Ample vector bundles and Bordiga surfaces

Let X be a smooth complex projective variety and let Z ? X be a smooth surface, which is the zero locus of a section of an ample vector bundle ? of rank dimX – 2 ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is a very ample line bundle and assume that (Z, H _Z ) is a Bordiga surface, i.e., a rational surface having (?², ?? (4)) as its minimal adjunction theoretic reduction. Triplets (X, ?, H) as above are discussed and classified. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the conormal bundle of a G-stable subvariety

Let G be a reductive algebraic group and X a smooth G-variety. For a smooth locally closed G-stable subvariety M⊂X, we prove that the G-complexity of the (co)normal bundle of M is equal to the G-complexity of X. In particular, if X is spherical, then all (co)normal bundles are again spherical G-varieties. If X is a G-module with finitely many orbits, the closures of the conormal bundles of the orbits coincide with the irreducible components of the commuting variety. We describe properties of these closures for the representations associated with short gradings of simple Lie algebras. Received: 22 April 1998     

On codimension one submersions of euclidean spaces

Summary It is shown that when (n–1) first integrals of a dynamical systemX inR ⁿ are known (and they are independent at any point ofR ⁿ ) then one can have (n3) that certain orbits ofX are diffeomorphic to circles and others diffeomorphic to straight-lines. An analytical criterion is also given (involving only the first derivatives of the first integrals) in order thatall the orbits be diffeomorphic to straight lines. Therefore the criterion is sufficient in order to avoid the presence of geometrical chaos among the orbits of the dynamical systemX.     

Donagi–Morrison’s examples on 2-elementary K3 surfaces

Let X be an algebraic K3 surface, and let L be a base point free and big line bundle on X. If X admits a map of degree 2 to the projective plane branched over a smooth sextic and L is the pullback of the line bundle O_\mathbbP²(3),{\mathcal{O}_{\mathbb{P}^{2}}(3),} then the gonality of the smooth curves of the complete linear system |L| is not constant. The polarized K3 surface (X, L) consisting of the K3 surface X and the line bundle L is called Donagi–Morrison’s example. In this paper, we give a necessary and sufficient condition for the polarized K3 surface (X, L) consisting of a 2-elementary K3 surface X and an ample line bundle L to be Donagi–Morrison’s example.     

Flot horosphérique des repères sur les variétés hyperboliques de dimension 3 et spectre des groupes kleiniens

Let Γ be a Kleinian group. The action of the upper unipotent subgroup by right multiplication on Γ\PSL(2,ℂ) is conjugated to a two-dimensional flow on the frame bundle of the hyperbolic manifold Γ\³. We show that the topology of orbits (compactness, divergence, density) is analogous to the topology of the horospherical foliation on hyperbolic manifolds. In order to study dense orbits, we prove a result of "non-arithmeticity" of the spectrum of Kleinian groups. Received: 8 January 2002     

BLOWING UP PROJECTIVELY NORMAL SPACE CURVES AND VERY AMPLE DIVISORS

Let C ? P ³ be a projectively normal curve and π : X (C) → P ³ the blowing-up of C,E the exceptional divisor of π and H ∈(X (C)) the total transform of the hyperplane line bundle. Let H (C,t) be the Hilbert function of C and Δ² H its second difference function. Let σ be the minimal integer such that Δ² H (C σ) = 0. Here we prove that is C has no σ-secant line, then |σ H – E| is very ample.     

On Rational Varieties Smooth Except at a Seminormal Singular Point

We construct a class of projective rational varieties X of any dimension m ≥ 1, which are smooth except at a point O, with the projective space ? ^m as normalization, having smooth branches, and reduced projectivized tangent cone in O. The Hilbert function of X is considered and is explicitly computed when the point O is seminormal. Indeed, we study seminormality, obtaining necessary and sufficient conditions for O to be seminormal and show that in such case the tangent cone is reduced and seminormal.     

Line bundles on non-primary Hopf manifolds

Let X be a Hopf manifolds with Abelian fundamental group, L any flat line bundle on X, we give a formula for computing explicitly the cohomology Hq(X,Ωp(L) using the method of group action and the generalized Douady sequence.     

Exponential rarefaction of real curves with many components

Given a positive real Hermitian holomorphic line bundle L over a smooth real projective manifold X, the space of real holomorphic sections of the bundle L ^d inherits for every d∈ℕ^∗ a L ²-scalar product which induces a Gaussian measure. When X is a curve or a surface, we estimate the volume of the cone of real sections whose vanishing locus contains many real components. In particular, the volume of the cone of maximal real sections decreases exponentially as d grows to infinity.     

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.     

Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Generalization of a criterion for semistable vector bundles

It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that both H⁰(X,EF) and H¹(X,EF) vanishes. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on X such that Hⁱ(X,EF)=0 for all i. We also give an explicit bound for the rank of F.     

Toute variété magnifique est sphérique

LetG be a (connected) reductive group (over C). An algebraicG-varietyX is called “wonderful”, if the following conditions are satisfied:X is (connected) smooth and complete;X containsr irreducible smoothG-invariant divisors having a non void transversal intersection;G has 2 ^r orbits inX. We show that wonderful varieties are necessarily spherical (i.e., they are almost homogeneous under any Borel subgroup ofG).      

On the singularities of foliations and of vector bundle maps

LetF be a (smooth) Γ _q ^∞ -stucture (often called a codimension-q Haefliger structure) on a compact manifoldX ⁿ . Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onX ⁿ . An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofX ⁿ , and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.     

Seshadri-exceptional foliations

 The maximal Seshadri number μ(L) of an ample line bundle L on a smooth projective variety X measures the local positivity of the line bundle L at a general point of X. By refining the method of Ein-Küchle-Lazarsfeld, lower bounds on μ(L) are obtained in terms of L ⁿ , n=dim(X), for a class of varieties. The main idea is to show that if a certain lower bound is violated, there exists a non-trivial foliation on the variety whose leaves are covered by special curves. In a number of examples, one can show that such foliations must be trivial and obtain lower bounds for μ(L). The examples include the hyperplane line bundle on a smooth surface in ℙ³ and ample line bundles on smooth threefolds of Picard number 1. Received: 29 June 2001 / Published online: 16 October 2002 RID="⋆" ID="⋆" Supported by Grant No. 98-0701-01-5-L from the KOSEF. RID="⋆⋆" ID="⋆⋆" Supported by Grant No. KRF-2001-041-D00025 from the KRF.     

Berezin transform on¶compact Hermitian symmetric spaces

Let X=G ^* be a compact Hermitian symmetric space. We study the Berezin transform on L ²(X) and calculate its spectrum under the decomposition of L ²(X) into the irreducible representations of G ^*. As applications we find the expansion of powers of the canonical polynomial (Bergman reproducing kernel for the canonical line bundle) in terms of the spherical polynomials on X, and we find the irreducible decomposition of tensor products of Bergman spaces on X. Received: 10 September 1996 / Revised version: 10 September 1997     

False Hyperelliptic Surfaces with Section

Let X be a nonsingular relatively minimal projective surface over an algebraically closed field of characteristic p > 0. We call X a false hyperelliptic surface if X satisfies the following conditions: (1) c₂(X) = 0, c₁(X)² = 0, dim Alb (X) = 1, and (2) All fibres of the Albanese mapping of X are rational curves with only one cusp of type xp^v + yⁿ = 0. In this article, we consider a false hyperelliptic surface whose Albanese mapping has a cross-section. We prove that every false hyperellyptic surface with section arises from an elliptic ruled surface and that every false hyperelliptic surface has an elliptic fibration with multiple fibre. Moreover, we construct an example of false hyperelliptic surface with section, whose elliptic fibration has a multiple fibre of supersingular elliptic curve of multiplicity p^v (v > 1).     

Ample Vector Bundles with Small g − q

Let X be a smooth complex projective variety and let Z ? X be a smooth submanifold of dimension ≥ 2, which is the zero locus of a section of an ample vector bundle ? of rank dim X ? dim Z ≥ 2 on X. Let H be an ample line bundle on X, whose restriction H _Z to Z is generated by global sections. The structure of triplets (X,?,H) as above is described under the assumption that the curve genus of the corank-1 vector bundle ? ⊕ H ^{⊕ (dim Z?1)} is ≤ h ¹( _X ) + 2.     

Regular Action in Rings

Let R be a ring with identity, X(R) the set of all nonzero non-units of R and G(R) the group of all units of R. By considering left and right regular actions of G(R) on X(R), the following are investigated: (1) For a local ring R such that X(R) is a union of n distinct orbits under the left (or right) regular action of G(R) on X(R), if J ⁿ  ≠ 0 = J ⁿ⁺¹ where J is the Jacobson radical of R, then the set of all the distinct ideals of R is exactly {R, J, J ²,…, J ⁿ , 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ring R is left (and right) duo ring. (3) For the full matrix ring S of n × n matrices over a commutative ring R, the number of orbits under left regular action of G(S) on X(S) is equal to the number of orbits under right regular action of G(S) on X(S); the result also holds for the ring of n × n upper triangular matrices over R.     

Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO _Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ³. For almost all such schemes the graded Betti numbers in ℙ³ depend only on the Hilbert function ofX in ℙ³. Work done with financial support of M.U.R.S.T., while the authors were members of C.N.R.