Holomorphic Hermitian vector bundles over the Riemann sphere

We give a complete classification of isomorphism classes of all SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹. We construct a canonical bijective correspondence between the isomorphism classes of SU(2)-equivariant holomorphic Hermitian vector bundles on CP¹ and the isomorphism classes of pairs ({Hn}_n∈Z,T), where each Hn is a finite dimensional Hilbert space with Hn=0 for all but finitely many n, and T is a linear operator on the direct sum _⊕n∈ZHn such that T(Hn)⊂Hn+2 for all n.     

Pseudoconvex domains spread over complex homogeneous manifolds

Using the concept of inner integral curves defined by Hirschowitz we generalize a recent result by Kim, Levenberg and Yamaguchi concerning the obstruction of a pseudoconvex domain spread over a complex homogeneous manifold to be Stein. This is then applied to study the holomorphic reduction of pseudoconvex complex homogeneous manifolds X = G/H. Under the assumption that G is solvable or reductive we prove that X is the total space of a G-equivariant holomorphic fiber bundle over a Stein manifold such that all holomorphic functions on the fiber are constant.     

Obstructions to toric integrable geodesic flows in dimension 3

The geodesic flow of a Riemannian metric on a compact manifold Q is said to be toric integrable if it is completely integrable and the first integrals of motion generate a homogeneous torus action on the punctured cotangent bundle T ^* Q\Q. If the geodesic flow is toric integrable, the cosphere bundle admits the structure of a contact toric manifold. By comparing the Betti numbers of contact toric manifolds and cosphere bundles, we are able to provide necessary conditions for the geodesic flow on a compact, connected 3-dimensional Riemannian manifold to be toric integrable.Mathematics Subject Classifications (2000): primary 53D25; secondary 53D10     

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

On the L 2-extension of twisted holomorphic sections of singular hermitian line bundles

We prove a sharp Ohsawa–Takegoshi–Manivel type L ²-extension result for twisted holomorphic sections of singular hermitian line bundles over almost Stein manifolds. We establish as corollaries some extension results for pluri-twisted holomorphic sections of singular hermitian line bundles over projective manifolds.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Strong Hypercontractivity and Relative Subharmonicity

A Hermitian metric, g, on a complex manifold, M, together with a smooth probability measure, μ, on M determine minimal and maximal Dirichlet forms, Q_D and Q_max, given by Q(f)=∫_M g(grad f(z), grad f(z)) dμ(z). Q_D is the form closure of Q on C^∞_c(M) and Q_max is the form closure of Q on C¹_b(M). The corresponding operators, A_D and A_max, generate semigroups having standard hypercontractivity properties in the scale of L^p spaces, p>1, when the corresponding form, Q, satisfies a logarithmic Sobolev inequality. It was shown by the author (1999, Acta Math.182, 159-206) that the semigroup e^−tA_D has even stronger hypercontractivity properties when restricted to certain holomorphic subspaces of L^p. These results are extended here to A_max. When (M, g) is not complete it is necessary that the elliptic differential operator A_max degenerate on the boundary of M. A second proof of these strong hypercontractive inequalities for both A_D and A_max is given, which depends on an extension of the submean value property of subharmonic functions. The Riemann surface for z^1/n and the weighted Bergman spaces in the unit disc are given as examples.     

Steinness of the Total Space of a Non‐Trivial Algebraic Affine ℂ‐Bundle on the Punctured Complex Affine Plane

We prove that the total space E of an algebraic affine ℂ‐bundle π : E → X on the punctured complex affine plane X ≔ ℂ² – {(0, 0)} is Stein if and only if it is not isomorphic to the trivial holomorphic line bundle X × ℂ.     

Path bundles on n-cubes

A path bundle is a set of 2^a paths in an n-cube, denoted Q_n, such that every path has the same length, the paths partition the vertices of Q_n, the endpoints of the paths induce two subcubes of Q_n, and the endpoints of each path are complements. This paper shows that a path bundle exists if and only if n>0 is odd and 0?a?n-⌈log₂(n+1)⌉.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Vector bundles on a nearly degenerate Riemann surface

We show that on a Riemann surface X close to a maximally degenerate complex curve, any semistable vector bundle of degree 0 is associated with a linear representation of the Schottky group ?? _X uniformizing X. Further, we study the relationship between the representation space of ?? _X and the moduli space of semistable bundles on X as a higher rank version of Abel?CJacobi??s theorem.     

Wiener–Tauberian type theorems for radial sections of homogenous vector bundles on certain rank one Riemannian symmetric spaces of noncompact type

We will show that an uniform treatment yields Wiener–Tauberian type results for various Banach algebras and modules consisting of radial sections of some homogenous vector bundles on rank one Riemannian symmetric spaces G/K of noncompact type. One example of such a vector bundle is the spinor bundle. The algebras and modules we consider are natural generalizations of the commutative Banach algebra of integrable radial functions on G/K. The first set of them are Beurling algebras with analytic weights, while the second set arises due to Kunze–Stein phenomenon for noncompact semisimple Lie groups.     

On the adjunction process over a surface in char.p

Let K_s be the canonical bundle on a non singular projective surface S (over an algebraically closed field F, char F=p) and L be a very ample line bundle on S. Suppose (S,L) is not one of the following pairs: (P²,O(e)), e=1,2, a quadric, a scroll, a Del Pezzo surface, a conic bundle. Then

1. (K_s?L)² is spanned at each point by global sections. Let \(\phi :S \to P^N _F \) be the map given by the sections Γ(K_s?L)², and let φ=s o r its Stein factorization.
2. r:S→S′=r(S) is the contraction of a finite number of lines, E_i for i=1,...r, such that E_i·E_i=K_S·E_i=?L·E_i=?1.
3. If h°(L)≥6 and L·L≥9, then s is an embedding.

     

Jumping conics on a smooth quadric in $${\mathbb {P}_3}$$

We investigate the jumping conics of stable vector bundles E of rank 2 on a smooth quadric surface Q with the first Chern class c₁ = O_Q(-1,-1){c_1= \mathcal{O}_Q(-1,-1)} with respect to the ample line bundle O_Q(1,1){\mathcal {O}_Q(1,1)} . We show that the set of jumping conics of E is a hypersurface of degree c ₂(E) − 1 in \mathbb P₃^*{\mathbb {P}_3^{*}} . Using these hypersurfaces, we describe moduli spaces of stable vector bundles of rank 2 on Q in the cases of lower c ₂(E).     

Lineare Komplexbündel im einfach isotropen Raum

According to K. Strubecker ([5]–[8]) a three dimensional real affine space with the metric ds²=dx²+dy² is called a simply iso-tropic space J₃(1). In J₃(1) exist 41 types of bundles of linear line complexes. In this paper we study the bundles of type 1, 2-called cyclic bundle and 38-called Strubecker-bundle. For a cyclic bundle and a Strubecker-bundle we give only the fundamental theorem and one metric result, however for a cyclic bundle the metric theory is completely developed.

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Herrn o. Prof. Dr. K. Strubecker zum 80. Geburtstag gewidmet     

Real parabolic vector bundles over a real curve

We define real parabolic structures on real vector bundles over a real curve. Let (X, σ _X ) be a real curve, and let S???X be a non-empty finite subset of X such that σ _X (S)?=?S. Let N?≥?2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y?∈?Y, one has $\sigma_Y(gy) = g^{-1}\sigma_Y(y)$ . We established an equivalence between the category of real parabolic vector bundles on (X, σ _X ) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σ _Y ).     

Quillen bundle and geometric prequantization of non-abelian vortices on a Riemann surface

In this paper we prequantize the moduli space of non-abelian vortices. We explicitly calculate the symplectic form arising from L ² metric and we construct a prequantum line bundle whose curvature is proportional to this symplectic form. The prequantum line bundle turns out to be Quillen’s determinant line bundle with a modified Quillen metric. Next, as in the case of abelian vortices, we construct line bundles over the moduli space whose curvatures form a family of symplectic forms which are parametrized by Ψ₀, a section of a certain bundle. The equivalence of these prequantum bundles are discussed.     

Transnormal functions on a Riemannian manifold

We extend theorems of É. Cartan, Nomizu, Münzner, Q.M. Wang, and Ge–Tang on isoparametric functions to transnormal functions on a general Riemannian manifold. We show that if a complete Riemannian manifold M admits a transnormal function, then M is diffeomorphic to either a vector bundle over a submanifold, or a union of two disk bundles over two submanifolds. Moreover, a singular level set Q is austere and minimal, if exists, and generic level sets are tubes over Q. We give a criterion for a transnormal function to be an isoparametric function.     

Uniform approximation by polynomials on compacta of special form

A lower estimate of the least deviations is obtained and polynomials of the best uniform approximation are found for some functions given on compact sets of the complex plane containing complete preimages Q ^?1(v _j ) of several points v _j ?? ? for some polynomial Q(z) of complex variable.