L2L^2-Riemann-Roch for singular complex curves

We present a comprehensive $L^2$-theory for the $\overline\partial$-operator on singular complex curves, including $L^2$-versions of the Riemann-Roch theorem and some applications.Comment: 19 page

Modifications of torsion-free coherent analytic sheaves

We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert-Riemenschneider canonical sheaf of holomorphic n-forms.Comment: 28 pages; the article has been completely rewritten due to a wrong statement in the first versio

A generalization of Takegoshi's relative vanishing theorem

We present a generalization of Takegoshi's relative version of the Grauert-Riemenschneider vanishing theorem. Under some natural assumptions, we extend Takegoshi's vanishing theorem to the case of Nakano semi-positive coherent analytic sheaves on singular complex spaces. We also obtain some new results about proper modifications of torsion-free coherent analytic sheaves.Comment: 21 pages - We fixed a mistake by adding Hausdorff-condition

Chern forms of singular metrics on vector bundles

We study singular hermitian metrics on holomorphic vector bundles, following Berndtsson-P{\u{a}}un. Previous work by Raufi has shown that for such metrics, it is in general not possible to define the curvature as a current with measure coefficients. In this paper we show that despite this, under appropriate codimension restrictions on the singular set of the metric, it is still possible to define Chern forms as closed currents of order 0 with locally finite mass, which represent the Chern classes of the vector bundle.Comment: 18

On Lelong Numbers of Generalized Monge-Amp\`ere Products

We consider generalized (mixed) Monge-Amp\`ere products of quasiplurisubharmonic functions (with and without analytic singularities) as they were introduced and studied in several articles written by subsets of M. Andersson, E. Wulcan, Z. B{\l}ocki, R. L\"ark\"ang, H. Raufi, J. Ruppenthal, and the author. We continue these studies and present estimates for the Lelong numbers of pushforwards of such products by proper holomorphic submersions. Furthermore, we apply these estimates to Chern and Segre currents of pseudoeffective vector bundles. Among other corollaries, we obtain the following generalization of a recent result by X. Wu. If the non-nef locus of a pseudoeffective vector bundle $E$ on a K\"ahler manifold is contained in a countable union of $k$-codimensional analytic sets, and if the $k$-power of the first Chern class of $E$ is trivial, then $E$ is nef.Comment: 32 pages; minor corrections in the second versio

Chern forms of hermitian metrics with analytic singularities on vector bundles

We define Chern and Segre forms, or rather currents, associated with a Griffiths positive singular hermitian metric $h$ with analytic singularities on a holomorphic vector bundle $E$. The currents are constructed as pushforwards of generalized Monge-Amp\`ere products on the projectivization of $E$. The Chern and Segre currents represent the Chern and Segre classes of $E$, respectively, and coincide with the Chern and Segre forms of $E$ and $h$, where $h$ is smooth. Moreover, our currents coincide with the Chern and Segre forms constructed by the first three authors and Ruppenthal in the cases when these are defined

Feasibility of state of the art PET/CT systems performance harmonisation

Purpose The objective of this study was to explore the feasibility of harmonising performance for PET/CT systems equipped with time-of-flight (ToF) and resolution modelling/point spread function (PSF) technologies. A second aim was producing a working prototype of new harmonising criteria with higher contrast recoveries than current EARL standards using various SUV metrics. Methods Four PET/CT systems with both ToF and PSF capabilities from three major vendors were used to acquire and reconstruct images of the NEMA NU2-2007 body phantom filled conforming EANM EARL guidelines. A total of 15 reconstruction parameter sets of varying pixel size, post filtering and reconstruction type, with three different acquisition durations were used to compare the quantitative performance of the systems. A target range for recovery curves was established such that it would accommodate the highest matching recoveries from all investigated systems. These updated criteria were validated on 18 additional scanners from 16 sites in order to demonstrate the scanners' ability to meet the new target range. Results Each of the four systems was found to be capable of producing harmonising reconstructions with similar recovery curves. The five reconstruction parameter sets producing harmonising results significantly increased SUVmean (25%) and SUVmax (26%) contrast recoveries compared with current EARL specifications. Additional prospective validation performed on 18 scanners from 16 EARL accredited sites demonstrated the feasibility of updated harmonising specifications. SUVpeak was found to significantly reduce the variability in quantitative results while producing lower recoveries in smaller ( Conclusions Harmonising PET/CT systems with ToF and PSF technologies from different vendors was found to be feasible. The harmonisation of such systems would require an update to the current multicentre accreditation program EARL in order to accommodate higher recoveries. SUVpeak should be further investigated as a noise resistant alternative quantitative metric to SUVmax

The Importance of Slow-roll Corrections During Multi-field Inflation

We re-examine the importance of slow-roll corrections during the evolution of cosmological perturbations in models of multi-field inflation. We find that in many instances the presence of light degrees of freedom leads to situations in which next to leading order slow-roll corrections become significant. Examples where we expect such corrections to be crucial include models in which modes exit the Hubble radius while the inflationary trajectory undergoes an abrupt turn in field space, or during a phase transition. We illustrate this with two examples -- hybrid inflation and double quadratic inflation. Utilizing both analytic estimates and full numerical results, we find that corrections can be as large as 20%. Our results have implications for many existing models in the literature, as these corrections must be included to obtain accurate observational predictions -- particularly given the level of accuracy expected from CMB experiments such as PlanckComment: v1: 21 pages, 3 figures, 1 appendix. v2: clarifications to {\S}{\S}2.1, 3.1 and 4, {\S}5.3 added, references added, results unchanged. Matches published version in JCA