We study the J-flow from the point of view of an algebrogeometric stability condition. In terms of this we give a lower bound for the natural associated energy functional, and we show that the blowup behavior found by Fang and Lai [11] is reflected by the optimal destabilizer. Finally we prove a general existence result on complex tori.

Abstract. We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kähler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.

We show that the almost complex structure underlying a non-Kähler, nearly Kähler 6-manifold (in particular, the standard almost complex structure of S 6 ) cannot be compatible with any symplectic form, even locally. To cite this article: M. Lejmi, C. R. Acad. Sci. Paris, Ser. I 343 (2006). RésuméLes variétés strictement approximativement kählérienne de dimension 6 et les formes symplectiques. Nous démontrons que la structure presque-complexe d'une variété nearly-kählérienne non-intégrable de dimension 6-en particulier la structure presquecomplexe standard sur la sphère S 6 -ne peut pas être compatible avec une forme symplectque. Pour citer cet article : M. Lejmi, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the Kähler case. Our main question is the existence of almost Kähler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory.

Abstract. Given a path of almost-Kähler metrics compatible with a fixed symplectic form on a compact 4-manifold such that at time zero the almost-Kähler metric is an extremal Kähler one, we prove, for a short time and under a certain hypothesis, the existence of a smooth family of extremal almost-Kähler metrics compatible with the same symplectic form, such that at each time the induced almost-complex structure is diffeomorphic to the one induced by the path.