A note on the splitting principle

We offer a new perspective on the splitting principle. We give an easy proof that applies to all classical types of vector bundles and in fact to G-bundles for any compact connected Lie group G. The perspective gives precise calculational information and directly ties the splitting principle to the specification of characteristic classes in terms of classifying spaces.     

A Splitting of Vector Bundles with Connection and Pseudo-Inverses of the Fundamental Form

We investigate a splitting of a short exact sequence of vector bundles with connection. We obtain an existence theorem of a unimodular splitting under a certain regularity condition on the fundamental form. In particular, we use a pseudo-inverse of the fundamental form to choose a canonical splitting, which is applied to an equiaffine immersion. Received: August 25, 2006. Revised: June 20, 2007.     

Bundles over the Fano Threefold V 5

ABSTRACT

Using an explicit resolution of the diagonal for the variety V ₅, we provide cohomological characterizations of the universal and quotient bundles. A splitting criterion for bundles over V ₅ is also proved.

The presentation of semistable aCM bundles is shown, together with a resolution–theoretic classification of low rank aCM bundles.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

A Sufficient Condition for Splitting of Arithmetically Cohen–Macaulay Bundles on General Hypersurfaces

A vector bundle on a hypersurface is arithmetically Cohen–Macaulay if its intermediate cohomologies vanish. On projective spaces, such bundles coincide with those which split into a direct sum of line bundles, but this fails on hypersurfaces of higher degree in general. In this article, we give an inequality which gives a sufficient condition for splitting of arithmetically Cohen–Macaulay bundles on general hypersurfaces.     

Remarks on groups of bundle automorphisms over the Riemann sphere

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff–Grothendieck bundle splitting \(\oplus _{i=1}^{r} \mathcal {O}(m_{i})\) over \(\mathbb {C}\mathbb {P}^{1}\) is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset \(S\subset \mathbb {C}\mathbb {P}^{1}\). The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types’ automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.     

A splitting theorem for affine skew‐product flows

Topological conjugacy for continuous skew product flows on topological group bundles and semidirect products is studied resulting in a splitting theorem. The results are used to discuss fiber entropy.     

A conjectural formula for genus one Gromov-Witten invariants of some local Calabi-Yau <Emphasis Type="Italic">n</Emphasis>-folds

We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special cases, and assuming the validity of our conjecture we check the integrality of genus one Bogomol’nyi-Prasad-Sommerfield (BPS) numbers of local Calabi-Yau 5-folds defined by Klemm and Pandharipande.     

A conjectural formula for genus one Gromov-Witten invariants of some local Calabi-Yau n-folds

We conjecture a formula for the generating function of genus one Gromov-Witten invariants of the local Calabi-Yau manifolds which are the total spaces of splitting bundles over projective spaces. We prove this conjecture in several special cases, and assuming the validity of our conjecture we check the integrality of genus one Bogomol'nyi-Prasad-Sommerfield(BPS) numbers of local Calabi-Yau 5-folds defined by Klemm and Pandharipande.     

On compact splitting complex submanifolds of quotients of bounded symmetric domains

We study compact complex submanifolds S of quotient manifolds X = ?/Γ of irreducible bounded symmetric domains by torsion free discrete lattices of automorphisms, and we are interested in the characterization of the totally geodesic submanifolds among compact splitting complex submanifolds S ? X, i.e., under the assumption that the tangent sequence over S splits holomorphically. We prove results of two types. The first type of results concerns S ? X which are characteristic complex submanifolds, i.e., embedding ? as an open subset of its compact dual manifold M by means of the Borel embedding, the non-zero(1, 0)-vectors tangent to S lift under a local inverse of the universal covering map π : ? → X to minimal rational tangents of M.We prove that a compact characteristic complex submanifold S ? X is necessarily totally geodesic whenever S is a splitting complex submanifold. Our proof generalizes the case of the characterization of totally geodesic complex submanifolds of quotients of the complex unit ball Bnobtained by Mok(2005). The proof given here is however new and it is based on a monotonic property of curvatures of Hermitian holomorphic vector subbundles of Hermitian holomorphic vector bundles and on exploiting the splitting of the tangent sequence to identify the holomorphic tangent bundle TSas a quotient bundle rather than as a subbundle of the restriction of the holomorphic tangent bundle TXto S. The second type of results concerns characterization of total geodesic submanifolds among compact splitting complex submanifolds S ? X deduced from the results of Aubin(1978)and Yau(1978) which imply the existence of K¨ahler-Einstein metrics on S ? X. We prove that compact splitting complex submanifolds S ? X of sufficiently large dimension(depending on ?) are necessarily totally geodesic. The proof relies on the Hermitian-Einstein property of holomorphic vector bundles associated to TS,which implies that endomorphisms of such bundles are parallel, and the construction of endomorphisms of these vector bundles by means of the splitting of the tangent sequence on S. We conclude with conjectures on the sharp lower bound on dim(S) guaranteeing total geodesy of S ? X for the case of the type-I domains of rank2 and the case of type-IV domains, and examine a case which is critical for both conjectures, i.e., on compact complex surfaces of quotients of the 4-dimensional Lie ball, equivalently the 4-dimensional type-I domain dual to the Grassmannian of 2-planes in C~4.     

VECTOR BUNDLES ON PROJECTIVE SPACES OVER A DISCRETE VALUATION RING

Let E be a vector bundle on P ⁿ _D, n ≥ 2, with D discrete valuation ring. Here we give a necessary and sufficient condition for the splitting of E as a direct sum of line bundles.     

Cohomology Splittings of Stiefel Manifolds

The complex Stiefel manifolds admit a stable decomposition asThom spaces of certain bundles over Grassmannians. The purposeof the paper is to identify the splitting in any complex orientedcohomology theory.     

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 splitting theorem for holomorphic Banach bundles

This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided .     

Non-Hermitian Yang-Mills connections

We study Yang-Mills connections on holomorphic bundles over complex K?hler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isomorphic to the complexification of the space of Hermitian Yang-Mills connections (which is, by Uhlenbeck and Yau, the same as the space of stable bundles). Further, we study the NHYM connections over hyperk?hler manifolds. We construct direct and inverse twistor transform from NHYM bundles on a hyperk?hler manifold to holomorphic bundles over its twistor space. We study the stability and the modular properties of holomorphic bundles over twistor spaces, and prove that work of Li and Yau, giving the notion of stability for bundles over non-K?hler manifolds, can be applied to the twistors. We identify locally the following two spaces: the space of stable holomorphic bundles on a twistor space of a hyperk?hler manifold and the space of rational curves in the twistor space of the ‘Mukai’ dual hyperk?hler manifold.     

Holomorphic bundles on O(− k) are algebraic

We show that holomorphic bundles on O(?k) for k>0 are algebraic. We also show that holomorphic bundles on O(?1) are trivial outside the zero section. A corollary is that bundles on the blow-up of a surface at a point are trivial on a neighborhood of the exceptional divisor minus the exceptional divisor.     

Finding ein components in the moduli spaces of stable rank 2 bundles on the projective 3-space

Some method is proposed for finding Ein components in moduli spaces of stable rank 2 vector bundles with first Chern class c₁ = 0 on the projective 3-space. We formulate and illustrate a conjecture on the growth rate of the number of Ein components in dependence on the numbers of the second Chern class. We present a method for calculating the spectra of the above bundles, a recurrent formula, and an explicit formula for computing the number of the spectra of these bundles.     

On 4-manifolds and span-related numbers for cat manifolds

LetM ⁿ denote any closed connected CAT manifold of positive dimensionn. We define CATs(Mⁿ) to be the smallest positive dimension of all closed connected CAT manifoldsN for which the CAT span ofM×N is strictly greater than the CAT span ofN. We determine a formula for this characteristic number which involves only the Kirby-Siebenmann numberks(M) ofM and a Stiefel-Whitney number. Several results on splitting the tangent bundles of closed 4-manifolds are obtained. For example, both the Euler number ofM ⁴ andks(M⁴) represent the total obstruction to positive CAT span for a non-smoothable closed connected 4-manifold. Dedicated to the memory of Professor Otto Endler     

Affiliated subspaces and infiniteness of von Neumann algebras

We show that the structural properties of von Neumann algebra s are connected with the metric and order theoretic properties of various classes of affiliated subspaces. Among others we show that properly infinite von Neumann algebra s always admit an affiliated subspace for which (1) closed and orthogonally closed affiliated subspaces are different; (2) splitting and quasi‐splitting affiliated subspaces do not coincide. We provide an involved construction showing that concepts of splitting and quasi‐splitting subspaces are non‐equivalent in any GNS representation space arising from a faithful normal state on a Type I factor. We are putting together the theory of quasi‐splitting subspaces developed for inner product spaces on one side and the modular theory of von Neumann algebra s on the other side.     

p-adic representations and vector bundles on Mumford curves

Christopher Deninger andAnnette Werner constructed a functor that associates representations of the algebraic fundamental group of an algebraic curve to a class of vector bundles on that curve. We compare this to a construction byFaltings for Mumford curves that associates representations of the Schottky group to semistable vector bundles of degree 0. We prove that for a certain class of vector bundles on Mumford curves the constructions induce isomorphic representations.