 

Nefton Pali
La connexion et la courbure de Chern du fibré tangent d'une variété presque complexe


Published: 
November 28, 2005

Keywords: 
Connexions de Chern, Courbure de Chern, Variétés presque complexes, Coordonnées presque complexes 
Subject: 
32C35 


Abstract
The \bar{\partial}_{J} operator over an almost complex manifold induces canonical
connections of type (0,1) over the bundles of (p,0)forms. If the almost complex structure
is integrable then the previous connections induce the canonical holomorphic structures of the
bundles of (p,0)forms. For p=1 we can extend the corresponding connection to all Schur
powers of the bundle of (1,0)forms. Moreover using the canonical Clinear isomorphism
betwen the bundle of (1,0)forms and the complex cotangent bundle T*_{X,J} we deduce
canonical connections of type (0,1) over the Schur powers of the complex cotangent bundle
T*_{X,J}. If the almost complex structure is integrable then the previous
(0,1)connections induce the canonical holomorphic structures of those bundles.
In the nonintegrable case those (0,1)connections induce just the holomorphic canonical
structures of the restrictions of the corresponding bundles to the images of smooth
Jholomorphic curves. We introduce the notion of Chern curvature for those bundles.
The geometrical meaning of this notion is a natural generalisation of the classical notion of
Chern curvature for the holomophic vector bundles over a complex manifold. We have a
particular interest for the case of the tangent bundle in view of applications concerning
the regularisation of Jplurisubharmonic functions by means of the geodesic flow induced
by a Chern connection on the tangent bundle. This method has been used by Demailly, 1994,
in the complex integrable case. Our specific study in the case of the tangent bundle gives an
asymptotic expanson of the Chern flow which relates in an optimal way the geometric obstructions
caused by the torsion of the almost complex structure, and the nonsymplectic nature of the
metric.


Author information
Department of Mathematics, Princeton University, Princeton, NJ 08544, USA
npali@math.princeton.edu

