Holomorphic Vector Bundles Research Articles

Coupled connections on a compact Riemann surface

We consider holomorphic differential operators on a compact Riemann surface X whose symbol is an isomorphism. Such a differential operator of order n on a vector bundle E sends E to K ⊗ n X ⊗ E, where K X is the holomorphic cotangent bundle. We classify all those holomorphic vector bundles E over X that admit such a differential operator. The space of all differential operators whose symbol is an isomorphism is in bijective correspondence with the collection of pairs consisting of a flat vector bundle E over X and a holomorphic subbundle of E satisfying a transversality condition with respect to the connection.

Equivariant moduli problems, branched manifolds, and the Euler class

The purpose of this paper is to explain the equivariant Euler class associated to an oriented G-equivariant Fredholm section S : B → E of a Hilbert space bundle over a Hilbert manifold. The key hypotheses are that the Lie group G is compact, the isotropy subgroups are finite, and the zero set of the section is compact. The present paper is motivated by our joint work with Gaio [9] on invariants of Hamiltonian group actions. In this work the Fredholm section arises from a version of the vortex equations, where the target space is a symplectic manifold with a Hamiltonian G-action [8, 19, 20]. In many interesting cases the resulting moduli spaces are compact and so the results of the present paper can be applied. Other examples of Fredholm sections with compact zero sets are the Seiberg–Witten equations over a four-manifold [25] or the harmonic map equations when the target space is a negatively curved manifold (see e.g. [14]). This is in sharp contrast to the Gromov–Witten invariants of general (compact) symplectic manifolds [11, 16, 17, 22] and to the Donaldson invariants of smooth four-manifolds [10], where the moduli spaces are noncompact and the compactifications are the source of some major difficulties of the theory. Since the unperturbed moduli space is compact, our framework is considerably simpler than the one required for the construction of the Gromov–Witten invariants. Our exposition follows closely the work of Li–Robbin–Ruan [16]. In the case G = {1l} similar results were proved in [6, 12, 21]. In [12] Fulton proved that, if B is a finite dimensional complex manifold, E → B is a holomorphic vector bundle, and S : B → E is a holomorphic section, then the zero set M := S−1(0) carries a fundamental cycle (in singular homology) which is Poincare dual to the Euler class. This was extended to the infinite ∗Supported by National Science Foundation grant DMS-0072267

Superpotentials for vector bundle moduli

We present a method for explicitly computing the non-perturbative superpotentials associated with the vector bundle moduli in heterotic superstrings and M-theory. This method is applicable to any stable, holomorphic vector bundle over an elliptically fibered Calabi–Yau threefold. For specificity, the vector bundle moduli superpotential, for a vector bundle with structure group G= SU(3), generated by a heterotic superstring wrapped once over an isolated curve in a Calabi–Yau threefold with base B= F 1 , is explicitly calculated. Its locus of critical points is discussed. Superpotentials of vector bundle moduli potentially have important implications for small instanton phase transitions and the vacuum stability and cosmology of superstrings and M-theory.

Holomprhic vector bundles on C2∖{0}

Here we show the existence of a rank 2 holomorphic vector bundleE on C2∖{0} without any holomorphic rank 1 subsheaf. Hence, contrary to the algebraic case, there are open subsets of dimension 2 Stein manifolds with holomorphic vector bundles which are not filtrable in any weak sense.

Holomorphic vector bundles on compact complex non-algebraic surfaces

LetX be a smooth complex compact surface without non-constant meromorphic functions. Here we prove the existence of rank holomorphic vector bundles onX containing exactly one rank one saturated subsheaf.

MONODROMY APPROACH TO QUANTUM COMPUTING

In this work, a gauge approach to quantum computing is considered. It is assumed that there exists a classical procedure for placing certain classical system in a state described by a holomorphic vector bundle with connection with logarithmic singularities. This bundle and its connection are constructed with the aid of unitary operators realizing the given algorithm using methods of the monodromic Riemann–Hilbert problem. Universality is understood in the sense that for any collection of unitary matrices there exists a connection with logarithmic singularities whose monodromy representation involves these matrices.

Asymptotic Behaviour of Tame Nilpotent Harmonic Bundles with Trivial Parabolic Structure

Let E be a holomorphic vector bundle. Let θ be a Higgs field, that is a holomorphic section of End (E) ⊗ Ω1,0X satisfying θ2 = 0. Let h be a pluriharmonic metric of the Higgs bundle (E, θ). The tuple (E, θ, h) is called a harmonic bundle. Let X be a complex manifold, and D be a normal crossing divisor of X. In this paper, we study the harmonic bundle (E, θ, h) over X − D. We regard D as the singularity of (E, θ, h), and we are particularly interested in the asymptotic behaviour of the harmonic bundle around D. We will see that it is similar to the asymptotic behaviour of complex variation of polarized Hodge structures, when the harmonic bundle is tame and nilpotent with the trivial parabolic structure. For example, we prove constantness of general monodromy weight filtrations, compatibility of the filtrations, norm estimates, and the purity theorem. For that purpose, we will obtain a limiting mixed twistor structure from a tame nilpotent harmonic bundle with trivial parabolic structure, on a punctured disc. It is a solution of a conjecture of Simpson.

Analog of secondary invariants for holomorphic vector bundles

Given a pair of holomorphic structures on a complex vector bundle over a complex manifold, Dolbeault cohomological invariants are constructed. The construction is based on the construction of Chern–Simons secondary classes.

A criterion for the existence of a flat connection on a parabolic vector bundle

We define holomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a holomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the para- bolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic connection if and only if every direct summand of it is of degree zero. 2000 Mathematics Subject Classification. 14H60, 32L05

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,ϕ), where E1 and E2 are holomorphic vector bundles over a fixed compact Riemann surfaceX, andϕ: 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 Kahler 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.

Abelianization of Wilson loop of non-Abelian gauge theory on complex manifold and Polyakov's conjecture

A gauge theory of GL(2) gauge group is considered on a complex manifold and hermitian connections of holomorphic vector bundles of rank 2 over a compact complex manifold M are abelianized using monoidal transformation. The abelianized connections of the line bundle over a manifold M̃ as a P 1 fibre over M are constructed explicitly. The Wilson loop of the GL(2, C) gauge theory is abelianized to that of C ∗ gauge theory. By this the Polyakov conjecture is verified for a GL(2) gauge theory on a complex manifold.

Nonperturbative flipped SU(5) vacua in heterotic M-theory

The evidence for neutrino masses in atmospheric and solar neutrino experiments provides further support for the embedding of the Standard Model fermions in the chiral 16 SO(10) representation. Such an embedding is afforded by the realistic free fermionic heterotic-string models. In this paper we advance the study of these string models toward a nonperturbative analysis by generalizing the work of Donagi, Pantev, Ovrut and Waldram from the case of G= SU(2 n+1) to G= SU(2 n) stable holomorphic vector bundles on elliptically fibered Calabi–Yau manifolds with fundamental group Z 2 . We demonstrate existence of G= SU(4) solutions with three generations and SO(10) observable gauge group over Hirzebruch base surface, whereas we show that certain classes of del Pezzo base surface do not admit such solutions. The SO(10) symmetry is broken to SU(5)× U(1) by a Wilson line. The overlap with the realistic free fermionic heterotic-string models is discussed.

Vector Bundle Moduli and Small Instanton Transitions

We give the general presciption for calculating the moduli of irreducible, stable SU(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi-Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B=F_r. The transition moduli that are produced by chirality changing small instanton phase transitions are defined and specifically enumerated. The origin of these moduli, as the deformations of the spectral cover restricted to the ``lift'' of the horizontal curve of the M5-brane, is discussed. We present an alternative description of the transition moduli as the sections of rank n holomorphic vector bundles over the M5-brane curve and give explicit examples. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.

Kähler Yamabe minimizers on minimal ruled surfaces

It is shown that if a minimal ruled surface $\mathrm{P}(E) \rightarrow \Sigma$ admits a Kähler Yamabe minimizer, then this metric is generalized Kähler-Einstein and the holomorphic vector bundle $E$ is quasi-stable.

On a splitting problem

We show that any short exact sequence 0→K→Ω× C n→Ω× C m→0 of holomorphic vector bundles splits over a pseudoconvex open subset Ω of a Banach space which has a countable unconditional basis.

Five-brane superpotentials in heterotic M-theory

The open supermembrane contribution to the non-perturbative superpotential of bulk space five-branes in heterotic M-theory is presented. We explicitly compute the superpotential for the modulus associated with the separation of a bulk five-brane from an end-of-the-world three-brane. The gauge and κ-invariant boundary strings of such open supermembranes are given and the role of the holomorphic vector bundle on the orbifold fixed plane boundary is discussed in detail.

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let X be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, X should be biholomorphic to a rational homogeneous manifold G/P, where G is a simple Lie group, and P ⊂ G is a maximal parabolic subgroup. In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents C x , and (b) recovering the structure of a rational homogeneous manifold from C x . The author proves that, when b 4 (X) = 1 and the generic variety of minimal rational tangents is 1-dimensional, X is biholomorphic to the projective plane P 2 , the 3-dimensional hyperquadric Q 3 , or the 5-dimensional Fano homogeneous contact manifold of type G 2 , to be denoted by K(G 2 ). The principal difficulty is part (a) of the scheme. We prove that C x C PT x (X) is a rational curve of degrees < 3, and show that d = 1 resp. 2 resp. 3 corresponds precisely to the cases of X = P 2 resp. Q 3 resp. K(G 2 ). Let κ be the normalization of a choice of a Chow component of minimal rational curves on X. Nefness of the tangent bundle implies that κ is smooth. Furthermore, it implies that at any point x ∈ X, the normalization κ x of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that κ x is a rational curve, our principal object of study is the universal family u of κ, giving a double fibration p: u → κ, μ: u → X, which gives P 1 -bundles. There is a rank-2 holomorphic vector bundle V on κ whose projectivization is isomorphic to p: u → κ. We prove that V is stable, and deduce the inequality d < 4 from the inequality c 2 1 (V) < 4c 2 (V) resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of d = 4 is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on V in the special case where c 2 1 (V) = 4c 2 (V).

Holomorphic vector bundles on non-algebraic surfaces

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is, in general, still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII. Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces. To cite this article: A. Teleman, M. Toma, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 383–388.

Vector Bundle Moduli

We give the general presciption for calculating the number of moduli of irreducible, stable U(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi–Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B = F r. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.

Projective Structure, Symplectic Connection and Quantization

Let X be a connected Riemann surface equipped with a projective structure \(\mathfrak{p}\). Let E be a holomorphic symplectic vector bundle over X equipped with a flat connection. There is a holomorphic symplectic structure on the total space of the pullback of E to the space of all nonzero holomorphic cotangent vectors on X. Using \(\mathfrak{p}\), this symplectic form is quantized. A moduli space of Higgs bundles on a compact Riemann surface has a natural holomorphic symplectic structure. Using \(\mathfrak{p}\), a quantization of this symplectic form over a Zariski open subset of the moduli space of Higgs bundles is constructed.