Searcharxiv⌕ Search

arXiv subjects

Laszlo Lempert

Publications and source records attributed to Laszlo Lempert.

At least 19 recordsLinked to original sources
arXiv2026

On a construction of hermitian metrics on holomorphic vector bundles

With any holomorphic vector bundle $E$ over a compact base one can associate a line bundle, denoted $O_{PE}(1)$. According to a conjecture of Griffiths, if $O_{PE}(1)$ admits a positively curved hermitian metric $k$, then $E$ also admits a positively curved hermitian metric, $h$. In this paper we show that, while the conjecture may be correct, it is not possible to obtain $h$ out of $k$ by a fiberwise construction that is functorial.

math.CV↗
arXiv2026

Ellipsoids in pseudoconvex domains II

We consider families of pseudoconvex domains $X[s]\subset \mathbb C^n$ parametrized by complex manifolds $S$, and the maximal hermitian ellipsoids inscribed in each $X[s]$. The paper investigates how these maximal ellipsoids and their volume vary with $s$.

math.CV↗
arXiv2026

Ellipsoids in pseudoconvex domains

We consider the problem of maximizing the volume of hermitian ellipsoids inscribed in a given pseudoconvex domain in complex Euclidean space. We prove existence and uniqueness, and give a characterization of the maximizer.

math.CV↗
arXiv2024

Two variational problems in Kähler geometry

On a Kähler manifold we consider the problems of maximizing/minimizing Monge--Ampère energy over certain subsets of the space of Kähler potentials. Under suitable assumptions we prove that solutions to these variational problems exist, are unique, and have a simple characterization. We then use the extremals to construct hermitian metrics on holomorphic vector bundles, and investigate their curvature.

math.CV↗
arXiv2021

The principle of least action in the space of Kähler potentials

Given a compact Kähler manifold, the space $\mathcal H$ of its (relative) Kähler potentials is an infinite dimensional Fréchet manifold, on which Mabuchi and Semmes have introduced a natural connection $\nabla$. We study certain Lagrangians on $T\mathcal H$, in particular Finsler metrics, that are parallel with respect to the connection. We show that geodesics of $\nabla$ are paths of least action; under suitable conditions the converse also holds; and prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.

math.CV↗
arXiv2021

On the adjoint action of the group of symplectic diffeomorphisms

We study the action of Hamiltonian diffeomorphisms of a compact symplectic manifold ($X,ω$) on $C^\infty(X)$ and on functions $C^\infty(X)\to \mathbb R$. We describe various properties of invariant convex functions on $C^\infty(X)$. Among other things we show that continuous convex functions $C^\infty(X)\to \mathbb R$ that are invariant under the action are automatically invariant under so called strict rearrangements and they are continuous in the sup norm topology of $C^\infty(X)$; but this is not generally true if the convexity condition is dropped.

math.SG↗
arXiv2019

Extrapolation, a technique to estimate

We introduce a technique to estimate a linear operator by embedding it in a family $A_t$ of operators, $t\in(σ_0,\infty)$, with suitable curvature properties. One can then estimate the norm of each $A_t$ by bounds that hold in the limit $t\toσ_0$, respectively, $t\to\infty$. We illustrate this technique on an extension problem that arises in complex geometry.

math.CV↗
arXiv2017

On complex Legendre duality

Complex Legendre duality is a generalization of Legendre transformation from Euclidean spaces to Kahler manifolds, that Berndtsson and collaborators have recently constructed. It is a local isometry of the space of Kahler potentials. We show that the fixed point of such a transformation must correspond to a real analytic Kahler metric.

math.CV↗
arXiv2016

Noncommutative potential theory

We propose to view hermitian metrics on trivial holomorphic vector bundles $E\toΩ$ as noncommutative analogs of functions defined on the base $Ω$, and curvature as the notion corresponding to the Laplace operator or $\partial\overline\partial$. We discuss noncommutative generalizations of basic results of ordinary potential theory, mean value properties, maximum principle, Harnack inequality, and the solvability of Dirichlet problems.

math.CV↗
arXiv2014

Modules of square integrable holomorphic germs

This paper was inspired by Guan and Zhou's recent proof of the so-called 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.

math.CV↗
arXiv2013

Representing analytic cohomology groups of complex manifolds

Consider a holomorphic vector bundle $L\to X$ and an open cover ${\frak U}=\{U_a\colon a\in A\}$ of $X$, parametrized by a complex manifold $A$. We prove that the sheaf cohomology groups $H^q(X,L)$ can be computed from the complex $C^{\bullet}_{\text{hol}}$ $({\frak U},L)$ of cochains $(f_{a_0\ldots a_q})_{a_0,\ldots, a_q\in A}$ that depend holomorphically on the $a_j$, provided $S=\{(a,x)\in A\times X\colon x\in U_a\}$ is a Stein open subset of $A\times X$. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomorphic action of a complex reductive Lie group on $L$.

math.CV↗
arXiv2013

Analytic cohomology groups of infinite dimensional complex manifolds

Given a cohesive sheaf $\Cal S$ over a complex Banach manifold $M$, we endow the cohomology groups $H^q(M,\Cal S)$ of $M$ and $H^q(\frak U,\Cal S)$ of open covers $\frak U$ of $M$ with a locally convex topology. Under certain assumptions we prove that the canonical map $H^q(\frak U,\Cal S)\to H^q(M,\Cal S)$ is an isomorphism of topological vector spaces.

math.CV↗
arXiv2008

On the cohomology groups of holomorphic Banach bundles

We consider a compact complex manifold $M$, and introduce the notion of two holomorphic Banach bundles $E,F$ over $M$ being compact perturbations of one another. Given two such bundles we show that if the cohomology groups $H^q(M,E)$ are finite dimensional then so are the cohomology groups $H^q(M,F)$; as well as a more precise result in the same spirit.

math.CV↗
arXiv2008

Coherent sheaves and cohesive sheaves

We consider coherent and cohesive sheaves of $\cO$--modules over open sets $Ω\subset\bC^n$. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat.

math.CV↗
arXiv2008

Analytic continuation in mapping spaces

We consider a Stein manifold $M$ of dimension $\geq 2$ and a compact subset $K\subset M$ such that $M'=M\backslash K$ is connected. Let $S$ be a compact differential manifold, and let $M_S$, resp. $M'_S$ stand for the complex manifold of maps $S\to M$, resp. $S\to M'$, of some specified regularity, that are homotopic to constant. We prove that any holomorphic function on $M'_S$ continues analytically to $M_S$ (perhaps as a multivalued function).

math.CV↗
arXiv2005

Analytic sheaves in Banach spaces

We introduce a class of analytic sheaves in a Banach space X, that we call cohesive sheaves. Cohesion is meant to generalize the notion of coherence from finite dimensional analysis. Accordingly, we prove the analog of Cartan's Theorems A and B for cohesive sheaves over pseudoconvex open subsets of X, provided X has an unconditional basis.

math.CV↗
arXiv2004

Dolbeault cohomology of a loop space

The loop space LP_1 of the Riemann sphere is an infinite dimensional complex manifold consisting of maps (loops) from S^1 to P_1 in some fixed C^k or Sobolev W^{k,p} space. In this paper we compute the Dolbeault cohomology groups H^{0,1}(LP_1).

math.CV↗