On holomorphic maps and Generalized Complex Geometry

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.     

Warped product Kähler manifolds and Bochner–Kähler metrics

Using as an underlying manifold an alpha-Sasakian manifold, we introduce warped product Kähler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kähler manifold is of quasi-constant holomorphic sectional curvatures with a special distribution. Conversely, we prove that any Kähler manifold of quasi-constant holomorphic sectional curvatures with a special distribution locally has the structure of a warped product Kähler manifold whose base is an alpha-Sasakian space form. As an application, we describe explicitly all Bochner–Kähler metrics of quasi-constant holomorphic sectional curvatures. We find four families of complete metrics of this type. As a consequence, we obtain Bochner–Kähler metrics generated by a potential function of distance in complex Euclidean space and of time-like distance in the flat Kähler–Lorentz space.     

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.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

Generalized complex structures on Kodaira surfaces

We compute the deformations in the sense of generalized complex structures of the standard classical complex structure on a primary Kodaira surface and we prove that the obtained family of deformations is a smooth locally complete family depending on four complex parameters. This family is the same as the extended deformations (in the sense of Kontsevich and Barannikov) in degree two, obtained by Poon using differential Gerstenhaber algebras.     

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.     

Generalized holomorphic bundles and the B-field action

On a generalized complex manifold, there is an associated definition of a generalized holomorphic bundle, introduced by Gualtieri. In the case of an ordinary complex structure, this notion yields an object which we call a co-Higgs bundle, and we consider the B-field action of a closed form of type (1,1)$(1,1)$, both local and global. The effect makes contact with both Nahm’s equations and holomorphic gerbes.     

Geometric quantization of weak-Hamiltonian functions

The paper presents an extension of the geometric quantization procedure to integrable, big-isotropic structures. We obtain a generalization of the cohomology integrality condition, we discuss geometric structures on the total space of the corresponding principal circle bundle and we extend the notion of polarization.     

Poisson Quasi-Nijenhuis Structures with Background

We define Poisson quasi-Nijenhuis structures with background on Lie algebroids and we prove that any generalized complex structure on a Courant algebroid which is the double of a Lie algebroid has an associated Poisson quasi-Nijenhuis structure with background. We prove that any Lie algebroid with a Poisson quasi-Nijenhuis structure with background constitutes, with its dual, a quasi-Lie bialgebroid. We also prove that any pair (π,ω) of a Poisson bivector and a 2-form induces a Poisson quasi-Nijenhuis structure with background and we observe that particular cases correspond to already known compatibilities between π and ω. This paper was presented as a poster in the conference “Poisson 2008”, EPFL, Lausanne, in July 2008.     

Deformations of generalized holomorphic structures

A deformation theory of generalized holomorphic structures in the setting of (generalized) principal fibre bundles is developed. It allows the underlying generalized complex structure to vary together with the generalized holomorphic structure. We study the related differential graded Lie algebra, which controls the deformation problem via the Maurer–Cartan equation. As examples, we check the content of the Maurer–Cartan equation in detail in the special cases where the underlying generalized complex structure is symplectic or complex. A deformation theorem, together with some non-obstructed examples, is also included.     

On surfaces whose twistor lifts are harmonic sections

We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant.     

On sufficient and necessary conditions for linearity of the transverse Poisson structure

We study the possibility of bringing the transverse Poisson structure to a coadjoint orbit (on the dual of a real Lie algebra) to a normal linear form. We study the relation between two sufficient conditions for linearity of such structures (P. Molino’s condition and our own). We then use these conditions to conclude that, if the isotropy subgroup of the (singular) point in question is compact, or if the isotropy subalgebra is semisimple, then there is a linear transverse Poisson structure to the corresponding coadjoint orbit.     

Flat principal bundles over an abelian variety

We prove that a principal G-bundle E_G over a complex abelian variety A, where G is a complex reductive algebraic group, admits a flat holomorphic connection if and only if E_G is isomorphic to all the translations of it by the group structure of A.     

Reduction of generalized complex structures

We study reduction of generalized complex structures. More precisely, we investigate the following question. Let J$J$ be a generalized complex structure on a manifold M$M$, which admits an action of a Lie group G$G$ preserving J$J$. Assume that _M0$M_0$ is a G$G$-invariant smooth submanifold and the G$G$-action on _M0$M_0$ is proper and free so that _MG?_M0/G$M_G?M_0/G$ is a smooth manifold. Under what condition does J$J$ descend to a generalized complex structure on _MG$M_G$? We describe a sufficient condition for the reduction to hold, which includes the Marsden–Weinstein reduction of symplectic manifolds and the reduction of the complex structures in Kähler manifolds as special cases. As an application, we study reduction of generalized Kähler manifolds.     

Differential operators and immersions of a Riemann surface into a Grassmannian

We consider equivariant holomorphic immersions of a universal cover of a compact Riemann surface X into a Grassmannian satisfying a nondegeneracy condition. The equivariance condition says that there is a homomorphism ρ of the Galois group to that takes the natural action of the Galois group on to the action of the Galois group on defined using ρ. We prove that the space of such embeddings are in bijective correspondence with the space of all holomorphic differential operators of order two on a rank n vector bundle over X with the property that the symbol of the operator is an isomorphism.     

Some comments on Chern-Simons gauge theory

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291     

Generalized holomorphic structures

We define the notion of generalized holomorphic principal bundles and establish that their associated vector bundles of holomorphic representations are generalized holomorphic vector bundles defined by M. Gualtieri. Motivated by our definition, several examples of generalized holomorphic structures are constructed. A reduction theorem of generalized holomorphic structures is also included.     

Holomorphic vector fields of compact pseudo-Kähler manifolds

It is well known that a pseudo-Kähler structure is a natural generalization of the Kähler structure. In this paper, we consider holomorphic vector fields of a compact pseudo-Kähler manifold from the viewpoint of Kähler manifolds.     

Normal Bundles to Laufer Rational Curves in Local Calabi–Yau Threefolds

We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank two holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.