Morse–Novikov cohomology of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering, with the monodromy acting on this covering by holomorphic homotheties. We define three cohomology invariants, the Lee class, the Morse–Novikov class, and the Bott–Chern class, of an LCK-structure. These invariants play together the same role as the Kähler class in Kähler geometry. If these classes coincide for two LCK-structures, the difference between these structures can be expressed by a smooth potential, similar to the Kähler case. We show that the Morse–Novikov class and the Bott–Chern class of a Vaisman manifold vanish. Moreover, for any LCK-structure on a manifold, admitting a Vaisman structure, we prove that its Morse–Novikov class vanishes. We show that a compact LCK-manifold M$M$ with vanishing Bott–Chern class admits a holomorphic embedding into a Hopf manifold, if _dimCM?3${dim}_{\mathbb{C}}M?3$, a result which parallels the Kodaira embedding theorem.     

The spectrum of the 4-generation Dirac–Kähler extension of the SM

We compute the mass spectrum of the fermionic sector of the Dirac–Kähler extension of the SM (DK-SM) by showing that there exists a Bogoliubov transformation that transforms the DK-SM into a flavor U(4)$U(4)$ extension of the SM (SM-4) with a particular choice of masses and mixing textures. Mass relations of the model allow determination of masses of the 4th generation. Tree level prediction for the mass of the 4th charged lepton is 370 GeV. The model selects the normal hierarchy for neutrino masses and reproduces naturally the near tri-bimaximal and quark mixing textures. The electron neutrino and the 4th neutrino masses are related via a see-saw-like mechanism.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

Kähler Reduction of Metrics with Holonomy G<Subscript>2</Subscript>

A torsion-free G₂ structure admitting an infinitesimal isometry such that the quotient is a Kähler manifold is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G₂.     

Pseudo-Hyperkähler Geometry and Generalized Kähler Geometry

We discuss the conditions for additional supersymmetry and twisted super-symmetry in N = (2, 2) supersymmetric nonlinear sigma models described by one left and one right semi-chiral superfield and carrying a pair of non-commuting complex structures. Focus is on linear non-manifest transformations of these fields that have an algebra that closes off-shell. We find that additional linear supersymmetry has no interesting solution, whereas additional linear twisted supersymmetry has solutions with interesting geometrical properties. We solve the conditions for invariance of the action and show that these solutions correspond to a bi-hermitian metric of signature (2, 2) and a pseudo-hyperkähler geometry of the target space.     

Conification of Kähler and Hyper-Kähler Manifolds

Given a Kähler manifold M endowed with a Hamiltonian Killing vector field Z, we construct a conical Kähler manifold ${\hat{M}}$ such that M is recovered as a Kähler quotient of ${\hat{M}}$ . Similarly, given a hyper-Kähler manifold (M, g, J ₁, J ₂, J ₃) endowed with a Killing vector field Z, Hamiltonian with respect to the Kähler form of J ₁ and satisfying ${\mathcal{L}_ZJ_2 = -2J_3}$ , we construct a hyper-Kähler cone ${\hat{M}}$ such that M is a certain hyper-Kähler quotient of ${\hat{M}}$ . In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-Kähler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-Kähler cone, which in turn defines a quaternionic Kähler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.     

On the Hyperkähler/Quaternion Kähler Correspondence

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established.     

On Certain Kähler Quotients of Quaternionic Kähler Manifolds

We prove that, given a certain isometric action of a two-dimensional Abelian group A on a quaternionic Kähler manifold M which preserves a submanifold N ? M, the quotient M′ = N/A has a natural Kähler structure. We verify that the assumptions on the group action and on the submanifold N ? M are satisfied for a large class of examples obtained from the supergravity c-map. In particular, we find that all quaternionic Kähler manifolds M in the image of the c-map admit an integrable complex structure compatible with the quaternionic structure, such that N ? M is a complex submanifold. Finally, we discuss how the existence of the Kähler structure on M′ is required by the consistency of spontaneous ${\mathcal{N} = 2}$ to ${\mathcal{N} = 1}$ supersymmetry breaking.     

Generalized Kähler Geometry

Generalized Kähler geometry is the natural analogue of Kähler 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 prove that generalized Kähler geometry is equivalent to the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2, 2) supersymmetry. We also prove the existence of natural holomorphic Courant algebroids for each of the underlying complex structures, and that these split into a sum of transverse holomorphic Dirac structures. Finally, we explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kähler geometry.     

Generalized Pseudo-Kähler Structures

In this paper we consider pseudo-bihermitian structures – pairs of complex structures compatible with a pseudo-Riemannian metric. We establish relations of these structures with generalized (pseudo-) Kähler geometry and holomorphic Poisson structures similar to that in the positive definite case. We provide a list of compact complex surfaces which could admit pseudo-bihermitian structures and give examples of such structures on some of them. We also consider a naturally defined null plane distribution on a generalized pseudo-Kähler 4-manifold and show that under a mild restriction it determines an Engel structure.     

Special Kähler Manifolds

We give an intrinsic definition of the special geometry which arises in global N= 2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special K?hler manifold is so related to an integrable system. The cotangent bundle of a special K?hler manifold carries a hyperk?hler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry. Received: 5 December 1997 / Accepted: 16 November 1998     

Almost-Kähler Deformation Quantization

We use a natural affine connection with nontrivial torsion on an arbitrary almost-Kähler manifold which respects the almost-Kähler structure in order to construct a Fedosov-type deformation quantization on this manifold.     

Kähler spinning particles

We construct the U(N) spinning particle theories, which describe particles moving on Kähler spaces. These particles have the same relation to the N = 2 string as usual spinning particles have to the NSR string. We find the restrictions on the target space of the theories coming from supersymmetry and from global anomalies. Finally, we show that the partition functions of the theories agree with what is expected from their spectra, unlike that of the N = 2 string in which there is an anomalous dependence on the proper time.     

Compact Scalar-flat Indefinite Kähler Surfaces with Hamiltonian <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-Symmetry

The existence of scalar-flat indefinite Kähler metrics on compact complex surfaces is discussed. In particular, a compact scalar-flat indefinite Kähler surface admitting a Hamiltonian S¹-symmetry is proved to be biholomorphic to the product of two complex projective lines, with the help of a generalization of the Bando-Calabi-Futaki character. In fact, it is shown that none of such metrics exist on Hirzebruch surfaces of positive degree. On the other hand, by employing an analogue of LeBruns hyperbolic ansatz, we construct a wealth of explicit scalar-flat indefinite Kähler metrics on the product of complex projective lines, and also prove that these explicit metrics provide infinitely many different isometry classes, by examining a necessary and sufficient condition for these metrics to be isometric to each other.Supported by JSPS–MEXT. Grant-in-Aid for Young Scientists (No. 13740053).Acknowledgement Several components of this work were carried out when the author was visiting the Mathematical Institute, Tôhoku University in the academic year 2000. I would like to thank the participants of the Geometry Seminar at the institute for their friendship. Thanks also go to Professors Henrik Pedersen, Kazuo Akutagawa, Hiroyasu Izeki and Keisuke Ueno for a variety of discussion, suggestions and comments. I am particularly grateful to Professors Akito Futaki and Yasuhiro Nakagawa for many valuable comments on the Bando-Calabi-Futaki character, and to Professor Shin Nayatani for many helpful suggestions and constant encouragement since the early stage of this work. I would also like to thank the referee for useful suggestion and especially for correcting a flaw in the earlier proof of Proposition 1. Last but not least, I wish to express my sincere gratitude to Professor Seiki Nishikawa for his insightful advice and continuous help as well as for the warm welcome he extended to me while I visited to Tohoku University.     

Mass in Kähler Geometry

We prove a simple, explicit formula for the mass of any asymptotically locally Euclidean (ALE) Kähler manifold, assuming only the sort of weak fall-off conditions required for the mass to actually be well-defined. For ALE scalar-flat Kähler manifolds, the mass turns out to be a topological invariant, depending only on the underlying smooth manifold, the first Chern class of the complex structure, and the Kähler class of the metric. When the metric is actually AE (asymptotically Euclidean), our formula not only implies a positive mass theorem for Kähler metrics, but also yields a Penrose-type inequality for the mass.     

Hyperkähler metrics and supersymmetry

We describe two constructions of hyperkähler manifolds, one based on a Legendre transform, and one on a sympletic quotient. These constructions arose in the context of supersymmetric nonlinear -models, but can be described entirely geometrically. In this general setting, we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry, along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience.Supported by the Swedish Natural Science Research CouncilResearch supported by the National Science Foundation under Contract No. PHY 81 09110 A-01 and PHY 85-07627     

Instantons and Kähler manifolds

It is shown that for two-dimensional Euclidean chiral models of the field theory with values in arbitrary Kähler manifold duality equations reduce to the Cauchy-Riemann equations on this manifold. A class of models is described possessing such type solutions, the so called instanton solutions.