Elliptic Bundles Research Articles

Three-webs from circles

A new geometric necessary condition for regularity of a curved tree-web is found. A class of tree-webs from circles generalizing the regular tree-web of W. Blaschke from three elliptic bundles of circles with pairwise coinciding vertices is considered, and it is shown that only webs equivalent to the Blaschke web are regular in this class.

The Continuity Equation, Hermitian Metrics and Elliptic Bundles

We extend the continuity equation of La Nave–Tian to Hermitian metrics and establish its interval of maximal existence. The equation is closely related to the Chern–Ricci flow, and we illustrate this in the case of elliptic bundles over a curve of genus at least two.

Collapsing of the Chern–Ricci flow on elliptic surfaces

We investigate the Chern–Ricci flow, an evolution equation of Hermitian metrics generalizing the Kähler–Ricci flow, on elliptic bundles over a Riemann surface of genus greater than one. We show that, starting at any Gauduchon metric, the flow collapses the elliptic fibers and the metrics converge to the pullback of a Kähler–Einstein metric from the base. Some of our estimates are new even for the Kähler–Ricci flow. A consequence of our result is that, on every minimal non-Kähler surface of Kodaira dimension one, the Chern–Ricci flow converges in the sense of Gromov–Hausdorff to an orbifold Kähler–Einstein metric on a Riemann surface.

Oka manifolds

We give a positive answer to Gromov's question [Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2 (1989) 851–897, 3.4.(D), p. 881]: If every holomorphic map from a compact convex set in a Euclidean space C n to a certain complex manifold Y is a uniform limit of entire maps C n → Y , then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion O ( X , Y ) ↪ C ( X , Y ) of the space of holomorphic maps into the space of continuous maps is a weak homotopy equivalence. To cite this article: F. Forstnerič, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Positive toric fibrations

A principal toric bundle M is a complex manifold equipped with a free holomorphic action of a compact complex torus T. Such a manifold is fibred over M/T, with fibre T. We discuss the notion of positivity in fibre bundles and define positive toric bundles. Given an irreducible complex subvariety X ⊂ M of a positive principal toric bundle, we show that either X is T-invariant, or it lies in an orbit of a T-action. For principal elliptic bundles, this theorem is known from Verbitsky [Math. Res. Lett. 12 (2005) 251–264]. As follows from the Borel–Remmert–Tits theorem, any simply connected compact homogeneous complex manifold is a principal toric bundle. We show that compact Lie groups with left-invariant complex structure I are positive toric bundles, if I is generic. Other examples of positive toric bundles are discussed.

A Note on General Blow-Ups of Elliptic Quasi-Bundles

In this paper we give some numerical conditions for a line bundle on a general blow-up of elliptic quasi bundles to give an embedding of order k.

The Oka principle for multivalued sections of ramified mappings

In this paper we establish a basic version of the Oka principle for multivalued sections of ramified holomorphic maps h from a complex manifold Z onto a Stein manifold X. If the ramification locus of h projects into a closed complex subvariety X' of X and if h admits a fiber dominating spray over a small neighborhood of any point in X\X' then any multivalued continuous section of h which is holomorphic in a neighborhood of X' and unramified in X\X' can be homotopically deformed to a global holomorphic multivalued section. The corresponding results for sections of unramified submersions were established by Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897 (1989)) and Forstneric and Prezelj (Math. Ann., to appear).

Oka's principle for holomorphic submersions with sprays

We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen, Math. Ann. 133, 450-472, 1957). Gromov's result has been used in the proof of the embedding theorems for Stein manifolds and Stein spaces into Euclidean spaces of minimal dimension (Y. Eliashberg and M. Gromov, Ann. Math. 136, 123-135, 1992; J. Schurmann, Math. Ann. 307, 381-399, 1997). For further extensions see the preprints math.CV/0101034, math.CV/0107039, and math.CV/0110201.

On the holomorphic rank-2 vector bundles with trivial discriminant over non-Kähler elliptic bundles

On the holomorphic rank-2 vector bundles with trivial discriminant over non-Kähler elliptic bundles

Oka's principle for holomorphic fiber bundles with sprays

We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction see the papers math.CV/0101040, math.CV/0101040, math.CV/0101034, math.CV/0101238, math.CV/0107039, math.CV/0110201.

Elliptic ray bundles in three-dimensional geometry for nonimaging optics: a new approach

A new method to obtain elliptic ray bundles in three-dimensional geometry is presented. This type of bundle is of special interest in nonimaging optics because the only known three-dimensional ideal concentrators with homogeneous media transmit elliptic bundles. First, we reformulate the condition for a bundle to be elliptic; that is, the rays passing through any point of the space form a cone with an elliptic base. The solution found includes all the elliptic bundles obtained by other methods as a particular case. Second, we look for the concentrators defined by the new elliptic bundles with the flow-line design method of nonimaging optics. Most of these bundles do not generate new devices but are transmitted ideally through the already known three-dimensional ideal concentrators.

Algorithmic construction of O(3) chiral field equation hierarchy and the Landau - Lifshitz equation hierarchy via polynomial bundle

We investigate new polynomial hierarchies of Lax pairs which contain the polynomial pairs for the system of O(3) chiral field equations and Landau - Lifshitz equation introduced recently and give an algorithmic construction of the corresponding hierarchies of soliton equations. We compare the Landau - Lifshitz equation hierarchy obtained via a polynomial bundle with the hierarchy obtained via an elliptic bundle.

Application of Lorentz geometry to nonimaging optics: new three-dimensional ideal concentrators

A new family of three-dimensional ideal nonimaging concentrators with rotational symmetry is presented. The flow-line concentrator and the cone concentrator are particular cases of this family. First, we looked for elliptic bundles of rays, i.e., bundles such that the subset of rays passing through any point of the space forms a cone with an elliptic base (this search was done with the Lorentz geometry formalism.) Second, the concentrators, defined by their reflectors and receiver shapes, were derived from these elliptic bundles with the flow-line design method.

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑Cn be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ¦D∖Z are either Hirzebruch surfaces (projective bundles overP1), Hopf surfaces (elliptic bundles overP1), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.

TheK-very ampleness on an elliptic quasi bundle

TheK-very ampleness on an elliptic quasi bundle

Neron-Severi group for nonalgebraic elliptic surfaces I Elliptic bundle case

Neron-Severi group for nonalgebraic elliptic surfaces I Elliptic bundle case

Oka's Principle for Holomorphic Sections of Elliptic Bundles

Oka's Principle for Holomorphic Sections of Elliptic Bundles

Elliptic bundle and generating operators

A hierarchy of Poisson Hamiltonian structures connected with an “elliptic” spectral problem is constructed. Generating operators connected with the Landau-Lifshits equations and asymmetric chiral 0(3)-field are found.

Oka’s principle for holomorphic sections of elliptic bundles

2. The h-principle for locally trivial fibrations 2.1. The h-principle over small subsets 2.2. The h-principle over totally real submanifolds 2.3. Local extension of the h-principle 2.4. Localizable extensions 2.5. The h-principle for Cm-bundles 2.6. Manifolds with totally real souls 2.7. Totally real extensions 2.8. Nicely localizable extensions 2.9. The basic h-principle for Z -* X 2.10. Homomorphisms and holomorphic maps of rank > r

Real algebrogeometric solutions of the Landau-Lifshits equation in prym theta functions

It is also shown in [I] that (0.1) in the "rapidly decreasing" case (IS31 ÷ I, x ÷ ±~) describes a completely integrable Hamiltonian system, and an infinite series Of commuting integrals of motion is calculated. However up until recently the difficulties in applying the MISP to (0.1) were not overcome. They were connected with the absence of an exact formulation of the matrix Riemann problem, corresponding to the elliptic bundle (0.2). The formulation of the Riemann problem needed was recently found by Mikhailov and it was investigated in [2, 3]. Based on this formulation, various procedures for "dressing" were proposed [4, 5] for (0.i).