Professor Lempert’s research is in complex analysis and in complex geometry. In the past 10 years he has worked on cohomological questions, mostly in infinite dimensional complex manifolds; on Kahler manifolds, and on curvature properties of holomorphic vector bundles and of similar objects.
1. Lempert has extended to infinite dimensional Banach manifolds the theory of coherent sheaves. This involved introducing the new class of what now are called cohesive sheaves---these generalize coherent sheaves, and are better adapted to infinite dimensional problems---, and proving Cartan's Theorems A & B for infinite dimensional Stein manifolds. He has also studied certain special infinite dimensional manifolds, loop spaces of finite dimensional complex manifolds. The simplest of these, that is unrelated to Stein manifolds, is the loop space of the Riemann sphere, whose first cohomology group he was able to compute (but not yet the higher cohomology groups).
2. With an eye on finding special metrics on Kahler manifolds, Lempert has studied the space of Kahler metrics on a given complex manifold. These spaces have a Riemannian manifold structure. He proved that in general two points in the space of Kahler metrics cannot be joined by geodesics, contrary to earlier expectations.
3. A generalization of hermitian vector bundles is the notion of Hilbert fields, an object that arises in complex geometry and mathematical physics. Lempert has defined the curvature of such objects and investigated the effect of curvature on the local structure of the Hilbert field. In the holomorphic framework, he proved a maximum principle for metrics under curvature conditions, and studied the implications of curvature on the solvability of certain analytical questions.
Many of the above results have been obtained in collaborations.