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Abstract. We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is −∞ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

In their study of the Yamabe problem in the presence of isometry groups, E. Hebey and M. Vaugon announced a conjecture. This conjecture generalizes T. Aubin's conjecture, which has already been proven and is sufficient to solve the Yamabe problem. In this paper, we generalize Aubin's theorem and we prove the Hebey-Vaugon conjecture in dimensions less or equal to 37. RésuméDans leur étude du probleme de Yamabe équivariant, E. Hebey et M. Vaugon annonçaient une conjecture. Cette conjecture généralise la conjecture de T. Aubin qui a été déjà démontrée et est suffisante pour résoudre le probleme de Yamabe. Dans cet article, nous généralisons un théoreme de T. Aubin et nous démontrons que cette conjecture de Hebey-Vaugon est vraie jusqu'à la dimesion 37.

Abstract. We define the generalized logarithmic Gauss map for algebraic varieties of the complex algebraic torus of any codimension. Moreover, we describe the set of critical points of the logarithmic mapping restricted to our variety, and we show an analogous of Mikhalkin's result on the critical points of the logarithmic map restricted to a hypersurfaces.

Abstract. In this paper, we study the amoeba volume of a given k−dimensional generic analytic variety V of the complex algebraic torus (C * ) n . When n ≥ 2k, we show that V is algebraic if and only if the volume of its amoeba is finite. In this precise case, we establish a comparison theorem for the volume of the amoeba and the coamoeba. Examples and applications to the k−linear spaces will be given.