Complex Manifold Of Dimension Research Articles

A generalization of Cayley submanifolds

Given a Kähler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4 whose Kähler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that: (a) if the ambient manifold is Calabi-Yau, the minimal Cayley submanifolds are exactly the Cayley submanifolds as defined by Harvey and Lawson [HL1]; (b) if the ambient is a Kähler-Einstein manifold of nonzero scalar curvature, then minimal Cayley submanifolds have to be either complex or Lagrangian.

Holomorphic maps into complex ellipsoids which are kobayashi isometries at one point

Let M be a connected taut complex manifold of dimension n and let D be a bounded balanced pseudoconvex domain in with continuous Minkowski function. Assume that there exist a finite number of complex hyperplanes H j through the origin such that every point of is an extreme point for . Let f:→D be a holomorphic map. Let p be a point of M. Assume that f(p) = 0 and that df p is an isometry for the infinitesimal Kobayashi metric. In this case, we will show that f is a biholomorphic map.

First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kahler spin manifold (W,g) with positive scalar curvature and has introduced, in this context, the notion of Kahler twistor-spinor. We prove here that if dimC W = p ≥ 4 is even, in the limiting case, (W, g) is the Kahler product of an odd-dimensional limiting case compact Kahler spin manifold of complex dimension (p-1), by a flat Kahler manifold which is a compact quotient of C.

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy–Riemann equations on affine spaces.

Analytic Discs Attached to Manifolds with Boundary

Let X be a complex manifold of dimension n, M a closed half-space with boundary M, A an analytic disc of X to M, tangent to M at some point z0 of dA n M, and intersecting M + in any neighbourhood of z0. Then holomorphic functions extend from M + to a full neighborhood of z0. This theorem refines the results of [1] where the boundary dA (instead of the whole A) was supposed to intersect M . The argument of the proof consists in constructing a (closed) manifold with boundary W, contained in the envelop of holomorphy of M and such that A c W but A <£ d W. In this situation it is easy to find a new small disc A i c A with 8A l <£ dW. We are therefore in a situation similar to [1], and get the conclusion by exhibiting a disc transversal to d W at z0. Extension of holomorphic functions by the aid of tangent discs attached to M and of 0 is a particular case of a general theorem of wedge extendibility of CR-functions by A. Tumanov; the new part of our theorem is that no assumptions on defect are made. This paper is tightly inspired to the results and the techniques by A. Tumanov [7]. We also owe to A. Tumanov a great help during private communications.

Holomorphic mappings into bounded complete reinhardt domains of holomorphy which are kobayashi isometries at one point

Let M be a connected taut complex manifold of dimension n such that the set H(M) of holomorphic functions on M separates points of M and let D be a bounded complete Reinhardt domain of holomorphy in Let f:M→D be a holomorphic mapping. Let p be a point of M. Assume that f(p)=0 and that df p is an isometry for the infinitesimal Kobayashi metric, in this case, we will show that f is a biholomorphic mapping

Prepotential and the Seiberg-Witten theory

Construction of the prepotential in the Seiberg-Witten theory/the Whitham hierarchy is presented. Consideration begins from the notion of quasi-classical τ-functions, which uses as an input a family of complex spectral curves with a meromorphic differential dS subject to the constraint ∂dS/ ∂(moduli) = holomorphic, and which gives as an output a homogeneous prepotential on extended moduli space. The reversed construction is then discussed, which is straightforwardly generalizable from spectral curves to certain complex manifolds of dimension d > 1 (like K3 and CY families). Examples of particular N = 2 supersymmetric gauge models are considered from the point of view of this formalism. We discuss the similarity between the WP 1,1,2,2,6 12 Calabi-Yau model with h 21 = 2 and the 1d SL(2) Calogero/Ruijsenaars model by deriving the respective Picard-Fuchs equations. We, however, stop short of the claim that they belong to the same Whitham universality class beyond the conifold limit.

On genericity for holomorphic curves in four-dimensional almost-complex manifolds

We consider spaces of immersed (pseudo-)holomorphic curves in an almost complex manifold of dimension four. We assume that they are either closed or compact with boundary in a fixed totally real surface, so that the equation for these curves is elliptic and has a Fredholm index. We prove that this equation is regular if the Chern class is ≥ 1 (in the case with boundary, if the ambient Maslov number is ≥ 1). Then the spaces of holomorphic curves considered will be manifolds of dimension equal to the index.

On the Spectral Rigidity of Hopf Manifolds

We characterize Hopf manifolds of complex dimension 3 (resp. 2) in the class of compact locally conformal Kaehler manifold ( resp. Hermitian manifold ) by the spectrum of the Laplacian acting on functions.

On the removable singularities for meromorphic mappings

If $E$ is a closed subset of locally finite Hausdorff $(2n-2)$-measure on an $n$-dimensional complex manifold $\Omega$ and all the points of $E$ are nonremovable for a meromorphic mapping of $\Omega \setminus E$ into a compact Kahler manifold, then $E$ is a pure $(n-1)$-dimensional complex analytic subset of $\Omega$

Higher dimensional uniformisation and W-geometry

We formulate the uniformisation problem underlying the geometry of W n -gravity using the differential equation approach to W-algebras. We construct W n -space (analogous to superspace in supersymmetry) as an ( n − 1)-dimensional complex manifold using isomonodromic deformations of linear differential equations. The W n -manifold is obtained by the quotient of a Fuchsian subgroup of PSL( n, R ) which acts properly discontinuously on a simply connected domain in C P n − 1. The requirement that a deformation be isomonodromic furnishes relations whic enable one to convert non-linear W-diffeomorphisms to (linear) diffeomorphisms on the W n -manifold. We discuss how the Teichmüller spaces introduced by Hitchin can then be interpreted as the space of complex structures or the space of projective structures with real holonomy on the W n -manifold. The projective structures are characterised by Halphen invariants which are appropriate generalisations of the Schwarzian. This construction will work for all “generic” W-algebras.

Moduli spaces of compact complex submanifolds of complex fibered manifolds

In 1962 Kodaira[11] proved that ifX ↪ Y is a compact complex submanifold with normal bundle N such that H1(X, N) = 0, then X belongs to a locally complete family {Xt: t ∈ M} of complex submanifolds Xt of Y with the moduli space M being a (dimcH0(X, N))-dimensional complex manifold, and there exists a canonical isomorphismbetween the tangent space of M at a point t ∈ M and the space of all global sections of the normal bundle Nt of the embedding Xt ↪ Y.

Foliations with complex leaves

Let X be a smooth foliation whose leaves are complex manifolds of dimension n and let D be the sheaf of germs of smooth functions, holomorphic along the leaves (namely the germs of CR-functions on X). The aim of this paper is to study the ringed space ( X, D) and in particular: function theory for the algebra D( X) and cohomology with values in D. We introduce the notion of q-completeness of a foliation and we prove a general result about the embedding of a real analytic q-complete foliation. As a consequence we obtain an approximation theorem for CR-functions and an embedding theorem for real analytic Stein foliations. Using these results, we prove a vanishing theorem for the groups H 0( X, D) when X is a real analytic 1-complete foliation.

An application of a theorem of Huber in holomorphic foliation theory

Introduction. In the following we are concerned with 1-codimensional holomorphic foliations on a connected paracompact complex manifold X of dimension n. Let U be an open subset of X and f : U→C a holomorphic submersion onto a 1-dimensional complex manifold C. f : U→C is called a regular local holomorphic foliation of codimension one. Two regular local holomorphic foliations f1 : U1 → C1, f2 : U2 → C2 are called compatible, if for every x ∈ U1 ∩ U2 there exist an open neighborhood W ⊂ U1∩U2 of x and a biholomorphic mapping g : f1(W )→ f2(W ) such that f2 = g ◦ f1 on W . A (global) regular holomorphic foliation F of codimension one on X is a system {fj : Uj → Cj : j ∈ J} of compatible regular local holomorphic foliations of codimension one such that ⋃ j∈J Uj = X. We identify two regular foliations F1,F2 on X if every local foliation of F1 is compatible with every local foliation of F2. In the following we assume that every regular foliation F on X contains every local foliation which is compatible with those of F . By a theorem of Frobenius there is a one to one correspondence between the system of regular holomorphic foliations F of codimension 1 on X and the system of subsheaves Ω′ of the sheaf Ω of holomorphic Pfaffian forms on X such that Ω1/Ω′ is a locally free O-sheaf of rank n− 1 and ω ∧ dω = 0 for every ω ∈ Ωx, x ∈ X. Let F be a regular holomorphic foliation on X of codimension 1. A subset L of X is called a local leaf or a plaque of F , if there is a local holomorphic foliation f : U → C of F such that L is a connected component of a fiber of f . The relatively open subsets of the local leaves of F constitute a base of a topology T on X. T is called the F-topology. (X, T ) is a complex manifold of dimension n − 1. It is not connected. The connected components L of (X, T ) are called

Critical superstring vacua from noncritical manifolds: A novel framework for string compactification and mirror symmetry.

A new framework is found for the compactification of supersymmetric string theory. It is shown that the massless spectra of critical string vacua with central charge c=3${\mathit{D}}_{\mathrm{crit}}$ can be derived from manifolds of complex dimension ${\mathit{D}}_{\mathrm{crit}}$+2(Q-1),Q\ensuremath{\ge}1, whose first Chern class is quantized in a particular way. This new class is more general than that of Calabi-Yau manifolds because it contains spaces corresponding to vacua with no K\"ahler deformations, i.e., no antigenerations, thus providing mirrors of rigid Calabi-Yau manifolds. The constructions introduced here lead to new insights into the relation between Landau-Ginzburg vacua on the one hand and Calabi-Yau manifolds on the other.

Neighborhood of a Rational Curve with a Node

Let C be an irreducible compact analytic set of dimension 1 in a complex manifold of dimension 2. Let (C) denote the self-intersection number of C. It is well known that C has a strongly pseudoconvex neighborhood if and only if (C) 0. When (C)=0, this topological condition alone is insufficient to derive analytic conclusions. For a smooth curve C with (C) = 0, we obtained some conditions for the existence of a fundamental system of strongly pseudoconcave or pseudoflat neighborhoods in [8] (see also Neeman [5]). It will be natural to investigate such complex analytic properties in the case where C has singularities. In the present paper we treat one of the simplest cases of such C. In the sequel, C will always stand for a rational curve with only one node (ordinary double point). To state the result, we note first that the Picard variety of C, i.e., the set of all topologically trivial line bundles over C, is isomorphic to H\C9 C*)^C* as multiplicative group (see Lemma 1). Suppose that C is imbedded in a complex manifold S of dimension 2 and (C)=0. Let [C] denote the line bundle over S associated to the divisor C. The normal bundle N of C is defined to be the restriction [C] | C of [C] to C. By the assumption, N is a topologically trivial line bundle over C. Let aeC* be the number corresponding to N by the above isomorphism. Let d(p) denote the distance of p e S from the curve C with respect to some Riemannian metric

On characteristic forms on complex manifolds

Let X be a complex manifold of dimension n ⩾ 2. In this paper, we define Weyl forms P k ( π), 0 ⩽ k ⩽ n, a kind of characteristic forms, on X by means of a C′-projective connection π and prove the formula c k= ∑ j=0 k n+1−j k−j ((n+1) −c 1(θ)) k−j P j(π) 0⩽k⩽n where the c k ( θ) are the Chern forms defined by a suitable C′-affine connection θ (Theorem 3.1). For example, we have P 0( π) = 1. p 1(π)= 0, P 2(π) = c 2(θ) − {2(n + D} 1nc 1 2(θ) . P 3(π) = c 3(θ) − (n + 1) 1(n − 1) c 1(θ)c 2(θ)+ {3(n + 1) 3} 1n(n − 1) c 1 3(θ) . P 4(π) = c 4(θ) − (n + 1) 1(n − 2)c 1(θ)c 3(θ)+ {2(n + 1) 2} 1(n − 1)(n − 2) c 1 2(θ)c 2(θ)− {8(n + 1) 3} 1(n − 1)(n − 2) c 1 4(θ) . If X is compact, Kachler, and admits a holomorphic projective connection, then the Weyl forms are d-exact. Hence, in this case, our formula reduces to the formula concerning the Chern classes: c k=(n+1) −k n+1 k c k 1, 1⩽k⩽n The latter was claimed by Gunning without the keahlerity assumption, but unfortunately his proof was not correct (cf. Remark 3.39). A direct rigorous proof of formula (∗) under the kaehlerity assumption can be found in the work of Kobayashi and Ochiai.

On Verdier's specialization formula for Chern classes

Let M be a complex manifold of complex dimension m, and let X C M be a proper analytic subvariety. Let D C ~ be the unit disk, and f : X ~ D a proper analytic morphism. Put X~: = f l(s), s e D. We know that, for r > 0 sufficiently small, the inclusion Xo C f I(D,) is a homotopy equivalence. Thus for small nonzero s e D we may define the specialization map on homology a. :H,(X~)~H.(Xo) as the composition ~, H,(X~) , H , ( f I(D,)) ~ n,(Xo).

On different possibilities offered by the Born approximation in inverse scattering problems

One of the most productive approximations used to obtain analytic results in inverse scattering problems is the Born approximation. When the exciting wave is a monochromatic plane wave, in an n-dimensional problem, this approximation allows the author to obtain the Fourier transform of the unknown function on a certain (2n-1)-dimensional complex manifold M. The inverse transform performed on each submanifold of M, which can be bijectively mapped to Rn, results in a different method capable of determining the unknown function. In this paper, the author first establishes a unified theory which permits one to correctly grasp the differences and relationships between various methods and then, by introducing the concepts of accessible and inaccessible parts of the submanifolds, he suggests a basis for comparing the effectiveness of possible methods in the case of n=2. This comparison reveals that a rather simple method which has not attracted sufficient attention until now is one of the best methods.

Holomorphic Mappings into Strictly Convex Domains Which are Kobayashi Isometries at One Point

Let $M$ be a complex manifold of dimension $n$ and let $\Omega$ be a domain in ${{\mathbf {C}}^n}$. Let $f:M \to \Omega$ be a holomorphic map which is an isometry for the infinitesimal Kobayashi metric at one point. We given conditions on $M$ and on $\Omega$ which imply that $F$ must be a biholomorphic map.