The canonical foliations of a locally conformal Kähler manifold

Summary Let M be a locally conformal Kähler manifold. Then the Kähler form of M satisfies d= for some closed 1 -form , called the Lee form of M. We show that M admits three canonical foliations (four if is parallel) and we prove several properties of them, improving previous results of I. Vaisman. In particular all of these foliations are totally geodesic and Riemannian, and one of them is also almost complex. If this latter foliation is regular on a compact M, then we prove that M is a locally trivial fiber bundle over a compact Kähler manifold M, and the fibers are totally geodesic flat 2-tori. Finally we study geometrical properties, the canonical class and the Godbillon-Vey class of the totally real foliation of a CR-submanifold N cM.Work done during a visit of the second author at Michigan State University; this visit was supported by C.N.R., Italy.     

A twistor construction of Kähler submanifolds of a quaternionic Kähler manifold

A class of minimal almost complex submanifolds of a Riemannian manifold with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold of non zero scalar curvature, in particular, when is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M²ⁿ of maximal possible dimension 2n. More precisely, we prove that any such Kähler submanifold M²ⁿ of is the projection of a holomorphic Legendrian submanifold of the twistor space of , considered as a complex contact manifold with the natural holomorphic contact structure . Any Legendrian submanifold of the twistor space is defined by a generating holomorphic function. This is a natural generalization of Bryants construction of superminimal surfaces in S⁴=P¹. Mathematics Subject Classification (1991) Primary: 53C40; Secondary: 53C55     

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

Some submanifolds of a parakähler manifold

SiaM una sottovarietà totalmente ombelicale di una varietà parakähleriana $\tilde M$ . SeM è anche debolmente antiolomorfa, le curvature bisezionali, ordinarie e normale, diM sono legate da una elegante relazione, da cui discendono interessanti conseguenze. L'ultimo risultato del lavoro si riferisce alle sottovarietà parakäleriane delle varietà a curvatura sezionale costante.     

A Kähler structure of the triplectic geometry

We study the geometry of the triplectic quantization of gauge theories. We show that the triplectic geometry is determined by the geometry of a Kähler manifoldN endowed with a pair of transversal polarizations. The antibrackets can be brought to the canonical form if and only ifN admits a flat symmetric connection that is compatible with the complex structure and the polarizations.     

A formally Kähler structure on a knot space of a G 2-manifold

A knot space in a manifold M is a space of oriented immersions ${S^{1} \hookrightarrow M}$ up to Diff(S ¹). J.-L. Brylinski has shown that a knot space of a Riemannian threefold is formally Kähler. We prove that a space of knots in a holonomy G ₂ manifold is formally Kähler.     

Some submersions of extrinsic hyperspheres of a Bochner-Kähler manifold

The purpose of this paper is to extend some known results on Riemannian submersions from extrinsic hyperspheres of an Einstein-Kähler manifold to the case of a Bochner-Kähler manifold.     

A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space

We study the spaces and and _Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that and _Lip are infinite dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL ₂() on the free loop space ofS ².Oblatum 15-X-1994 & 5-VII-1995This research was supported in part by NSF grant DMS-92-05154.This research was partially supported by AFOSR grant F49620-92-J-0093.     

A note on compact Kähler manifolds with nef tangent bundles

In this paper, we study the compact Kähler manifolds whose tangent bundles are numerically effective and whose anti-Kodaira dimensions are equal to one. LetX be a compact Kähler manifold with nef tangent bundle and semiample anti-canonical bundle. We prove that κ(?K _X )=1 if and only if there exists a finite étale coverY→X such thatY??¹×A, whereA is a complex torus. As a consequence, we are able to improve upon a result of T. Fujiwara [3, 4].     

Schwarz lemma from a Kähler manifold into a complex Finsler manifold

Science China Mathematics - Suppose that M is a complete Kähler manifold such that its holomorphic sectional curvature is bounded from below by a constant and its radial sectional curvature is...     

On the structure of co-Kähler manifolds

By the work of Li, a compact co-Kähler manifold $M$ is a mapping torus $K_\varphi $ , where $K$ is a Kähler manifold and $\varphi $ is a Hermitian isometry. We show here that there is always a finite cyclic cover $\overline{M}$ of the form $\overline{M} \cong K \times S^1$ , where $\cong $ is equivariant diffeomorphism with respect to an action of $S^1$ on $M$ and the action of $S^1$ on $K \times S^1$ by translation on the second factor. Furthermore, the covering transformations act diagonally on $S^1, K$ and are translations on the $S^1$ factor. In this way, we see that, up to a finite cover, all compact co-Kähler manifolds arise as the product of a Kähler manifold and a circle.     

A comparison theorem for the mean exit time from a domain in a Kähler manifold

Let M be a Kähler manifold with Ricci and antiholomorphic Ricci curvature bounded from below. Let be a domain in M with some bounds on the mean and JN-mean curvatures of its boundary . The main result of this paper is a comparison theorem between the Mean Exit Time function defined on and the Mean Exit Time from a geodesic ball of the complex projective space ⁿ () which involves a characterization of the geodesic balls among the domain . In order to achieve this, we prove a comparison theorem for the mean curvatures of hypersurfaces parallel to the boundary of , using the Index Lemma for Submanifolds.Work partially supported by a DGICYT Grant No. PS87-0115-C03-01.     

A Schwarz lemma for Kähler affine metrics and the canonical potential of a proper convex cone

Annali di Matematica Pura ed Applicata (1923 -) - This is an account of some aspects of the geometry of Kähler affine metrics based on considering them as smooth metric measure spaces and...     

On the structure of nearly pseudo-Kähler manifolds

Firstly we give a condition to split off the Kähler factor from a nearly pseudo-Kähler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly Kähler manifolds from twistor spaces to negatively curved quaternionic Kähler manifolds and para-quaternionic Kähler manifolds. The class of nearly pseudo-Kähler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-Kähler six-manifold is Einstein.     

Picard number of the generic fiber of an abelian fibered hyperkähler manifold

We shall show that the Picard number of the generic fiber of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.     

On the metric structure of non-Kähler complex surfaces

We give a characterization of a locally conformally K?hler (l.c.K.) metric with parallel Lee form on a compact complex surface. Using the Kodaira classification of surfaces, we classify the compact complex surfaces admitting such structures. This gives a classification of Sasakian structures on compact three-manifolds. A weak version of the above mentioned characterization leads to an explicit construction of l.c.K. metrics on all Hopf surfaces. We characterize the locally homogeneous l.c.K. metrics on geometric complex surfaces, and we prove that some Inoue surfaces do not admit any l.c.K. metric. Received: 23 July 1998 / Revised: 2 June 1999