Coherent sheaves and cohesive sheaves

We consider coherent and cohesive sheaves of \cO--modules over open sets \Omega\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.Comment: 18 pages. In replacement the proof of Lemma 6.2 improved; typos correcte

A maximum principle for hermitian (and other) metrics

We consider homomorphisms of hermitian holomorphic Hilbert bundles. Assuming the homomorphism decreases curvature, we prove that its pointwise norm is plurisubharmonic.Comment: 10 page

Extrapolation, a technique to estimate

We introduce a technique to estimate a linear operator by embedding it in a family $A_t$ of operators, $t\in(\sigma_0,\infty)$, with suitable curvature properties. One can then estimate the norm of each $A_t$ by bounds that hold in the limit $t\to\sigma_0$, respectively, $t\to\infty$. We illustrate this technique on an extension problem that arises in complex geometry

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.Comment: Typos corrected, reference adde

Curvature of fields of quantum Hilbert spaces

We show that using the family of adapted K\"ahler 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

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

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 $L$

On Riemannian submersions

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

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).Comment: 26 page

Weak geodesics in the space of K\"ahler metrics

Given a compact K\"ahler manifold (X,\omega_0), according to Mabuchi, the set of K\"ahler forms cohomologous to \omega_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