Generalized Complex Geometry Research Articles

Generalized metrics and generalized twistor spaces

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry a la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms.

Generalised contact geometry as reduced generalised Complex geometry

Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the reduction procedure, generalised coK\"ahler structures are defined in a way which extends the K\"ahler/coK\"ahler correspondence.

Stable generalized complex structures

A stable generalized complex structure is one that is generically symplectic but degenerates along a real codimension two submanifold, where it defines a generalized Calabi–Yau structure. We introduce a formalism which allows us to view such structures as symplectic forms with singularities of logarithmic or elliptic type. This allows us to define two period maps: one for deformations in which the background 3-form flux is fixed, and one for which the flux is allowed to vary. As a result, we prove the unobstructedness of each of these deformation problems. We use the same approach to establish local classification theorems for the degeneracy locus as well as for analogues of Lagrangian submanifolds called Lagrangian branes.

Vibration Analysis of Composite Beams with Sinusoidal Periodically Varying Interfaces

Abstract As an increasing variety of composite materials with complex interfaces are emerging, we develop a theory to investigate composite beams and shed some light on new physical insights into composite beams with sinusoidal periodically varying interfaces. For the natural vibration of composite beams with continuous or periodically varying interfaces, the governing equation has been derived according to the generalised Hamiltonian principle. For composite beams having different boundary conditions, we transform the governing equations into integral equations and solve them by using the sinusoidal functions as test functions as well as the basis of the vibration modes. Due to the orthogonality of the sinusoidal functions, expansion coefficients in closed form can be found. Therefore, the proposed iterative schemes, with the help of the Rayleigh quotient and boundary functions, can quickly find the eigenvalues and free vibration modes. The obtained natural frequencies agree well with those obtained using the finite element method. In addition, the proposed method can be extended easily to laminated composite beams in more general cases or complex components and geometries in vibration engineering. The effects of different material properties of the upper and lower components and varying interface geometry function on the frequency of the composite beams are examined. According to our investigation, the natural frequency of a laminated beam with a continuous or periodically varying interface can be changed by altering the density or elastic modulus. We also show the responses of the frequencies of the components to the varying periodic interface.

Blow-Ups in Generalized K\xe4hler Geometry

We study blow-ups in generalized Kähler geometry. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson submanifolds for one of the two generalized complex structures and can be blown up in a generalized complex manner. We show that the bi-Hermitian structure underlying the generalized Kähler pair lifts to a degenerate bi-Hermitian structure on this blow-up. Then, using a deformation procedure based on potentials in Kähler geometry, we identify two concrete situations in which one can deform the degenerate structure on the blow-up into a non-degenerate one. We end with a study of generalized Kähler Lie groups and give a concrete example on {(S^1)^n times (S^3)^m}, for n + m even.

Courant algebroids and generalized complex geometry

In this survey, we recall the background and major developments in Courant algebroids and their applications. We mainly recall the origin and motivation of the definition of Courant algebroids. A brief introduction to the theories of Dirac structures, Lie bialgebroids, the Clifford construction, spinor representation, generalized complex geometry and regular Courant algebroids are given.

Generalized almost product structures and generalized CRF-structures

We give several equivalent characterizations of orthogonal subbundles of the generalized tangent bundle defined, up to B-field transform, by almost product and local product structures. We also introduce a pure spinor formalism for generalized CRF-structure and investigate the resulting decomposition of the de Rham operator. As applications we give a characterization of generalized complex manifolds that are locally the product of generalized complex factors and discuss infinitesimal deformations of generalized CRF-structures.

On a Complex-Symplectic Mirror Pair

We study the canonical Poisson structure on the loop space of the super-double-twisted-torus and its quantization. As a consequence we obtain a rigorous construction of mirror symmetry as an intertwiner of the N=2 super-conformal structures on the super-symmetric sigma-models on the Kodaira-Thurston nilmanifold and a gerby torus of complex dimension 2. As an application we are able to identify global moduli of equivariant generalized complex structures on these target spaces with moduli of equivariant orthogonal complex structures on the doubled geometry.

Global Optimization of Heat Exchanger Networks. Part 1: Stages/Substages Superstructure

We solve globally an extension of the isothermal mixing stages model (Synheat) proposed by Yee and Grossmann, later extended to nonisothermal mixing by Bjork and Weterlund and solved globally by Faria et al. This new extension of that model was recently presented in a conference proceedings. The model allows specific structures that are more commonly accepted in industry than generalized complex structures that suffer from somewhat disorganized branched and sub-branches which present several practical challenges. This stages/substages superstructure has limited branching and several matches in series in each branch. Our industrial experience shows us that these are acceptable in practice. The novel contribution of this article is then solving such a model globally, adding the control of temperature differences upon mixing

Type one generalized Calabi–Yaus

We study type one generalized complex and generalized Calabi--Yau manifolds. We introduce a cohomology class that obstructs the existence of a globally defined, closed 2-form which agrees with the symplectic form on the leaves of the generalized complex structure, the twisting class. We prove that in a compact, type one, 4n-dimensional generalized complex manifold the Euler characteristic must be even and equal to the signature modulo four. The generalized Calabi--Yau condition places much stronger constrains: a compact type one generalized Calabi--Yau fibers over the 2-torus and if the structure has one compact leaf, then this fibration can be chosen to be the fibration by the symplectic leaves of the generalized complex structure. If the twisting class vanishes, one can always deform the structure so that it has a compact leaf. Finally we prove that every symplectic fibration over the 2-torus admits a type one generalized Calabi--Yau structure.

Some results of quantum correction on compact Calabi–Yau manifolds of dimension 3 and 4

It is shown that for compact Calabi–Yau threefolds, “no strong quantum correction” is equivalent to the condition that, with respect to the Hodge metric, the image Φ(T) of the Teichmüller space T under the period map Φ is an open submanifold W of the same dimension as T. In this paper, we extend this result to the case of dimension 4. We also obtained similar results for compact generalized Calabi–Yau manifolds in dimension 3 and 4, under the assumption that their Teichmüller space T exists and is a simply connected complex manifold.

Local analytic geometry of generalized complex structures

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the holomorphic Poisson structure occurring in this local model is independent of the choice of gauge equivalence, and is hence the unique local invariant of generalized complex manifolds. We use this result to prove that the complex locus of a generalized complex manifold naturally inherits the structure of a complex analytic space.

Analytical and variational numerical methods for unstable miscible displacement flows in porous media

The miscible displacement of one fluid by another in a porous medium has received considerable attention in subsurface, environmental and petroleum engineering applications. When a fluid of higher mobility displaces another of lower mobility, unstable patterns – referred to as viscous fingering – may arise. Their physical and mathematical study has been the object of numerous investigations over the past century. The objective of this paper is to present a review of these contributions with particular emphasis on variational methods. These algorithms are tailored to real field applications thanks to their advanced features: handling of general complex geometries, robustness in the presence of rough tensor coefficients, low sensitivity to mesh orientation in advection dominated scenarios, and provable convergence with fully unstructured grids.This paper is dedicated to the memory of Dr. Jim Douglas Jr., for his seminal contributions to miscible displacement and variational numerical methods.

Espace des twisteurs d’une variété quaternionique Kähler généralisée

Specifying an almost quaternionic structure Q on a 4n-manifold M is equivalent to specifying a twistor bundle Z(Q)⟶M. When Q is invariant under a torsion free connection, Z(Q) can be endowed with an almost complex structure 𝒥. For n=1 Atiyah, Hitchin and Singer [2] have related the integrability of 𝒥 to the geometry of (M,Q). For n>1 Salamon [23, 24] showed that the almost complex structure 𝒥 on Z(Q) is always integrable. Recently, Pantilie [21] introduced the concept of a generalized quaternionic Kähler structure 𝒬 on M; defined a generalized twistor space 𝒵(𝒬); and showed that 𝒵(𝒬) comes naturally equiped with a tautological almost generalized complex structure, but leaves open the problem of the integrability. The purpose of this paper is precisely to fill this gap by showing that the almost generalized complex structure on 𝒵(𝒬) is always integrable for n>1. We will conclude by giving several examples.

Mirror symmetry & supersymmetry on SU(4)-structure backgrounds

We revisit the backgrounds of type IIB on manifolds with $SU(4)$-structure and discuss two sets of solutions arising from internal geometries that are complex and symplectic respectively. Both can be realized in terms of generalized complex geometry. We identify a map which relates the complex and symplectic supersymmetric systems. In the semi-flat torus bundle setting this map corresponds to T-duality and suggest a way of extending the mirror transform to non-K\"ahler geometries.

AN ABSTRACT MORIMOTO THEOREM FOR GENERALIZEDF-STRUCTURES

We abstract Morimoto's construction of complex structures on product manifolds to pairs of certain generalized $F$-structures on manifolds that are not necessarily global products. As applications we characterize invariant generalized complex structures on product manifolds in which one factor is a Lie group and we generalize a theorem of Blair, Ludden and Yano on Hermitian bicontact manifolds.

Generalized contact bundles

In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of the integrability condition for generalized contact structures; (3) in light of new results on multiplicative forms and Spencer operators [8], it allows a simple interpretation of the defining equations of a generalized contact structure in terms of Lie algebroids and Lie groupoids.

Cohomologies of Generalized Complex Manifolds and Nilmanifolds

We study generalized complex cohomologies of generalized complex structures constructed from certain symplectic fibre bundles over complex manifolds. We apply our results in the case of left-invariant generalized complex structures on nilmanifolds and to their space of small deformations.

On cotangent manifolds, complex structures and generalized geometry

We develop various properties of symmetric generalized complex structures (in connection with their holomorphic space and B-field transformations), which are analogous to the well-known results of Gualtieri on skew-symmetric generalized complex structures. Given a symmetric or skew-symmetric generalized complex structure \mathcal J and a connection D on a manifold M, we construct an almost complex structure J^{\mathcal J,D} on the cotangent manifold T^{*}M and we study its integrability. For \mathcal J skew-symmetric, we relate the Courant integrability of \mathcal J with the integrability of J^{\mathcal J, D}. We consider in detail the case when M=G is a Lie group and \mathcal J , D are left-invariant. We also show that our approach generalizes various well-known results from special complex geometry.

$C^{\infty}$-logarithmic transformations and generalized complex structures

We provide a new construction of generalized complex manifolds by every logarithmic transformations. Applying a technique of broken Lefschetz fibrations, we obtain twisted generalized complex structures with arbitrary large numbers of connected components of type changing loci on the manifold which is obtained from a symplectic manifold by logarithmic transformations of multiplicity 0 on a symplectic 2-torus with trivial normal bundle. Elliptic surfaces with non-zero euler characteristic and the connected sums (2m 1)S 2 × S 2 , (2m 1)CP 2 #lCP 2 and S 1 × S 3 admit twisted generalized complex structures Jn with n type changing loci for arbitrary large n.