Showing papers on "Complex dimension published in 2001"

Bochner-Kahler metrics

Abstract: A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kahler metric will be given. It is shown that any Bochner-Kahler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kahler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kahler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kahler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kahler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kahler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.

A splitting theorem for kähler manifolds whose ricci tensors have constant eigenvalues

Abstract: It is proved that a compact Kahler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kahler–Einstein manifolds. A stronger result is established for the case of Kahler surfaces. Without the compactness assumption, irreducible Kahler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situations: there are homogeneous examples of any complex dimension n ≥ 2 with one eigenvalue negative and the other one positive or zero; there are homogeneous examples of any complex dimension n ≥ 3 with two negative eigenvalues; there are non-homogeneous examples of complex dimension 2 with one of the eigenvalues zero. The problem of existence of Kahler metrics whose Ricci tensor has two distinct constant eigenvalues is related to the celebrated (still open) conjecture of Goldberg [24]. Consequently, the irreducible homogeneous examples with negative eigenvalues give rise to complete Einstein strictly almost Kahler metrics of any even real dimension greater than 4.

On the Betti Numbers of Irreducible Compact Hyperkähler Manifolds of Complex Dimension Four

Abstract: The study of higher dimensional hyperkahler manifolds has attracted much attention: we have [Wk], [Bg1,2,3,4], [Fj1,2], [Bv1], [Vb1,2], [Sl1,2], [HS], [Huy], [Gu3,4,5] etc. It is evident that there are only a few known examples of these manifolds and the obvious question is: can we classify them as in the case of complex dimension 4? The Riemann-Roch formula plays an important role in the surface case, which yields K-3 surfaces as the only irreducible examples. However, for the higher dimensional case, the Riemann-Roch formula is not enough to give a picture of both the Hodge diamond and the existence of holomorphic sections of line bundles. In [Gu5], we combined the results of the Riemann-Roch formula in [Sl1,2] (see also [LW]) and the representations generated by the Kahler classes (see [Vb2], [LL], [Bg4]) to give a picture of the Hodge diamonds of irreducible compact hyperkahler manifolds of complex dimension 4. Theorem 1 (reproduced here) gives an upper bound b2 ≤ 23 for the second Betti number and was obtained independently by Beauville [Bv2] (unpublished). He kindly let me publish alone. The bound is obtained by applying Verbitsky’s work. In [HS] there is also an upper bound for the Euler characteristic but there seems as yet no lower bound, nor any bound for the Betti numbers. However, the method in [HS] actually gives us a way to calculate what we call generalized Chern numbers (which are only defined on hyperkahler manifolds) by Rozansky-Witten invariants, some of which in turn can be calculated as Chern numbers. Combining this approach with the method in [Bg4] we obtain an inequality in the opposite direction to the one in [HS] and apply it to our situation. Surprisingly, once we already have the bound on b2 this gives a more natural and much stronger inequality than the one we manipulated from the Riemann-Roch formula in [Gu5]. Therefore, we obtain our: Main Theorem. If M is an irreducible compact hyperkahler manifold of complex dimension 4, then 3 ≤ b2 ≤ 23. Moreover,

On the automorphism groups of hyperbolic manifolds

Abstract: We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n e 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism groups of dimension $\ge n^2+2$.

Closed conformal vector fields and Lagrangian submanifolds in complex space forms

Abstract: We study a wide family of Lagrangian submanifolds in non flat complex space forms that we will call pseudoumbilical because of their geometric properties. They are determined by admitting a closed and conformal vector field X such that X is a principal direction of the shape operator AJX , being J the complex structure of the ambient manifold. We emphasize the case X = JH, where H is the mean curvature vector of the immersion, which are known as Lagrangian submanifolds with conformal Maslov form. In this family we offer different global characterizations of the Whitney spheres in the complex projective and hyperbolic spaces. Let M be a Kaehler manifold of complex dimension n. The Kaehler form Ω on M is given by Ω(v, w) = 〈v, Jw〉, being 〈, 〉 the metric and J the complex structure on M . An immersion φ : M −→ M of an n-dimensional manifold M is called Lagrangian if φ∗Ω ≡ 0. This property involves only the symplectic structure of M . In this family of Lagrangian submanifolds, one can study properties of the submanifold involving the Riemannian structure of M . One ∗Research partially supported by a DGICYT grant No. PB97-0785.

Einstein deformations of hyperbolic metrics

Abstract: The simplest nontrivial examples of Einstein metrics are certainly rank one symmetric spaces. We are interested in those of negative curvature, that is the hyperbolic spaces KH (m > 2), where K is the field of real numbers (R), complex numbers (C), quaternions (H) or the algebra of octonions (O); in the last case we have only the Cayley hyperbolic plane OH. They are the noncompact duals of the projective spaces KP. We normalize the metric so that the maximum of the sectional curvature is −1. We denote by d the real dimension of K (so d = 1, 2, 4 or 8) and by n = md the real dimension of KH. The boundary sphere Sn−1 of a hyperbolic space carries a rich geometric structure, namely a conformal Carnot-Caratheodory metric. Let see this first in the real and complex examples. The real hyperbolic space (with constant sectional curvature −1) is the unit ball B in R, with the metric

Chern numbers of almost complex manifolds

Abstract: It is shown that any system of numbers that can be realised as the system of Chern numbers of an almost complex manifold of dimension 2n, n > 2, can also be realised in this way by a connected almost complex manifold. This answers an old question posed by Hirzebruch. Let -F(n) denote the number of partitions of the natural number n. A theorem of Milnor (cf. [5]) states that a system of ir(n) numbers can be realised as the system of Chern numbers of an almost complex manifold of (real) dimension 2rn if and only if it can be realised in this way by some algebraic manifold of (complex) dimension n belonging to some class A. (Manifolds are understood to be oriented, differentiable, and compact without boundary.) The class A is generated (under cartesian product and disjoint union) by the complex projective spaces, the hypersurfaces H(r,t) of double degree (1, 1) in Cpr x CPS with r, s > 1, and certain algebraic manifolds which realise the negative of the Chern numbers of the manifolds already listed. Thus, at least in principle, it is known which systems of ir(n) numbers can be realised as the Chern numbers of a 2n-dimensional almost complex manifold. In low dimensions, a complete set of restrictions is given as follows (cf. [5]): n=1: c1lOmod2, n = 2: c2+ C2-i0 mod 12, n = 3: C1C2=0 mod 24, c3lO-C3-0 mod 2, n =4: -c4l + 4cC2 +1CC3c+ 3C2 Cc4= _ mod720, 2c4 + cIc2 _ 0 mod 12, cic32c4 _ 0 mod 4. In [5] Hirzebruch raised the question whether a system of ir(n) numbers satisfying the necessary restrictions can be realised as the system of Chern numbers of a connected almost complex manifold of dimension 2n, and speculated that the connectedness assumption might impose additional inequalities between the Chern numbers. If the question is asked for complex or algebraic manifolds, there are indeed additional restrictions on the Chern numbers in the form of inequalities, as was first shown by Van de Ven [13] for complex dimension 2 (cf. [2]). In that paper Van de Ven also proved that no additional restrictions occur for connected almost complex manifolds of real dimension 4. Received by the editors May 2, 2000. 2000 Mathematics Subject Classification. Primary 57R20, 32Q60. (D2001 American Mathematical Society

Equivariant deformations of meromorphic actions on compact complex manifolds

Abstract: Meromorphicity is the most basic property for holomorphic ${\mathbb C}^*$ -actions on compact complex manifolds. We prove that the meromorphicity of ${\mathbb C}^*$ -actions on compact complex manifolds are not necessarily preserved by small deformations, if the complex dimension of complex manifolds is greater than two. In contrast, we also show that the meromorphicity of ${\mathbb C}^*$ -actions on compact complex surface depends only on the topology (the first Betti number) of the surface. We construct such examples of dimension greater than two by studying an equivariant deformation of certain complex threefold, so called a twistor space.

Napoleon in isolation

Abstract: Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R 2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic 3-manifolds, where hyperbolic Dehn surgeries on some collection of cusps leave the geometric structure at some other collection of cusps unchanged. 1. GEOMETRIC ISOLATION 1.1. Definition. Let M be a complete, finite volume hyperbolic 3-manifold with n torus cusps, which we denote c1, . . , cn. The following definition is found in [7]: Definition 1.1. A collection of cusps Cjj . ... ) Cjm is geometrically isolated from a collection cil, .. , cin if any hyperbolic Dehn surgery on any collection of the Cik leaves the geometric structure on all the cj1 invariant. Note that this definition is not symmetric in the collections cik and cj,, and in fact there are examples which show that a symmetrized definition is strictly stronger (see [7]). More generally, we can ask for some prescribed set of fillings on the Cik which leave the cj, invariant. Generalized (non-integral) hyperbolic surgeries on a cusp are a holomorphic parameter for the space of all (not necessarily complete) hyperbolic structures on M with a particular kind of allowable singularities (i.e. generalized cone structures) in a neighborhood of the complete structure. Moreover, the complex dimension of the space of geometric shapes on a complete cusp is 1. Consequently, for dimension reasons whenever n > m there will be families of generalized surgeries leaving the geometric structures on the cjl invariant. There is no particular reason to expect, however, that any of these points will correspond to an integral surgery on the Cik. When there is a 1 complex dimensional holomorphic family of isolated generalized surgeries which contains infinitely many integral surgeries, we say that we have an example of an isolation phenomenon. Neumann and Reid describe other qualities of isolation in [7] including the following: Definition 1.2. A collection of cusps cjl .... , Cjm is strongly isolated from a collection cil . .. . Cim if after any hyperbolic Dehn surgeries on ally collection of the cjl, Received by the editors June 15, 1999 and, in revised form, March 6, 2000. 2000 Mathematics Subject Classification. Primary 57M50, 57M25. (?)2001 American Mathematical Society

Hausdorff Dimension of Non-Hyperbolic Repellers. I: Maps with Holes

Abstract: This is the first paper in a two-part series devoted to studying the Hausdorff dimension of invariant sets of non-uniformly hyperbolic, non-conformal maps Here we consider a general abstract model, that we call piecewise smooth maps with holes We show that the Hausdorff dimension of the repeller is strictly less than the dimension of the ambient manifold Our approach also provides information on escape rates and dynamical dimension of the repeller

Eigenvalues of the Kählerian Dirac Operator

Abstract: We get estimates on the eigenvalues of the Kahlerian Dirac operator in terms of the eigenvalues of the scalar Laplace–Beltrami operator. In odd complex dimension, these estimates are sharp, in the sense that, for the first eigenvalue, they reduce to Kirchberg's inequality.

Non-Linear Sigma Model on Conifolds

Abstract: Explicit solutions to the conifold equations with complex dimension $n=3,4$ in terms of {\it{complex coordinates (fields)}} are employed to construct the Ricci-flat K\"{a}hler metrics on these manifolds. The K\"{a}hler 2-forms are found to be closed. The complex realization of these conifold metrics are used in the construction of 2-dimensional non-linear sigma model with the conifolds as target spaces. The action for the sigma model is shown to be bounded from below. By a suitable choice of the 'integration constants', arising in the solution of Ricci flatness requirement, the metric and the equations of motion are found to be {\it{non-singular}}. As the target space is Ricci flat, the perturbative 1-loop counter terms being absent, the model becomes topological. The inherent U(1) fibre over the base of the conifolds is shown to correspond to a gauge connection in the sigma model. The same procedure is employed to construct the metric for the resolved conifold, in terms of complex coordinates and the action for a non-linear sigma model with resolved conifold as target space, is found to have a minimum value, which is topological. The metric is expressed in terms of the six real coordinates and compared with earlier works. The harmonic function, which is the warp factor in Type II-B string theory, is obtained and the ten-dimensional warped metric has the $AdS_{5}\times X_{5}$ geometry.

Effective freeness of adjoint line bundles

Abstract: In this note we establish a new Fujita-type effective bound for the base point freeness of adjoint line bundles on a compact complex projective manifold of complex dimension $n$. The bound we obtain (approximately) differs from the linear bound conjectured by Fujita only by a factor of the cube root of $n$. As an application, a new effective statement for pluricanonical embeddings is derived.

Arrangement managing device and arrangement simulation method

Abstract: PROBLEM TO BE SOLVED: To provide a device and a method for management with which suitable arrangement management is possible by displaying a symbol reflecting a real dimensional relation corresponding to applied layout. SOLUTION: An arrangement managing device for synthesizing and displaying a picture showing the layout of a managing object and the symbols of elements to be arranged on the layout on the display screen of a display device is provided with a real dimension setting part for finding a dimension per display pixel of the display screen as unit pixel dimension on the basis of the real dimension between two points on the layout, a symbol preparing part for preparing the symbol, and a symbol arranging part for displaying the symbol at the designated arranging position on the layout on the basis of the real dimension while utilizing the unit pixel dimension of the display pixel.

Rigidity at infinity for even-dimensional asymptotically complex hyperbolic spaces

Abstract: Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dimension.

Rank one micro-locally free Ê-modules and non-integrable connexions in dimension two

Abstract: In [3], it is shown that above a complex manifold X of dimension ⩾3 , every micro-locally free E -module of rank 1 is of the form E ⊗ π −1 O π −1 L for a line bundle L on X . This result is false in dimension 2 , and the purpose of this Note is to determine the structure of the micro-locally free E - and D -modules of rank 1 in this case. One of the main result is the description of micro-locally free D -modules of rank 1 in terms of certain vector bundles on X with a non-integrable connexion.

The dual Kaehler cone of compact Kaehler threefolds

Abstract: We consider a compact Kaehler manifold whose dual Kaehler cone contains a rational interior point. The general problem we have in mind is how far the manifold is from being projective; i.e. we ask for the algebraic dimension. We prove e.g. that if the interior point is represented by an effective curve in dimension 3, then the manifold has algebraic dimension at least 2 unless the variety is a so-called simple non-Kummer 3-fold. We also investigate 3-folds containing curves with ample normal bundle. It seems likely that examples with algebraic dimension 2 really exist.

On Fractals in Information Systems: The First Step

Abstract: We introduce the notion of a fractal in an information system and we define a dimension function of a fractal in an information system parallel to the Minkowski dimension in Euclidean spaces. We prove basic properties of this new dimension.