Deformations Of Complex Structure Research Articles

DEFORMATION OF COMPLEX STRUCTURE ON NORMAL ISOLATED SINGULARITIES

We give an analytic construction of the versal deformation of normal isolated singularities of dimension greater than or equal to three, by means of deformation of complex structure on its regular part.

Stress Distribution in Bone: Single-Unit Implant Prostheses Veneered with Porcelain or a New Composite Material

The purpose of this study was to simulate dynamic and static occlusal loading on one unit fixed, implant-supported prostheses veneered with porcelain and with GRADIA in a 3-dimensional finite element model of the human mandible to analyze and compare the resultant stresses in the superstructures and in the supporting bone. Calculation and visualization of stress, deformation, and displacement of complex structures under simulated forces were evaluated by finite element analysis (FEA) using ANSYS. The device employed was from the OSTEOFIX Dental Implant System (Oulu, Finland), and the veneering materials used were standard dental porcelain and GC GRADIA (Tokyo, Japan), a new composite material. Two different loading conditions were considered: static and transitional or impact, each delivered in three different directions: horizontal (Fh) at 0 degrees , vertical (Fv) at 90 degrees , and oblique (Fo) at 120 degrees . The proportion of the force: magnitude was fh:fv:fo = 1:3.5:7. A vertical load of 500 N, a horizontal load of 143 N, and an oblique load of 1000 N were applied. The results showed that the highest stresses in the bone-implant interface occurred in the region of cortical bone adjacent to the first thread of implants in all models and varied within 6.5%. Maximum stresses and displacements were higher (7%) in those models with statically loaded implants as compared with those that had been dynamically loading. The direction of loading played a major role in determining stress levels and they varied at up to 85%. It was shown that with dynamic loads, the peak of 1.568 mm was registered in the model with the GRADIA veneering material. This displacement was 6.5% higher than that found with the Vita VMK 68 veneers. These results suggest that the implant superstructure-fixed single crown veneering materials-porcelain and GRADIA played minor influences to the displacements and stresses in the implant supported bone with a 1% variance. One of the reasons for this element resulting in so minor a contribution to stress relief was the relatively small volume of the veneers as compared with the far greater volume and stiffness of the superstructures and implants. One of the advantages of GC GRADIA is that it reduces impact force under dynamic load up to a 6.5%.

Conifold Type Singularities,N=2 Liouville and SL(2;R)/U(1) Theories

In this paper we discuss various aspects of non-compact models of CFT of the type: $ \prod_{j=1}^{N_L} {N=2 Liouville theory}_j \otimes \prod_{i=1}^{N_M} {N=2 minimal model}_i $ and $ \prod_{j=1}^{N_L}{SL(2;R)/U(1) supercoset}_j \otimes \prod_{i=1}^{N_M} {N=2 minimal model}_i $. These models are related to each other by T-duality. Such string vacua are expected to describe non-compact Calabi-Yau compactifications, typically ALE fibrations over (weighted) projective spaces. We find that when the Liouville ($SL(2;R)/U(1)$) theory is coupled to minimal models, there exist only (c,c), (a,a) ((c,a), (a,c))-type of massless states in CY 3 and 4-folds and the theory possesses only complex (K\"{a}hler) structure deformations. Thus the space-time has the characteristic feature of a conifold type singularity whose deformation (resolution) is given by the N=2 Liouville (SL(2;R)/U(1)) theory. Spectra of compact BPS D-branes determined from the open string sector are compared with those of massless moduli. We compute the open string Witten index and determine intersection numbers of vanishing cycles. We also study non-BPS branes of the theory that are natural extensions of the ``unstable B-branes'' of the SU(2) WZW model in hep-th/0105038.

Heterotic compactifications and nearly Kähler manifolds

We propose that under certain conditions heterotic string compactifications on half-flat and nearly-Kahler manifolds are equivalent. Based on this correspondence we argue that the moduli space of the nearly-Kahler manifolds under discussion consists only of the Kahler deformations moduli space and there is no correspondent for the complex structure deformations.

Complex structures on affine motion groups

We study existence of complex structures on semidirect products g ⊕ρ v, where g is a real Lie algebra and ρ is a representation of g on v. Our first examples, the Euclidean algebra e(3) and the Poincaré algebra e(2, 1), carry complex structures obtained by deformation of a regular complex structure on sl (2, C). We also exhibit a complex structure on the Galilean algebra G(3, 1). We construct next a complex structure on g ⊕ρ v starting with one on g under certain compatibility assumptions on ρ. As an application of our results we obtain that there exists k ∈ {0, 1} such that (S1)k × E(n) admits a left invariant complex structure, where S1 is the circle and E(n) denotes the Euclidean group. We also prove that the Poincaré group P4k+3 has a natural left invariant complex structure. In case dim g = dim v, there is an adapted complex structure on g ⊕ρ v precisely when ρ determines a flat, torsion-free connection on g. If ρ is self-dual, g ⊕ρ v carries a natural symplectic structure as well. If, moreover, ρ comes from a metric connection then g ⊕ρ v possesses a pseudo-Kähler structure. We prove that the tangent bundle TG of a Lie group G carrying a flat torsion-free connection ∇ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

Complex structures on affine motion groups

We study existence of complex structures on semidirect products $\g \oplus_{\rho} \v$ where $\g$ is a real Lie algebra and $\rho$ is a representation of $\g$ on $\v$. Our first examples, the Euclidean algebra $\e(3)$ and the Poincar\'e algebra $ \e(2,1)$, carry complex structures obtained by deformation of a regular complex structure on $\sl (2, \c)$. We also exhibit a complex structure on the Galilean algebra $\G(3,1)$. We construct next a complex structure on $\g \oplus_{\rho} \v$ starting with one on $\g$ under certain compatibility assumptions on $\rho$. As an application of our results we obtain that there exists $k\in \{0,1\}$ such that $(S^1)^k \times E(n)$ admits a left invariant complex structure, where $S^1$ is the circle and E(n) denotes the Euclidean group. We also prove that the Poincar\'e group $P^{4k+3}$ has a natural left invariant complex structure. In case $\dim \g= \dim \v$, then there is an adapted complex structure on $\g\oplus_{\rho} \v$ precisely when $\rho$ determines a flat, torsion-free connection on $\g$. If $\rho$ is self-dual, $\g \oplus_{\rho}\v$ carries a natural symplectic structure as well. If, moreover, $\rho$ comes from a metric connection then $\g\oplus_{\rho} \v$ possesses a pseudo-K\"ahler structure. We prove that the tangent bundle $TG$ of a Lie group $G$ carrying a flat torsion free connection $\nabla$ and a parallel complex structure possesses a hypercomplex structure. More generally, by an iterative procedure, we can obtain Lie groups carrying a family of left invariant complex structures which generate any prescribed real Clifford algebra.

A REMARK ON THE DOUADY SEQUENCE FOR NON-PRIMARY HOPF MANIFOLDS

To determine cohomology groups of holomorphic vector bundles or more general coherent analytic sheaves on complex manifolds is very important in several complex variables and complex geometry. For example, Cartan-Serre's theorem B and Kodaira's vanishing theorem are fundamental respectively in the studies of two classes of complex manifolds: Stein manifolds and projective algebraic manifolds. Deformation and moduli space of complex structures are also closely related to the determination of the cohomology groups of the bundles according to Kodaira-Spencer's theory

Regular Systems on Riemann Surfaces

In this paper, we consider the current status of the Riemann–Hilbert problem and problems closely related to it. In the global theory of differential equations with regular singular points, the Riemann– Hilbert problem is one of the central topics. Two-dimensional models in contemporary theoretical physics use fundamental facts of the theory of Riemann surfaces. One of the central problems of this theory is the Riemann–Hilbert problem and questions related to it, for example, the linear conjugation problem. A natural language for the investigation of the Riemann–Hilbert problem is the language of holomorphic bundles with connections on Riemann surfaces. It makes clear the relationship with the linear conjugation problem, which is usually considered for Holder-class functions. We develop a different approach; namely, we consider the Riemann–Hilbert problem for the Carleman– Bers–Vekua system, replacing the Wiener–Hopf factorization of a matrix-valued function by the Φ-factorization. To study deformations of complex structures on Riemann surfaces, we also consider the Beltrami equation as a particular case of the Carleman–Bers–Vekua equation. We also discuss the algebraic formulation of the Riemann–Hilbert problem in the framework of the differential Galois theory, where this problem is known as the inverse problem and is interesting for us because of its constructive nature.

Prediction of Butt Welding Deformation of Curved Shell Plates by Inherent Strain Method

In order to achieve high productivity of assembly blocks of hull structure, it is important to predict welding deformations accurately and to apply these data to the production planning. For this purpose the transient thermal-elastic-plastic analysis by FEM (finite-element method) may be utilized. However, this method is not a practical approach to analyze the deformation of large and complex structures, such as ship hull structures, in view of time and cost. Whereas the inherent strain method deals the residual strain near the weld line as the inherent strain, therefore the residual deformation can be simply calculated by elastic analysis. This method is much more practical and efficient than the transient thermal-elastic-plastic analysis. Thus, in this paper, the inherent strain method is used to predict the welding deformation of hull structures. Past research activities of the inherent strain method have been focused on the prediction of the welding deformation of small size specimens and simple shape specimens; however, very few of them treat the problem of large and complex structures. Hence, in this paper, prediction of welding deformation of curved shell plate, of which size is approximately full scale, is conducted by the inherent strain method. In applying elastic analysis to predict the welding deformation of actual hull blocks, contact condition between the plate and the positioning jig should be adequately realized in the analysis. Therefore, a practical solution is proposed to con- sider the contact condition between the plate and the jig by judging reaction forces of the jig at calculation steps. Furthermore, since the welded plates may slip on the positioning jig due to the welding deformation, rigid body motion of plates is caused. Hence this motion should be taken into account of the estimation of calculated results of FEM. In addition to the rigid body motion, coordinate system of the measured data is usually different from the calculated data, therefore, the deviation of the coordinate system must be involved. These calibrations to assess the calculated deformation by FEM are also discussed. The above prediction method is confirmed to be successful by comparing the calculated deformations and the measured ones of large size specimen of curved shell plates due to butt welding.

Dolbeault cohomology of compact nilmanifolds

LetM=G/Γ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space $$\mathcal{C}\left( \mathfrak{g} \right)$$ of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification $$\mathfrak{g}^\mathbb{C} $$ of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.

Non-commutative Calabi–Yau manifolds

We discuss aspects of the algebraic geometry of compact non-commutative Calabi–Yau manifolds. In this setting, it is appropriate to consider local holomorphic algebras which can be glued together into a compact Calabi–Yau algebra. We consider two examples: a toroidal orbifold T 6/ Z 2× Z 2 , and an orbifold of the quintic in CP 4 , each with discrete torsion. The non-commutative geometry tools are enough to describe various properties of the orbifolds. First, one describes correctly the fractionation of branes at singularities. Secondly, for the first example we show that one can recover explicitly a large slice of the moduli space of complex structures which deform the orbifold. For this example we also show that we get the correct counting of complex structure deformations at the orbifold point by using traces of non-commutative differential forms (cyclic homology).

Deformations of open stein manifolds

We consider the deformation of the complex structure on an open Stein manifold. We show that a tame, compactly supported deformation of a Stein manifold is trivial. The remainder of our results are for deformations of the standard complex structure on Cn. A deformation of Cn which tends to a constant deformation faster that r-3 is trivial. Harmonic deformation tensors (w.r.t to the standard Euclidean metric) which are regular at infinity are constant.

Differential Graded Cohomology and Lie Algebras¶of Holomorphic Vector Fields

The Dolbeault resolution of the sheaf of holomorphic vector fields $Lie$ on a complex manifold $M$ relates $Lie$ to a sheaf of differential graded Lie algebras, known as the Fr\"olicher-Nijenhuis algebra $g$. We establish - following B. L. Feigin - an isomorphism between the differential graded cohomology of the space of global sections of $g$ and the hypercohomology of the sheaf of continuous cochain complexes of $Lie$. We calculate this cohomology up to the singular cohomology of some mapping space. We use and generalize results of N. Kawazumi on complex Gelfand-Fuks cohomology. Applications are - again following B. L. Feigin - in conformal field theory, and in the theory of deformations of complex structures. In an erratum to this paper, we admit that the sheaf of continuous cochains of a sheaf of vector fields with values in the ground fields does not make much sense. The most important cochains (like evaluations in a point or integrations over the manifold) do not come from sheaf homomorphisms. The main result of the above article (theorem 7) remains true.

CUTTING AND PASTING OF RIEMANN SURFACES WITH ABELIAN DIFFERENTIALS I

We introduce a method of constructing once punctured Riemann surfaces by cutting the complex plane along "line segments" and pasting by "parallel transformations". The advantage of this construction is to give a good visualization of the deformation of complex structures of Riemann surfaces. In fact, given a positive integer g, there appears a family of once punctured Riemann surfaces of genus g which is complete and effectively parametrized at any point. Our construction naturally gives each of the resulting surfaces what we call a Lagrangian lattice Λ, a certain subgroup of the first homology. Furthermore Λ and the puncture determine an Abelian differential ωΛ of the second kind on the Riemann surface. Using Λ and ωΛ we consider the Kodaira–Spencer maps and some extension of the family to obtain any once punctured Riemann surface with a Lagrangian lattice. In particular we describe the moduli space of once punctured elliptic curves with Lagrangian lattices.

Shared-money computing: Crash analysis on the Tera MTA

Programs that simulate complex-structure deformation scale poorly on most vector multiprocessors and MPPs. The authors worked on an alternative architecture — the Tera MTA — and tested performance results for a model crash problem using LS-DYNA code.

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Programs that simulate complex-structure deformation scale poorly on most vector multiprocessors and MPPs. The authors worked on an alternative architecture-the Tera MTA-and tested performance results for a model crash problem using LS-DYNA code.

Kodaira-Spencer deformation of complex structures and Lagrangian field theory

Similar to the Beltrami parametrization of complex structures on a (compact) Riemann surface, we use in this paper the Kodaira-Spencer deformation theory of complex structures on a (compact) complex manifold of higher dimension. According to the Newlander-Nirenberg theorem, a smooth change of local complex coordinates is implemented with respect to an integrable complex structure parametrize by a Beltrami differential. The question of constructing a local field theory on a complex compact manifold is addressed and the action of smooth diffeomorphisms is studied in the BRS algebraic approach. The BRS cohomology for the diffeomorphisms gives generalized Gel’fand-Fuchs cocycles provided that the Kodaira-Spencer integrability condition is satisfied. The diffeomorphism anomaly is computed and turns out to be holomorphically split as in the bidimensional Lagrangian conformal models. Moreover, its algebraic structure is much more complicated than the one proposed in a recent paper [Losev et al. Nuc. Phys. B 484, 196 (1997)].

A regularity theorem for deformations of a compact complex surface

Let ƒ:M →D ⊑Cn be a holomorphic family of compact, complex surfaces, which is locally trivial onD∖Z, for an analytic subsetZ. Conditions are found under which ƒ extends trivially toD, if the fibers of ƒ¦D∖Z are either Hirzebruch surfaces (projective bundles overP1), Hopf surfaces (elliptic bundles overP1), hyperelliptic bundles, or any compact complex surface having one of these as minimal model under blowing-down. The results of this paper are motivated by the existence of non-Hausdorff moduli spaces in the deformation of complex structure for certain complex manifolds.

O( d, d;R) deformations of complex structures and extended worldsheet supersymmetry

It is shown that the O( d, d;R) deformations of the superstring vacua and the O( d, d + 16;R) deformations of the heterotic string vacua preserve extended worldsheet supersymmetry and, hence, generate superconformal deformations. The transformations of the complex structures are given explicitly and the action of the discrete duality subgroup is discussed. The results are valid when the complex structures are independent of the d coordinates which appear in the transformations. It is shown that generic deformations do not preserve the known superfield formulations of (2,2) extended supersymmetry. The analysis is performed by decomposing the transformations in terms of the metric vielbein and by introducing space-time connections induced due to the non-linear action of the O( d, d;R) and O( d, d + 16;R) deformations on the background fields.

String theory near a conifold singularity

We demonstrate that type II string theory compactified on a singular Calabi-Yau manifold is related to c = 1 string theory compactified at the self-dual radius. We establish this result in two ways. First we show that complex structure deformations of the conifold correspond, on the mirror manifold, to the problem of maps from two dimensional surfaces to S 2. Using two dimensional QCD we show that this problem is identical to c = 1 string theory. We then give an alternative derivation of this correspondence by mapping the theory of complex structure deformations of the conifold to Chern-Simons theory on S 3. These results, in conjunction with similar results obtained for the compactification of the heterotic string on K 3 ×T 2 , provide strong evidence in favour of S-duality between type II strings compactified on a Calabi-Yau manifold and the heterotic string on K 3 × T 2.