  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 $\C$-linear 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 induces the canonical holomorphic
structures of those bundles. In the non integrable case those
$(0,1)$-connections induces just the holomorphic canonical structures of the
restrictions of the corresponding bundles to the images of smooth
$J$-holomorphic 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
$J$-plurisubharmonic fonctions by means of the geodesic flow induced by a Chern
connection on the tangent bundle. This method has been used by Demailly 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 a optimal way
the geometric obstructions caused by the torsion of the almost complex
structure, and the non symplectic nature of the metric.
