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László Lempert

Publications and source records attributed to László Lempert.

13 recordsLinked to original sources
arXiv2023

Aron--Berner--type extension in complex Banach manifolds

Let $S$ be a compact Hausdorff space and $X$ a complex manifold. We consider the space $C(S,X)$ of continuous maps $S\to X$, and prove that any bounded holomorphic function on this space can be continued to a holomorphic function, possibly multivalued, on a larger space $B(S,X)$ of Borel maps. As an application we prove two theorems about bounded holomorphic functions on $C(S,X)$, one reminiscent of the Monodromy Theorem, the other of Liouville's Theorem.

math.CV↗
arXiv2022

To the geometry of spaces of plurisubharmonic functions on a Kähler manifold

Consider a compact Kähler manifold $(X,ω)$ and the space $\cal E(X,ω)=\cal E$ of $ω$--plurisubharmonic functions of full Monge--Ampère mass on it. We introduce a quantity $ρ[u,v]$ to measure the distance between $u, v\in\cal E$; $ρ[u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset\mathbb R$. We explore properties of $ρ[u,v]$, and using them we study Lagrangians and associated energy spaces of $ω$--plurisubharmonic functions. Many results here generalize Darvas's findings about his metrics $d_χ$.

math.CV↗
arXiv2021

On the Bergman kernels of holomorphic vector bundles

Consider a very ample line bundle $ E \to X$ over a compact complex manifold, endowed with a hermitian metric of curvature $-i ω$, and the space $\mathcal{O}(E)$ of its holomorphic sections. The Fubini--Study map associates with positive definite inner products $\langle \, , \rangle$ on $\mathcal{O}(E)$ functions FS$(\langle \, ,\rangle) \in \mathcal{H}_ω=\{u \in C^{\infty}(X):ω+i\partial\overline{\partial} u >0\}$. We prove that FS is an injective immersion, but its image in general is not closed in $\mathcal{H}_ω$. To obtain a closed range, FS has to be extended to certain degenerate inner products. This we do by associating Bergman kernels with general inner products on the dual $\mathcal{O}(E)^*$, and the paper describes some simple properties of this association.

math.CV↗
arXiv2019

Isometries in spaces of Kähler potentials

The space of Kähler potentials in a compact Kähler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of Kähler potentials, and prove existence and uniqueness for such isometries.

math.CV↗
arXiv2019

On Riemannian submersions

We prove that the image of a real analytic Riemannian manifold under a smooth Riemannian submersion is necessarily real analytic.

math.DG↗
arXiv2019

Adapted complex and involutive structures

We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.

math.DG↗
arXiv2013

Direct images, fields of Hilbert spaces, and geometric quantization

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family $H_s$ of Hilbert spaces, and the question arises if the spaces $H_s$ are canonically isomorphic. [ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$, introduce a connection on $H$, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle $E-->Y$ along a non-proper map $Y-->S$. We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold $M$ by endowing $TM$ with the family of adapted Kähler structures from arxiv:1004.4069 [LSz]. This leads to a direct image problem. When $M$ is homogeneous, we prove the direct image is an analytic field of Hilbert spaces. For certain such $M$---but not all---the direct image is even flat; which means that in those cases quantization is unique.

math-ph↗
arXiv2012

Weak geodesics in the space of Kähler metrics

Given a compact Kähler manifold (X,ω_0), according to Mabuchi, the set of Kähler forms cohomologous to ω_0 has the natural structure of an infinite dimensional Riemannian manifold. We address the question whether points in this space can be joined by a geodesic, and strengthening previous findings of the second author with Vivas, we show that this cannot always be done even with a certain type of generalized geodesics.

math.CV↗
arXiv2012

Curvature of fields of quantum Hilbert spaces

We show that using the family of adapted Kähler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank-1, the obtained field $H^{corr}$ of quantum Hilbert spaces produced by geometric quantization including the half-form correction is flat if $M$ is the 3-dimensional sphere and not even projectively flat otherwise.

math-ph↗
arXiv2011

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 H_0 of Kähler forms cohomologous to ωhas the natural structure of an infinite dimensional Riemannian manifold. We address the question whether any two points in H_0 can be connected by a smooth geodesic, and show that the answer, in general, is "no".

math.CV↗
arXiv2010

A new look at adapted complex structures

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

math.CV↗