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1.
We prove that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild
conditions. This results is applied to the study of ramified endomorphisms of Fano manifolds with b
2 = 1. It is conjectured that is the only Fano manifold admitting an endomorphism of degree d ≥ 2, and we verify this conjecture in several cases. An important ingredient is a generalization of a theorem of Andreatta–Wisniewski,
characterizing projective space via the existence of an ample subsheaf in the tangent bundle.
Marian Aprodu was supported in part by a Humboldt Research Fellowship and a Humboldt Return Fellowship. He expresses his special
thanks to the Mathematical Institute of Bayreuth University for hospitality during the first stage of this work. Stefan Kebekus
and Thomas Peternell were supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie” and the DFG-Forschergruppe
“Classification of Algebraic Surfaces and Compact Complex Manifolds”. A part of this paper was worked out while Stefan Kebekus
visited the Korea Institute for Advanced Study. He would like to thank Jun-Muk Hwang for the invitation. 相似文献
2.
Marco Andreatta 《manuscripta mathematica》2003,110(4):505-512
We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector
bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves
on projective manifolds.
Received: 16 May 2002 / Revised version: 18 November 2002 Published online: 3 March 2003
Mathematics Subject Classification (2000): 14E30, 14J40, 14J45 相似文献
3.
Muhammad Zafrullah 《manuscripta mathematica》1993,79(1):225-238
In this paper we investigate projective 4-dimensional manifolds X whose tangent bundles TX are numerically effective and give an almost complete classification. An important technical tool is the “Mori theory” of
projective manifolds X whose canonical bundles KX are not numerically effective. 相似文献
4.
Jean-Pierre Demailly 《Inventiones Mathematicae》1996,124(1-3):243-261
Let be an ample line bundle
on a non singular projective
-fold .
It is first shown that
is very ample for
.
The proof develops an original idea of Y.T. Siu
and is based on a combination of the Riemann-Roch
theorem together with an
improved Noetherian induction technique for the Nadel multiplier ideal
sheaves. In the second part, an effective version of the big Matsusaka
theorem is obtained, refining an earlier version of Y.T. Siu: there is
an explicit polynomial bound
of
degree in the arguments,
such that is very ample for
. The refinement is obtained through
a new sharp upper bound for the dualizing sheaves of algebraic
varieties embedded in projective space.
Oblatum 30-I-1995 & 18-V-1995 相似文献
5.
The purpose of this note is initially to present an elementarybut surprising connectedness principle pertaining to the intersectionof a fixed subvariety X of some ambient space Z with anothersubvariety Y which is mobile (in the sense ofbeing movable, rather than actually moving). It is via thismobility that monodromy enters the picture, permitting the crucialpassage from relative or total-space irreducibilityto absolute or fibrewise connectedness (and sometimesirreducibility). A general form of this principle is given inTheorem 2 below. 1991 Mathematics Subject Classification 14C99,15N05. 相似文献
6.
The paper contains a final identification theorem for the genericK*-groups of finite Morley rank. 相似文献
7.
Nous étudions l'homotopie d'une variétéquasi-projective dans un espace projectif complexe selon laméthode de Lefschetz, c'est-à-dire en considérantses sections par les hyperplans d'un pinceau (tomographie).En particulier, nous aboutissons à un théorèmedu type de Lefschetz qui généralise dans une certainedirection les meilleurs résultats connus dus àHamm, Lê, Goresky et MacPherson. Ce théorèmeest démontré par récurrence sur la dimensionde l'espace projectif ambiant à partir d'un théorèmesur les pinceaux d'axe générique qui constituele résultat principal de l'article. Ce dernier comparela topologie de la variété à celle de sasection par un hyperplan générique du pinceausur la base des comparaisons (section hyperplane générique section par l'axe du pinceau) et (sections hyperplanesexceptionnelles section par l'axe); l'incidence dessingularités est mesurée par un invariant appeléprofondeur homotopique rectifiée globale(analogue global de la notion de profondeur homotopique rectifiéede Grothendieck). We study the homotopy of a quasi-projectivevariety in a complex projective space following Lefschetz'smethod, that is, by considering its sections by the hyperplanesof a pencil (tomography). Specifically, we obtain a theoremof Lefschetz type which generalizes in a certain direction thebest-known results due to Hamm, Lê, Goresky and MacPherson.This theorem is proved by induction on the dimension of theambient projective space with the help of a theorem on pencilswith generic axis which is the main result of the paper. Thelatter compares the topology of the variety with that of itssection by a generic hyperplane of the pencil, on the basisof the following comparisons: section by a generic hyperplanewith section by the axis of the pencil; and sections by theexceptional hyperplanes with section by the axis. The effectof the singularities is measured by an invariant called globalrectified homotopical depth (a global analogue of thenotion of rectified homotopical depth of Grothendieck). E-mail: eyral{at}cmi.univ-mrs.fr 2000 Mathematics Subject Classification:32S50, 14F35, 14F17. 相似文献
8.
Ravi Vakil 《Inventiones Mathematicae》2006,164(3):569-590
We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ? (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some
a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli
spaces.
More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves
in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane
curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological,
and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical
bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies
Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori
reason otherwise.
Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each
with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over
that lifts to ℤ/p7 but not ℤ/p8. (Of course the results hold in the holomorphic category as well.)
It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them
to more tractable deformation spaces via smooth morphisms. The essential starting point is Mn?v’s universality theorem.
Mathematics Subject Classification (2000) 14B12, 14C05, 14J10, 14H50, 14B07, 14N20, 14D22, 14B05 相似文献
9.
On sait associer à certaines structures de Poisson surRn, de 1-jet nul en 0, des actions de R2 sur Rn, donnéespar le rotationnel de leur partie quadratiqueet un autre champ de vecteurs. Lorsque ces actions sont nonrésonantes et hyperboliques, onmontre que ces structures sont quadratisables,en ce sens qu'il existe des coordonnées dans lesquelles,elles sont quadratiques. Dans le cas de la dimension 3, nosrésultats mènent à la non-dégénérescencegénérique des structures de Poisson quadratiquesà rotationnels inversibles. We can associate with some Poisson structures defined on Rnwith a zero 1-jet at zero, actions from R2 on Rn, given by thecurl of their quadratic part and another vectorfield. Assuming that those actions are hyperbolicsand without resonances, we give a normal formfor those structures. On R3, we prove that every quadratic Poissonstructure with invertible curl, is generically non degenerate. 相似文献
10.
Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order 总被引:4,自引:0,他引:4
Bourgain J.; Glibichuk A. A.; Konyagin S. V. 《Journal London Mathematical Society》2006,73(2):380-398
Our first result is a sum-product theorem forsubsets A of the finite field Fp, p prime, providing a lowerbound on max (|A + A|, |A · A|). The second and mainresult provides new bounds on exponential sums 相似文献
11.
12.
Masahiro Ohno 《manuscripta mathematica》1993,81(1):437-443
We give an example of a nondegeneraten-dimensional smooth projective varietyX inP
2n+1 with the canonical bundle ample a varietyX whose tangent variety TanX has dimension less than 2n over an algebraically closed field of any characteristic whenn≥9. This varietyX is not ruled by lines and the embedded tangent space at a general point ofX intersectsX at some other points, so that this yields an affirmative answer to a question of Ciliberto. 相似文献
13.
We make precise some properties of the Hermite function in relationwith the Morse theory introduced by Avner Ash in his papersOn eutactic forms,Canad. J. Math. 29 (1977) 10401054and On the existence of eutactic forms,Bull. LondonMath. Soc. 12 (1980) 192196, and with the cellular decompositionof the space of positive definite quadratic forms. We also establisha link between Ash's and Bavard's mass formulae. 相似文献
14.
We introduce orbifold Euler numbers for normal surfaces withboundary Q-divisors. These numbers behave multiplicatively underfinite maps and in the log canonical case we prove that theysatisfy the BogomolovMiyaokaYau type inequality.Existence of such a generalization was earlier conjectured byG. Megyesi [Proc. London Math. Soc. (3) 78 (1999) 241282].Most of the paper is devoted to properties of local orbifoldEuler numbers and to their computation. As a first application we show that our results imply a generalizedversion of R. Holzapfel's proportionality theorem[Ball and surface arithmetics, Aspects of Mathematics E29 (Vieweg,Braunschweig, 1998)]. Then we show a simple proof of a necessarycondition for the logarithmic comparison theorem which recoversan earlier result by F. Calderón-Moreno, F. Castro-Jiménez,D. Mond and L. Narváez-Macarro [Comment. Math. Helv.77 (2002) 2438]. Then we prove effective versions of Bogomolov's result on boundednessof rational curves in some surfaces of general type (conjecturedby G. Tian [Springer Lecture Notes in Mathematics 1646 (1996)143185)]. Finally, we give some applications to singularitiesof plane curves; for example, we improve F. Hirzebruch's boundon the maximal number of cusps of a plane curve. 2000 MathematicalSubject Classification: 14J17, 14J29, 14C17. 相似文献
15.
Motivated in part by the first author's work [23] on the Weyl-Berryconjecture for the vibrations of fractal drums(that is, drums with fractal boundary), M. L.Lapidus and C. Pomerance [31] have studied a direct spectralproblem for the vibrations of fractal strings(that is, one-dimensional fractal drums) and establishedin the process some unexpected connections with the Riemannzeta-function = (s) in the critical interval0 < s < 1. In this paper we show, in particular, thatthe converse of their theorem (suitably interpreted as a naturalinverse spectral problem for fractal strings, with boundaryof Minkowski fractal dimension D (0,1)) is not true in themidfractal case when D = , but that it is true for all other D in the criticalinterval (0,1) if and only if the Riemann hypothesis is true.We thus obtain a new characterization of the Riemann hypothesisby means of an inverse spectral problem. (Actually, we provethe following stronger result: for a given D (0,1), the aboveinverse spectral problem is equivalent to the partialRiemann hypothesis for D, according to which = (s)does not have any zero on the vertical line Re s = D.) Therefore,in some very precise sense, our work shows that the question(à la Marc Kac) "Can one hear the shape of a fractalstring?" – now interpreted as a suitable converse (namely,the above inverse problem) – is intimately connected withthe existence of zeros of = (s) in the critical strip 0 <Res < 1, and hence to the Riemann hypothesis. 相似文献
16.
S Subramanian 《Proceedings Mathematical Sciences》1989,99(3):197-208
We construct a line bundle on a complex projective manifold (a general ruled variety over a curve) which is not ample, but
whose restriction to every proper subvariety is ample. This example is of interest in connection with ampleness questions
of vector bundles on varieties of dimension greater than one. The method of construction shows that a stable bundle of positive
degree on a curve is ample. The example can be used to show that there is no restriction theorem for Bogomolov stability. 相似文献
17.
A theorem of K. W. Roggenkamp and L. L. Scott shows that fora finite group G with a normal p-subgroup containing its owncentralizer, any two group bases of the integral group ringZG are conjugate in the units of ZpG. Though the theorem presentsitself in the work of others and appears to be needed, thereis no published account. There seems to be a flaw in the proof,because a theorem appearing in the survey [K.W. Roggenkamp, The isomorphism problem for integral grouprings of finite groups, Progress in Mathematics 95 (Birkhäuser,Basel, 1991), pp. 193--220], where the main ingredients of aproof are given, is false. In this paper, it is shown how toclose this gap, at least if one is only interested in the conclusionmentioned above. Therefore, some consequences of the resultsof A. Weiss on permutation modules are stated. The basic stepsof which any proof should consist are discussed in some detail.In doing so, a complete, yet short, proof of the theorem isgiven in the case that G has a normal Sylow p-subgroup. 2000Mathematical Subject Classification: primary 16U60; secondary20C05. 相似文献
18.
The paper is an addendum to D. Andrica and L. Funar, Onsmooth maps with finitely many critical points, J. LondonMath. Soc. (2) 69 (2004) 783800. 相似文献
19.
This paper investigates the tightness property of the capacityinduced by the reduction operator with respect to the resolventof a right Markov process. Tightness is verified in two particularsituations: under the weak duality hypothesis,and if a substitute for the axiom of polarity for thedual theory holds. In the second context, the quasi-continuityproperty for the excessive functions is derived. These are extensionsand improvements of results of Lyons and Röckner, Ma andRöckner, Le Jan, and Fitzsimmons, mainly obtained in thecontext of Dirichlet forms. 2000 Mathematics Subject Classification31C25, 60J45 (primary), 31C15, 60J40 (secondary). 相似文献
20.
Borovik Alexandre V.; Burdges Jeffrey; Nesin Ali 《Journal London Mathematical Society》2008,77(1):240-252
There is a longstanding conjecture, due to Gregory Cherlin andBoris Zilber, that all simple groups of finite Morley rank aresimple algebraic groups. One of the major theorems in the areais Borovik's trichotomy theorem. The trichotomyhere is a case division of the generic minimal counterexampleswithin odd type, that is, groups with a large and divisibleSylow° 2-subgroup. The so-called uniqueness casein the trichotomy theorem is the existence of a proper 2-generatedcore. It is our aim to drive the presence of a proper 2-generatedcore to a contradiction, and hence bind the complexity of theSylow° 2-subgroup of a minimal counterexample to the Cherlin–Zilberconjecture. This paper shows that the group in question is aminimal connected simple group and has a strongly embedded subgroup,a far stronger uniqueness case. As a corollary, a tame counterexampleto the Cherlin–Zilber conjecture has Prüfer rankat most two. 相似文献