Flat holomorphic connections on principal bundles over a projective manifold

Let be a connected complex linear algebraic group and its unipotent radical. A principal -bundle over a projective manifold will be called polystable if the associated principal -bundle is so. A -bundle over is polystable with vanishing characteristic classes of degrees one and two if and only if admits a flat holomorphic connection with the property that the image in of the monodromy of the connection is contained in a maximal compact subgroup of .

     

Critical orbits of holomorphic maps on projective spaces

We study the dynamics of iterated holomorphic maps of a complex projective space onto itself. Relations between the Fatou set and the orbits of critical points are investigated. In particular, results concerning critically finite maps on the Riemann sphere are generalized to higher dimensional case.     

On the holomorphic length of a complex projective variety

We wish to thank R. Piergallini who called [2] to our attention.     

Topologically trivial holomorphic vector bundles on infinite-dimensional projective varieties

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuos homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.     

Affine and projective transformations of Berwald connections

We discuss the infinitesimal affine transformations of the Berwald connection of a spray, and the relation between the projective transformations of a spray and the affine transformations of its Berwald-Thomas-Whitehead connection.     

Threefolds with vanishing Hodge cohomology

We consider algebraic manifolds of dimension 3 over with for all and 0$">. Let be a smooth completion of with , an effective divisor on with normal crossings. If the -dimension of is not zero, then is a fibre space over a smooth affine curve (i.e., we have a surjective morphism from to such that the general fibre is smooth and irreducible) such that every fibre satisfies the same vanishing condition. If an irreducible smooth fibre is not affine, then the Kodaira dimension of is and the -dimension of is 1. We also discuss sufficient conditions from the behavior of fibres or higher direct images to guarantee the global vanishing of Hodge cohomology and the affineness of .

     

Characterization of the complex projective space by holomorphic vector fields

Received 22 June 1994; in final form 16 August 1994     

A note on complex projective threefolds admitting holomorphic contact structures

Oblatum 20-VI-1992 & 20-IV-1993     

Invariant connections and invariant holomorphic bundles on homogeneous manifolds

Let X be a differentiable manifold endowed with a transitive action α: A×X→X of a Lie group A. Let K be a Lie group. Under suitable technical assumptions, we give explicit classification theorems, in terms of explicit finite dimensional quotients, of three classes of objects:

• equivalence classes of α-invariant K-connections on X
• α-invariant gauge classes of K-connections on X, and
• α-invariant isomorphism classes of pairs (Q,P) consisting of a holomorphic K ^?-bundle Q → X and a K-reduction P of Q (when X has an α-invariant complex structure).

     

The Schwarzian derivative for maps between manifolds with complex projective connections

In this paper we define, in two equivalent ways, the Schwarzian derivative of a map between complex manifolds equipped with complex projective connections. Also, a new, coordinate-free definition of complex projective connections is given. We show how the Schwarzian derivative is related to the projective structure of the manifolds, to projective linear transformations, and to complex geodesics.

     

Vector bundles with holomorphic connection over a projective manifold with tangent bundle of nonnegative degree

For a projective manifold whose tangent bundle is of nonnegative degree, a vector bundle on it with a holomorphic connection actually admits a compatible flat holomorphic connection, if the manifold satisfies certain conditions. The conditions in question are on the Harder-Narasimhan filtration of the tangent bundle, and on the Neron-Severi group.

     

Deformation of holomorphic maps onto the blow-up of the projective plane

Let S be the blow-up of the projective plane at d distinct points and be any surjective holomorphic map from a compact complex manifold S^′. We will show that all deformations of ψ come from automorphisms of S if d?3. The result is optimal in the sense that it is not true if d?2. The strategy of the proof is to use the infinitesimal automorphisms of the web geometry on S arising from the natural foliations of S induced by the pencils of the lines through the blow-up centers.     

Twistor holomorphic Lagrangian surfaces in the complex projective and hyperbolic planes

We completely classify all the twistor holomorphic Lagrangian immersions in the complex projective plane ², i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over ² are holomorphic. This classification provides a one-parameter family of examples of Lagrangian spheres in ².Research partially supported by a DGICYT grant No. PB91-0731.     

Dual linear connections on fitted manifolds of a space with projective connection

This paper presents an investigation of dual linear connections (projective and affine), induced by different fittings of a space with a projective connection P_n,n, a regular hypersurface V_n-1P _n,n , and a regular hyperbandH _m P _n,n .Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 25–46, 1977.