László Lempert

Geodesics in the space of Kähler metrics

Let (X,ω) be a compact Kähler manifold. As discovered in the late 1980s by Mabuchi, the set H0 of Kähler forms cohomologous to ω has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H0 can be connected by a smooth geodesic, and show that the answer, in general, is “no”.

Symmetries and other transformations of the complex Monge-Ampère equation

On definit des transformations tangentes en utilisant la structure symplectique holomorphe du fibre cotangent holomorphe

The Dolbeault complex in infinite dimensions III. Sheaf cohomology in Banach spaces

Abstract.We prove that the sheaf cohomology groups Hq(Ω,?) vanish if Ω is a pseudoconvex open subset of a Banach space with unconditional basis, and q≥1.

The Tangent Bundle of an Almost Complex Manifold

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complex manifold with an almost complex structure. We describe various properties of this structure.

Modules of Square Integrable Holomorphic Germs

This paper was inspired by Guan and Zhou’s recent proof of the socalled strong openness conjecture for plurisubharmonic functions. We give a proof shorter than theirs and extend the result to possibly singular Hermitian metrics on vector bundles.

Complex Structures on the Tangent Bundle of Riemannian Manifolds

It is well known that any (paracompact) differentiable manifold M has a complexification, i.e., a complex manifold X ⊃ M, dimc X = dimℝ M, such that M is totally real in X (see Ref. 8). It is also known that a small neighborhood U of M in X is diffeomorphic to the tangent bundle TM of M. Thus, the tangent bundle TM of any differentiable manifold carries a complex manifold structure. This complex structure is, of course, not unique. One way of finding a “canonical” complex structure is to endow M with some extra structure and require that the complex structure on TM interact with the structure of M. Here we consider smooth (meaning infinitely differentiable) Riemannian manifolds M. When M = ℝ, there is a natural identification Tℝ ≅ ℂ given by

An independence result in several complex variables

{{T}_{\sigma }}\mathbb{R} \mathrel\backepsilon \tau \frac{\partial }{{\partial \sigma }} \leftrightarrow \sigma + i\tau \in \mathbb{C},

A new look at adapted complex structures

(1.1) and this endows Tℝ with a complex structure. In (1.1) σ denotes the coordinate on R. This coordinate depends on the algebraic structure of the identification (1.1); however, the complex structure on Tℝ depends only on the metric of ℝ. In other words, an isometry of ℝ induces a biholomorphic mapping on Tℝ.