Principal bundles on the affine line

We prove that any principal bundle on the affine line over a perfect field with a reductive group as structure group comes from the base field by base change.     

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

In 1981, Davis, Gitler, and Mahowald determined the geometric dimension of stable vector bundles of order over if is even and sufficiently large and . In this paper, we use the Bendersky-Davis computation of to show that the 1981 result extends to all (still provided that is sufficiently large). If , the result is often different due to anomalies in the formula for when , but we also determine the stable geometric dimension in these cases.

     

Globalization of Holomorphic Actions on Principal Bundles

Suppose G is a Lie group acting as a group of holomorphic automorphisms on a holomorphic principal bundle P → X. We show that if there is a holomorphic action of the complexification G^C of G on. X, this lifts to a holomorphic action of G^C on the bundle P → X. Two applications are presented. We prove that given any connected homogeneous complex manifold G/H with more than one end, the complexification G^C of G acts holomorphically and transitively on G/H. We also show that the ends of a homogeneous complex manifold G/H with more than two ends essentially come from a space of the form S/Γ, where Γ is a Zariski dense discrete subgroup of a semisimple complex Lie group S with S and Γ being explicitly constructed in terms of G and H.     

Principal bundles over a projective scheme

We prove the existence of a quasi-projective moduli scheme for principal bundles over an arbitrary projective scheme.

     

Extremal Kähler metrics on projective bundles over a curve

Let M=P(E) be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle E→Σ over a compact complex curve Σ of genus ?2. Building on ideas of Fujiki (1992) [27], we prove that M admits a Kähler metric of constant scalar curvature if and only if E is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of E as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in Kähler classes sufficiently far from the boundary of the Kähler cone. The methods used to prove the above results apply to a wider class of manifolds, called rigid toric bundles over a semisimple base, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds.     

A Generalisation of Nagata's Theorem on Ruled Surfaces

We prove a generalisation of a theorem of Nagata on ruled surface to the case of the fiber bundle E/P X, associated to a principal G-bundle E. Using this we prove boundedness for the isomorphism classes of semi-stable G-bundles in all characteristics.     

Principal parts bundles on projective spaces and quiver representations

We study the principal parts bundles \(\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)\) as homogeneous bundles and we describe their associated quiver representations. With this technique we show that if n≥2 and 0≤d<k then there exists an invariant decomposition \(\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)=Q_{k,d}\oplus(S^{d}V\otimes \mathcal {O}_{\mathbb {P}^{n}})\) with Q _k,d a stable homogeneous vector bundle. The decomposition properties of such bundles were previously known only for n=1 or k≤d or d<0. Moreover we show that the Taylor truncation maps \(H^{0}\mathcal {P}^{k}\mathcal {O}_{\mathbb {P}^{n}}(d)\to H^{0}\mathcal {P}^{h}\mathcal {O}_{\mathbb {P}^{n}}(d)\), defined for any h≤k and any d, have maximal rank.     

On multiply of sections of line bundles on compact Riemann surfaces

In this paper,we give some conditions on the surjective of multiply maps H~0(R,L)×H~0(R,K)→H~0(R,L(?)K).Here R is a compact Riemann surface,L a line bundle on R and K is the canonical line bundle.     

Exceptional collections of line bundles on projective homogeneous varieties

We construct new examples of exceptional collections of line bundles on the variety of Borel subgroups of a split semisimple linear algebraic group G$G$ of rank 2 over a field. We exhibit exceptional collections of the expected length for types _A2$A_2$ and _B2=_C2$B_2=C_2$ and prove that no such collection exists for type _G2$G_2$. This settles the question of the existence of full exceptional collections of line bundles on projective homogeneous G$G$-varieties for split linear algebraic groups G$G$ of rank at most 2.     

Nonnegatively and positively curved invariant metrics on circle bundles

We derive and study necessary and sufficient conditions for an -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial bundle that is not a product metric.

     

Characteristic classes of flat bundles, II

On a smooth varietyX defined over a fieldK of characteristic zero, one defines characteristic classes of bundles with an integrableK-connection in a group lifting the Chow group, which map, whenK is the field of complex numbers andX is proper, to Cheeger-Simons' secondary analytic invariants, compatibly with the cycle map in the Deligne cohomology.     

On foliated circle bundles over closed orientable 3-manifolds

We show that there exists a family of smooth orientable circle bundles over closed orientable 3-manifolds each of which has a codimension-one foliation transverse to the fibres of class C ⁰ but has none of class C ³ . There arises a necessary condition induced from the Milnor-Wood inequality for the existence of a foliation transverse to the fibres of an orientable circle bundle over a closed orientable 3-manifold. We show that with some exceptions this necessary condition is also sufficient for the existence of a smooth transverse foliation if the base space is a closed Seifert fibred manifold. Received: May 13, 1996     

On the projective normality of line bundles of extremal degree

We propose a new method, using deformation theory, to study the maximal rank conjecture. For line bundles of extremal degree, which can be viewed as the first case to test the conjecture, we prove that maximal rank conjecture holds by our new method.     

Monads and rank 3 vector bundles on quadrics

In this paper we give the classification of rank 3 vector bundles without “inner” cohomology on a quadric hypersurface (n > 3) by studying the associated monads.      

A note on principal parts on projective space and linear representations

Let be a closed subgroup of a linear algebraic group defined over a field of characteristic zero. There is an equivalence of categories between the category of linear finite-dimensional representations of , and the category of finite rank -homogeneous vector bundles on . In this paper we will study this correspondence for the sheaves of principal parts on projective space, and we describe the representation corresponding to the principal parts of a line bundle on projective space.

     

Another proof of Bianchi's identity in arbitrary bundles

In this paper we relate the group of automorphisms of a principal fibre bundle with the (second) Bianchi identity generalizing a result of Kazdan.During the preparation of this paper the author was supported by a fellowship of theGraduiertenkolleg Geometrie und Mathematische Physik at the Ruhr-Universität Bochum.     

On the Mislin Genus of Symplectic Groups

In this paper we give lower bounds for the Mislin genus of thesymplectic groups Sp(m). This result appears to be the exactanalogue of Zabrodsky's theorem concerning the special unitarygroups SU(n). It is achieved by the determination of the stablegenus of the quasi-projective quaternionic spaces QH(m), followingthe approach of McGibbon. It leads to a symplectic version ofZabrodsky's conjecture, saying that these lower bounds are infact the exact cardinality of the genus sets. The genus of Sp(2)is well known to contain exactly two elements. We show thatthe genus of Sp(3) has exactly 32 elements and see that theconjecture is true in these two cases. Independently, we also show that any homotopy type in the genusof Sp(m) fibers over the sphere S⁴m–1 with fiber in thegenus of Sp(m–1), and that any homotopy type in the genusof SU(n) fibers over the sphere S²n–1 with fiber in thegenus of SU(n–1). Moreover, these fibrations are principalwith respect to some appropriate loop structures on the fibers.These constructions permit us to produce particular spaces realizingthe lower bounds obtained. 2000 Mathematics Subject Classification55P60 (primary), 55P15, 55R35 (secondary)