Examples of this class of manifolds are the quaternionic flag manifolds, and particularly the quaternionic projective spaces. For an S 1_action which preserves a quasi-symplectic structure it is easy to compute the rotation numbers of the normal bundle over each connected component of the fixed point set from the rotation numbers of the quaternionic bundles naturally given by the quasi-symplectic structure. This is used to deduce a vanishing theorem for characteristic numbers from a result which was proved in a common paper with R. Schwarzenberger [11], and which is a consequence of the AtiyahSinger fixed point theorem. In Sect. 2 we recall a result from [11]. In Sect. 3 we prove the announced theorem. Sect. 4 contains applications to the Pontryagin classes of cohomology quaternionic projective spaces.

Exotic characteristic classes of quaternionic bundles

For aC ∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kahler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].

Remarks on the Coefficient Ring of Quaternionic Oriented Cohomology Theories

Let h* be a complex (or real) oriented cohomology theory. It has recently been observed that the of h* plays a very important role in such a theory. Above all, D. Quillen [6] showed that the formal group FU of the complex cobordism theory MU* ( ) is isomorphic to the Lazard universal formal group, and its coefficients generate the ground ring MU* (pt). Unfortunately, in the case of quaternionic oriented cohomology theory (see § 2), there exists no such formal group, since the tensor product of quaternionic line bundles does not yield a quaternionic bundle. This situation makes it difficult, for example, to produce enough generators of MSp*(pt). However there are some substitutes for the formal group. In particular, using the total Pontrjagin class of a certain quaternionic vector bundle 2C (see § 3), N. Ja. Gozman [5] defined a subring A of MU* (pt) which is contained in the image of the forgetful homomorphism (p: MSp* (pt}-*MU*(pt). In this paper, we will generalize his approach to arbitrary quaternionic oriented cohomology theory 7i*, and define a subring Ah of h* (pt) which is generated by the coefficients of certain power series. Then it will be shown that