Generalized complex geometry, as developed by Hitchin, contains complex and symplectic geometry as its extremal special cases. In this thesis, we explore novel phenomena exhibited by this geometry, such as the natural action of a B-field. We provide new examples, including some on manifolds admitting no known complex or symplectic structure. We prove a generalized Darboux theorem which yields a local normal form for the geometry. We show that there is an elliptic deformation theory and establish the existence of a Kuranishi moduli space.
We then define the concept of a generalized Kahler manifold. We prove that generalized Kahler geometry is equivalent to a bi-Hermitian geometry with torsion first discovered by physicists. We then use this result to solve an outstanding problem in 4-dimensional bi-Hermitian geometry: we prove that there exists a Riemannian metric on the complex projective plane which admits exactly two distinct Hermitian complex structures with equal orientation.
Finally, we introduce the concept of generalized complex submanifold, and show that such sub-objects correspond to D-branes in the topological A- and B-models of string theory.
We introduce natural deformation classes of generalized Kähler structures using the Courant symmetry group. We show that these yield natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. Lastly we show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.
These are lecture notes mainly aimed at graduate students on selected aspects of generalized geometry: in particular generalized complex and Kaehler structures and generalized holomorphic bundles. They are based on lectures given in March 2010 at the Chinese University of Hong Kong.
We present some fundamental facts about a class of generalized Kähler structures defined by invariant complex structures on compact Lie groups. The main computational tool is the BH-to-GK spectral sequences that relate the bi-Hermitian data to generalized geometry data. The relationship between generalized Hodge decomposition and generalized canonical bundles for generalized Kähler manifolds is also clarified.
In this lecture, we review some of the concepts of generalized geometry, as introduced by Hitchin and developed in the speaker's thesis. We also prove a Hodge decomposition for the twisted cohomology of a compact generalized Kähler manifold, as well as a generalization of the $dd^c$-lemma of Kähler geometry.
The purpose of this paper is to extend the mean curvature comparison and volume comparison estimates by Petersen, Sprouse, and Wei to integral generalized quasi-Einstein tensor. Moreover, we use our comparison results to get global diameter estimate.
In generalized complex geometry, we revisit linear subspaces and submanifolds that have an induced generalized complex structure. We give an expression of the induced structure that allows us to deduce a smoothness criteria, we dualize the results to submersions and we make a few comments on generalized complex mappings. Then, we discuss submanifolds of generalized Kaehler manifolds that have an induced generalized Kaehler structure. These turn out to be the common invariant submanifolds of the two classical complex structures of the generalized Kaehler manifold.
We prove that no nilpotent Lie algebra admits an invariant generalized Kaehler structure. This is done by showing that a certain differential graded algebra associated to a generalized complex manifold is formal in the generalized Kaehler case, while it is never formal for a generalized complex structure on a nilpotent Lie algebra.
We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Ampère equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating function of such equation as generalized holomorphic objects.
We continue the study of blow-ups in generalized complex geometry with the blow-up theory for generalized Kähler manifolds. The natural candidates for submanifolds to be blown-up are those which are generalized Poisson 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 t…
In this short note we prove that in the case of elliptic curves, the isomorphism of generalized complex structure between $T$-dual manifolds described by Cavalcanti-Gualtieri coincides with the mirror map for elliptic curves described by Polishchuk and Zaslow.
We collect the recent results on invariant f-structures in the generalized Hermitian geometry. Here the canonical f-structures on homogeneous k-symmetric spaces play a remarkable role. Specifically, these structures provide a wealth of invariant examples for the classes of nearly Kaehler f-structures, Hermitian f-structures and some others. Finally, we consider all invariant f-structures on the complex flag manifold SU(3)/T_{max} and describe them in the sense of generalized Hermitian geometry. In particular, it presents first invariant examples of Killing f-structures.
In this article we introduce a generalization of the Newton transformation to the case of a system of endomorphisms. We show that it can be used in the context of extrinsic geometry of foliations and distributions yielding new integral formulas containing generalized extrinsic curvatures.
We introduce the concept of G2(2)-structure on an orientable 3-manifold M using the setting of generalized geometry of type Bn, study their local deformation by making use of a Moser-type argument, and give a description of the cone of G2(2)-structures.
The aim of this paper is to develop on the 1-jet space J^1(R,M^4) the jet Generalized Lagrange Geometry for the rheonomic Chernov metric. The associated gravitational and electromagnetic field models based on the rheonomic Finsler Chernov metric tensor are developed and discussed.
It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite dimensional on a compact manifo…
We prove some structure results for \emph{transverse reducible} Sasaki manifolds. In particular, we show Sasaki manifolds with positive Ricci curvature is transversely irreducible, and so there is no join (product) construction for irregular Sasaki-Einstein manifolds, as opposed to the quasi-regular case done by Wang-Ziller and Boyer-Galicki. As an application, we classify compact Sasaki manifolds with non-negative transverse bisectional curvature, which can be viewed as the generalized Frankel conjecture (N. Mok's theorem) in Sasaki geometry.
The concept of quasi generalized CR-lightlike was first introduced by the authors in [18]. In this paper, we focus on ascreen and co-screen quasi generalized CR-lightlike submanifolds of indefinite nearly $μ$-Sasakian manifold. We prove an existence theorem for minimal ascreen quasi generalized CR-lightlike submanifolds admitting a metric connection. Classification theorems on nearly parallel and auto-parallel distributions on a co-screen quasi generalized CR-lightlike submanifold are also given. Several examples are also constructed, where necessary, to illustrate the main ideas.
We show that symplectic forms taming complex structures on compact manifolds are related to special types of almost generalized Kähler structures. By considering the commutator $Q$ of the two associated almost complex structures $J_{\pm}$, we prove that if either the manifold is 4-dimensional or the distribution ${Im} \, Q$ is involutive, then the manifold can be expressed locally as a disjoint union of twisted Poisson leaves.
We study neighbourhoods of submanifolds in generalized complex geometry. Our first main result provides sufficient criteria for such a submanifold to admit a neighbourhood on which the generalized complex structure is B-field equivalent to a holomorphic Poisson structure. This is intimately tied with our second main result, which is a rigidity theorem for generalized complex deformations of holomorphic Poisson structures. Specifically, on a compact manifold with boundary we provide explicit conditions under which any generalized complex perturbation of a holomorphic Poisson structure is B-fiel…
We prove long time existence and convergence results for the pluriclosed flow, which imply geometric and topological classification theorems for generalized Kähler structures. Our approach centers on the reduction of pluriclosed flow to a degenerate parabolic equation for a $(1,0)$-form, introduced in \cite{ST2}. We observe a number of differential inequalities satisfied by this system which lead to a priori $L^{\infty}$ estimates for the metric along the flow. Moreover we observe an unexpected connection to "Born-Infeld geometry" which leads to a sharp differential inequality which can be use…
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we exploit the impact of the general results in the almost quaternionic geometry. In particular we show, that the natural class of H--planar curves coincides with the class of all geodesics of the so called Weyl connections and preserving this class turns out to be the necessary and sufficient condition on diffeomorphisms to become morphisms of almost quaterni…
We study the asymptotic behavior of the generalized Bergman kernel of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on a symplectic manifold of bounded geometry. First, we establish the off-diagonal exponential estimate for the generalized Bergman kernel. As an application, we obtain the relation between the generalized Bergman kernel on a Galois covering of a compact symplectic manifold and the generalized Bergman kernel on the base. Then we state the full off-diagonal asymptotic expansion of the generalized Bergman kernel, improving the remainder estimate…
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 …
In [21], the authors initiated the study of quasi generalized CR (QGCR)-null submanifolds. In this paper, attention is drawn to some distributions on ascreen QGCR-null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifold.
In this paper we define the notion of a generalized coKähler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, Φ_i,E_{\pm,i}, G_i)$, $ i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $Φ_1$ and $Φ_2$, is generalized Kähler if and only if $(M_i, Φ_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coKähler structures. We also prove that products of generalized coKähler and generalized Kähler manifolds admit a generalized coKähler structure. We use these product constructions to give nontri…
The paper developes a geometrization of a Kronecker $h$-regular vertical fundamental metrical d-tensor $G^{(α)(β)}_{(i)(j)}$ on the jet fibre bundle of order one $J^1(T,M)$. This geometrization gives a mathematical model for both gravitational and electromagnetic field theory, in a general setting. In this context, the Einstein and Maxwell equations are described.
In this paper, we show the existence of a sequence of invariant differential operators on a particular homogeneous model $G/P$ of a Cartan geometry. The first operator in this sequence can be locally identified with the Dirac operator in $k$ Clifford variables, $D=(D_1,..., D_k)$, where $D_i=\sum_j e_j\cdot \partial_{ij}: C^\infty((\R^n)^k,§)\to C^\infty((\R^n)^k,§)$. We describe the structure of these sequences in case the dimension $n$ is odd. It follows from the construction that all these operators are invariant with respect to the action of the group $G$. These results are obtained by con…
In this study, we investigate generalized quasi-Einstein structure for normal metric contact pair manifolds. Firstly, we deal with elementary properties and examine, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifold. Secondly, the generalized quasi-constant curvature of normal metric contact pair manifolds are studied and it is proven that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi typ…
The fibre bundles adjoint to generalized almost quaternionic structures are studied. The most important classes of generalized almost quaternionic manifolds are considered.
We introduce linear Dirac and generalized complex structures on Cartan geometries and give criteria for Dirac subalgebras of $\frkg\ltimes\frkg^*$ representing Dirac structures on a Cartan geometry. We prove that there is a bijection between the linear generalized structures on a torsion free Cartan geometry and the equivariant generalized structures on its corresponding homogeneous space.
These are the lecture notes from the 26th Winter School "Geometry and Physics", Czech Republic, Srni, January 14 - 21, 2006. These lectures are an introduction into the realm of generalized geometry based on the tangent plus the cotangent bundle. In particular we discuss the relation of this geometry to physics, namely to two-dimensional field theories. We explain in detail the relation between generalized complex geometry and supersymmetry. We briefly review the generalized Kahler and generalized Calabi-Yau manifolds and explain their appearance in physics.
We introduce and study the notion of a biholomorphic gerbe with connection. The biholomorphic gerbe provides a natural geometrical framework for generalized Kahler geometry in a manner analogous to the way a holomorphic line bundle is related to Kahler geometry. The relation between the gerbe and the generalized Kahler potential is discussed.
We propose a new topological field theory on generalized complex geometry in two dimension using AKSZ formulation. Zucchini's model is $A$ model in the case that the generalized complex structuredepends on only a symplectic structure. Our new model is $B$ model in the case that the generalized complex structure depends on only a complex structure.
We give a physical derivation of generalized Kahler geometry. Starting from a supersymmetric nonlinear sigma model, we rederive and explain the results of Gualtieri regarding the equivalence between generalized Kahler geometry and the bi-hermitean geometry of Gates-Hull-Rocek. When cast in the language of supersymmetric sigma models, this relation maps precisely to that between the Lagrangian and the Hamiltonian formalisms. We also discuss topological twist in this context.
A closed 3-form $H \in Ω^3_0(M)$ defines an extension of $Γ(TM)$ by $Ω^2_0(M)$. This fact leads to the definition of the group of $H$-twisted Hamiltonian symmetries $\Ham(M, \JJ; H)$ as well as Hamiltonian action of Lie group and moment map in the category of (twisted) generalized complex manifold. The Hamiltonian reduction in the category of generalized complex geometry is then constructed. The definitions and constructions are natural extensions of the corresponding ones in the symplectic geometry. We describe cutting in generalized complex geometry to show that it's a general phenomenon in …
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some past and recent results on the generalized complex structure of supersymmetric type II vacua in various dimensions.
Notions of self-dual and anti self-dual almost quaternionic structures are introduced. The complete classification of self-dual and anti self-dual generalized Kaehler manifolds is obtained.
Algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in non-smooth differential geometry. This article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics.
The geometry of the target space of an N=(2,2) supersymmetry sigma-model carries a generalized Kahler structure. There always exists a real function, the generalized Kahler potential K, that encodes all the relevant local differential geometry data: the metric, the B-field, etc. Generically this data is given by nonlinear functions of the second derivatives of K. We show that, at least locally, the nonlinearity on any generalized Kahler manifold can be explained as arising from a quotient of a space without this nonlinearity.
Generalized complex geometry, introduced by Hitchin, encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry, including its enhanced symmetry group, elliptic deformation theory, relation to Poisson geometry, and local structure theory. We also define and study generalized complex branes, which interpolate between flat bundles on Lagrangian submanifolds and holomorphic bundles on complex submanifolds.
We define the notion of a moment map and reduction in both generalized complex geometry and generalized Kähler geometry. As an application, we give very simple explicit constructions of bi-Hermitian structures on $\C¶^n$, Hirzebruch surfaces, the blow up of $\CP^2$ at arbitrarily many points, and other toric varieties, as well as complex Grassmannians.
We use the formalism of generalized geometry to study the generic supersymmetric AdS_5 solutions of type IIB supergravity that are dual to N=1 superconformal field theories (SCFTs) in d=4. Such solutions have an associated six-dimensional generalized complex cone geometry that is an extension of Calabi-Yau cone geometry. We identify generalized vector fields dual to the dilatation and R-symmetry of the dual SCFT and show that they are generalized holomorphic on the cone. We carry out a generalized reduction of the cone to a transverse four-dimensional space and show that this is also a general…
Generalized tensor analysis in the sense of Colombeau's construction is employed to introduce a nonlinear distributional pseudo-Riemannian geometry. In particular, after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo-Riemannian metric, generalized connection and generalized curvature tensor. We prove a ``Fundamental Lemma of (pseudo-)Riemannian geometry'' in this setting and define the notion of geodesics of a generalized metric. Finally, we present applications of the resulting theory to general relativit…
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to ho…
The Serre construction of rank two holomorphic bundles with a section is adapted to construct generalized holomorphic bundles on a generalized complex 4-manifold from the data of a set of points on an elliptic curve. The motivation is the special case of rank two Poisson modules on a complex surface with a holomorphic Poisson structure.
This work is based on the talk delivered at Poisson 2008. We review the recent advances in Generalized Kahler geometry while stressing the use of Poisson and symplectic geometry. The derivation of the generalized Kahler potential is sketched and the relevant global issues are discussed.
We show that, locally, all geometric objects of Generalized Kahler Geometry can be derived from a function K, the "generalized Kahler potential''. The metric g and two-form B are determined as nonlinear functions of second derivatives of K. These nonlinearities are shown to arise via a quotient construction from an auxiliary local product (ALP) space.
We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kaehler manifolds.
On the generalized tangent bundle of a smooth manifold, we study skew-symmetric endomorphism satisfying an arbitrary polynomial equation with real constant coefficients. We study the compatibility of these structures with the de Rham operator and the Courant-Dorfman bracket. In particular we isolate several conditions that when restricted to the motivating example of generalized almost complex structure are equivalent to the notion of integrability.
Using the idea of a generalized Kaehler structure, which is a pair of commuting generalized complex structures, we construct bihermitian metrics on the projective plane and the product of two projective lines, and show that any such structure on a compact 4-manifold M defines one on the moduli space of anti-self-dual connections on a fixed principal bundle over M. We highlight the role of holomorphic Poisson structures in all these constructions.
We survey the concept of multiplicativity from its initial appearance in the theory of Poisson-Lie groups to the far-reaching generalizations, for multivectors and differential forms in the geometry and the generalized geometry of Lie groupoids, as well as their infinitesimal counterparts in the theory of Lie algebroids.
