Let ${\overline{L}}_i\to X_i$ be a holomorphic line bundle over a compact complex manifold for $i=1,2$. Let $S_i$ denote the associated principal circle-bundle with respect to some hermitian inner product on ${\overline{L}}_i$. We construct complex structures on $S=S_1\times S_2$ which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that ${\overline{L}}_i$ are equivariant ${\left({ℂ}^{*}\right)}^{n_i}$-bundles satisfying some additional conditions. The linear type complex structures...

Soit $E$, un fibré linéaire positif au-dessus d’une variété complexe compacte. Nous montrons que la fonction de distorsion définie par le rapport entre la métrique initiale et la métrique de Fubini-Study de $E^{\otimes k}$ admet un équivalent lorsque $k$ tend vers l’infini. Ceci améliore les encadrements de Kempf et Ji sur les variétés abéliennes, et les étend à toute variété projective. La démonstration repose sur le calcul d’un équivalent pour le noyau de la chaleur, avec contrôle de la convergence par rapport...

The space $\mathscr{H}$ of Kähler metrics in a fixed Kähler class on a projective Kähler manifold $X$ is an infinite dimensional symmetric space whose geodesics ${\omega }_t$ are solutions of a homogeneous complex Monge-Ampère equation in $A\times X$, where $A\subset ℂ$ is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials $\varphi (t,z)$ of ${\omega }_t$ may be approximated in a weak $C^0$ sense by geodesics ${\varphi }_N(t,z)$ of the finite dimensional symmetric space of Bergman metrics of height $N$. In this article we prove that ${\varphi }_N(t,z)\to \varphi (t,z)$ in $C^2([0,1]\times X)$ in the case of...

A statement in the paper “Holomorphic Morse inequalities on manifolds with boundary” saying that the holomorphic Morse inequalities for an hermitian line bundle $L$ over $X$ are sharp as long as $L$ extends as semi-positive bundle over a Stein-filling is corrected, by adding certain assumptions. A more general situation is also treated.