Algebraic Manifolds Research Articles

Families of Normal Maps in Several Complex Variables and Hyperbolicity of Complex Spaces

In this article, generalizations of the theorem of Zalcmal to complex spaces are given. Moreover, a generalization of Five-Point Theorem of Lappan to complex projective algebraicmanifolds also is shown.

Topological rigidity of algebraic ℙ3-bundles over curves

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus g≥1 is itself a ruled surface over a curve of genus g. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in the case g≥2.

G-Elements of matroid complexes

A g-element for a graded R-module is a one-form with properties similar to a Lefschetz class in the cohomology ring of a compact complex projective manifold, except that the induced multiplication maps are injections instead of bijections. We show that if k( Δ) is the face ring of the independence complex of a matroid and the characteristic of k is zero, then there is a non-empty Zariski open subset of pairs ( Θ, ω) such that Θ is a linear set of parameters for k( Δ) and ω is a g-element for k( Δ)/〈 Θ〉. This leads to an inequality on the first half of the h-vector of the complex similar to the g-theorem for simplicial polytopes.

Some remarks on multiplier ideals and vector bundles

In this paper, we give two applications of the theory of multiplier ideals to vector bundles over complex projective manifolds, generalizing to higher rank results already established for line bundles. The first addresses the existence of sections of (suitable twists) of symmetric powers of a very ample vector bundle, vanishing on a given subvariety. The second is a vanishing theorem of Gri‰ths type, adjusted with an additional multiplier ideal term as in the standard Nadel vanishing theorem. To begin with, we recall that, starting with work of Bombieri and Skoda, there has been considerable interest in going from hypersurfaces in P highly singular at a given set of points S to hypersurfaces through S of relatively low degree; there is now a very direct and terse approach based on multiplier ideals [3], [4], [5], [6]. Here, the same method is shown to yield a statement in the same spirit about sections of very ample vector bundles on a projective manifold. One says that a rank r holomorphic vector bundle on a complex projective manifold is very ample if the relative hyperplane line bundle OPE ð1Þ on the projectivised dual PE is very ample. If X is a projective manifold and ZJX an irreducible subvariety, for every integer pd 1 the p-th symbolic power I hpi Z JOZ of the ideal sheaf of Z is the ideal sheaf of the holomorphic functions vanishing with multiplicity dp along Z (that is, at a generic point of Z ).

Algebraic dependence of meromorphic mappings in value distribution theory

AbstractIn this paper we first prove some criteria for the propagation of algebraic dependence of dominant meromorphic mappings from an analytic finite covering space X over the complex m-space into a projective algebraic manifold. We study this problem under a condition on the existence of meromorphic mappings separating the generic fibers of X. We next give applications of these criteria to the uniqueness problem of meromorphic mappings. We deduce unicity theorems for meromorphic mappings and also give some other applications. In particular, we study holomorphic mappings into a smooth elliptic curve E and give conditions under which two holomorphic mappings from X into E are algebraically related.

Sur l'algébrisabilité locale de sous-variétés analytiques réelles génériques de [formula omitted

We prove that the local (pseudo)group of biholomorphisms stabilizing a minimal, finitely nondegenerate real algebraic submanifold in C n is a real algebraic local Lie group. We deduce necessary conditions for the local algebraizability of real analytic rigid tubes of arbitrary codimension in C n . To cite this article: H. Gaussier, J. Merker, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

On some problems of Kobayashi and Lang; algebraic approaches

The goal of this survey paper is to present results on hyperbolicity of complex algebraic manifolds, which appeared after the papers [48] and [42], where a number of important and intriguing conjectures were proposed. Since the paper puts the accent on measure hyperbolicity on one hand, and on algebraic methods on the other hands, we hope it will not overlap too much with the beautiful paper [21] by Demailly, except for the basic definitions and starting points. The basic questions asked in [42], and in a different spirit in [48] concern the relationships between curvature properties of a given complex manifold (or complex algebraic variety) on one hand, and the behaviour of holomorphic maps from disks or polydisks to them, on the other hand, or, from a more algebraic point of view, the (non) existence of subvarieties of certain types.

Period Map for Non-Compact Holomorphically Symplectic Manifolds

We study the deformations of a holomorphic symplectic manifold X, not necessarily compact, over a formal ring. We always assume both X and the symplectic form Ω to be algebraic over $\mathbb{C}.$ We show (under some additional, but mild, assumptions on X) that the coarse deformation space of the pair $\left\langle {X,\Omega } \right\rangle $ exists and is smooth, finite-dimensional and naturally embedded into H2(X). In particular, for an algebraic holomorphic symplectic manifold X which satisfies $H^i (\mathcal{O}_X ) = 0$ for all i > 0, the coarse moduli of formal deformations is isomorphic to Spec $\mathbb{C}\left[\kern-0.15em\left[ {t_1 , \ldots ,t_n } \right]\kern-0.15em\right],$ where $t_1 , \ldots t_n $ are coordinates in H2(X).

Young Tableaux and the Frobenius–Pullback of Tensor Differential Forms

In the case of prime characteristic char k = p > the pullback of the Frobenius provides an opportunity to define new discrete birational invariants of algebraic manifolds using the ps–th powers (df) instead of the differentials df. Earlier results concerning T–symmetrical tensor forms on complete intersections will be generalized with regard to these differential forms.

The Hard Lefschetz Theorem and the topology of semismall maps

A line bundle on a complex projective manifold is said to be lef if one of its powers is globally generated and defines a semismall map in the sense of Goresky–MacPherson. As in the case of ample bundles the first Chern class of lef line bundles satisfies the Hard Lefschetz Theorem and the Hodge–Riemann Bilinear Relations. As a consequence, we prove a generalization of the Grauert contractibility criterion: the Hodge Index Theorem for semismall maps, Theorem 2.4.1. For these maps the Decomposition Theorem of Beilinson, Bernstein and Deligne is equivalent to the non-degeneracy of certain intersection forms associated with a stratification. This observation, joint with the Hodge Index Theorem for semismall maps gives a new proof of the Decomposition Theorem for the direct image of the constant sheaf. A new feature uncovered by our proof is that the intersection forms involved are definite.

The structure of a class of finite ramified coverings and canonical forms of analytic matrix-functions in a neighborhood of a ramified turning point.

Let X be the germ of a complex- or real-analytic manifold M at a point x(o) in M, or the henselian germ of an algebraic manifold M over a field k of characteristic zero at a point x(o) in M(k), D subset X a divisor. Under some assumptions on D and its singularities we give a description of the structure, the singularities, and the divisor class group of all finite normal coverings of X ramified over D. Let g : X --> gl(n) be an analytic or a k-algebraic family, respectively, of semisimple matrices, the eigenvalues of which are ramified on D as functions of x in X. Put U = X - D. Using the above results under some quite general assumptions on g and D we construct an irreducible nonsingular variety U(c), a finite etale morphism a(c) : U(c) --> U, and a morphism u(c) : U(c) --> GL(n) (all in the same category as X and g), such that t(c)(x) = u(c)(x)g(x)u(c)(x)(-1) is a diagonal matrix, for all x in U(c). This construction gives, among other things, an extension in a refined form (on the level of U(c)-sections) of the classical one-parameter Perturbation Theory of matrices to the case of many parameters, ramified eigenvalues, not necessarily hermitian matrices, etc. We also prove the stable triviality of the eigenbundles of g on U and vanishing of their Chern classes.

Extending extremal contractions from an ample section

Let E be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section s A GðEÞ whose zero locus, Z¼ðs¼ 0Þ, is a smooth submanifold of the expected dimension dim Xr. We study the problem of extending birational contractions of Z to the ambient variety proving an extension property for blow-ups and we apply our results to classify X as above when Z is a P-bundle on a surface with nonnegative Kodaira dimension.

Fuchsian bispectral operators

The aim of this paper is to classify the bispectral operators of any rank with regular singular points (the infinite point is the most important one). We characterise them in several ways. Probably the most important result is that they are all Darboux transformations of powers of generalised Bessel operators (in the terminology of [4]). For this reason they can be effectively parametrised by the points of a certain (infinite) family of algebraic manifolds as pointed out in [4].

Cohomology Theory in Birational Geometry

This is a continuation of [9], where it was shown that K-equivalent complex projective manifolds have the same Betti numbers by using the theory of p-adic integrals and Deligne's solution to the Weil conjecture. The aim of this note is to show that with a little more book-keeping work, namely by applying Faltings' p-adic Hodge Theory, our p-adic method also leads to the equivalence of Hodge numbers — a result which was previously known via motivic integration.

Reduction of structure group of principal bundles over aprojective manifold with Picard number one

Let X be a complex projective manifold with Pic(X) ≅ Z. Let G be a connected reductive algebraic group. A principal G-bundle will becalled split if it admits a reductionof structure group to a maximal torus of G. This corresponds tothe usual definition of a split vector bundle as a direct sum ofline bundles. Let E be a principal G-bundle on X, and let ρ G→ G* be an injective homomorphism.We prove that if the associated principal G*-bundle E(G*) is split, then E admits a reduction to a Borel subgroupof G. Furthermore, if we assume that X is Fano or has trivialcanonical bundle, then E is also split. We apply this to obtain generalizations for principal bundles of two results about vector bundles on projective spaces. Namely, a principal G-bundle on CP is split if its restriction to a plane CP CPis split. Also, a principal G-bundle E on CP is trivial if there is a point p such that the restriction of E to every line through p is trivial.

Numerical integration of differential algebraic systems and invariant manifolds

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245-257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305-323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to an index-2 problem of Hessenberg form there is an (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation. AMS subject classification: 65L, 34C.

Variétés complexes dont l'éclatée en un point est de Fano

We classify complex projective manifolds X for which there exists a point a such that the blow-up of X at a is Fano. As a consequence, we get that, in dimension greater or equal than three, the quadric is the only complex manifold X for which there exists two distinct points a and b such that the blow-up of X with center { a, b} is Fano. To cite this article: L. Bonavero et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 463–468.

The visual core of a hyperbolic 3-manifold

In this note we introduce the notion of the visual core of a hyperbolic 3-manifold \(N={\bf H}^3/\Gamma\), and explore some of its basic properties. We investigate circumstances under which the visual core \({\cal V}(N')\) of a cover \(N'={\bf H}^3/\Gamma'\) of N embeds in N, via the usual covering map \(\pi: N'\rightarrow N\). We go on to show that if the algebraic limit of a sequence of isomorphic Kleinian groups is a generalized web group, then the visual core of the algebraic limit manifold embeds in the geometric limit manifold. Finally, we discuss the relationship between the visual core and Klein-Maskit combination along component subgroups.

On reducible hyperplane sections of 4-folds

We describe 4-dimensional complex projective manifolds X admitting a simple normal crossing divisor of the form A+B among their hyperplane sections, both components A and B having sectional genus zero. Let L be the hyperplane bundle. Up to exchanging the two components, (X,L,A,B) is one of the following: 1)(X,L) is a scroll over P1 with A itself a scroll and B a fibre, 2)(X,L)=(P2×P2,OP2×P2(1,1)) with A∈|OP2×P2(1,0)|,B∈|OP2×P2(0,1)|,3)X=PP2(V) where V=OP2(1)⊕2⊕OP2(2), L is the tautological line bundle, A=PP2 (OP2(1)⊕2, and B∈π*|OP2(2)| , where π : X→P2 is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.

On reducible hyperplane sections of 4-folds

We describe 4-dimensional complex projective manifolds X admitting a simple normal crossing divisor of the form A+B among their hyperplane sections, both components A and B having sectional genus zero. Let L be the hyperplane bundle. Up to exchanging the two components, (X,L,A,B) is one of the following: 1)(X,L) is a scroll over P1 with A itself a scroll and B a fibre, 2)(X,L)=(P2×P2,OP2×P2(1,1)) with A∈|OP2×P2(1,0)|,B∈|OP2×P2(0,1)|,3)X=PP2(V) where V=OP2(1)⊕2⊕OP2(2), L is the tautological line bundle, A=PP2 (OP2(1)⊕2, and B∈π*|OP2(2)| , where π : X→P2 is the scroll projection. This supplements a recent result of Chandler, Howard, and Sommese.