The openness conjecture and complex Brunn-Minkowski inequalities

We discuss recent versions of the Brunn-Minkowski inequality in the complex setting, and use it to prove the openness conjecture of Demailly and Koll\'ar.Comment: This is an account of the results in arXiv:1305.5781 together with some background material. It is based on a lecture given at the Abel symposium in Trondheim, June 2013. 13 page

Interpolation in non-positively curved K\"ahler manifolds

We extend to any simply connected K\"ahler manifold with non-positive sectional curvature some conditions for interpolation in $\mathbb{C}$ and in the unit disk given by Berndtsson, Ortega-Cerd\`a and Seip. The main tool is a comparison theorem for the Hessian in K\"ahler geometry due to Greene, Wu and Siu, Yau.Comment: 9 pages, Late

Extension of holomorphic functions and cohomology classes from non reduced analytic subvarieties

The goal of this survey is to describe some recent results concerning the L 2 extension of holomorphic sections or cohomology classes with values in vector bundles satisfying weak semi-positivity properties. The results presented here are generalized versions of the Ohsawa-Takegoshi extension theorem, and borrow many techniques from the long series of papers by T. Ohsawa. The recent achievement that we want to point out is that the surjectivity property holds true for restriction morphisms to non necessarily reduced subvarieties, provided these are defined as zero varieties of multiplier ideal sheaves. The new idea involved to approach the existence problem is to make use of L 2 approximation in the Bochner-Kodaira technique. The extension results hold under curvature conditions that look pretty optimal. However, a major unsolved problem is to obtain natural (and hopefully best possible) L 2 estimates for the extension in the case of non reduced subvarieties -- the case when Y has singularities or several irreducible components is also a substantial issue.Comment: arXiv admin note: text overlap with arXiv:1703.00292, arXiv:1510.0523

Simulating river flow to the Baltic Sea from climate simulations over the past millennium

The aim of this study was to reconstruct river flow to the Baltic Sea using data from different periods during the past thousand years. A hydrological model coupled to simulations from climate models was used to estimate river flow. A "millennium" simulation of past climate from the ECHO-G coupled atmosphere-ocean global climate model provided climatological inputs. Results from this global model were downscaled with the RCA3 regional climate model over northern Europe. Temperature and precipitation from the downscaled simulation results were then used in the HBV hydrological model to simulate river flows to the Baltic Sea for the periods 1000-1199 and 1551-1929. These were compared with observations for the period 1921-2002. A general conclusion from this work is that although climate has varied during the past millennium, variability in annual river flow to the Baltic Sea does not appear more pronounced in recent years than during the previous millennium, or vice versa

Szeg\"o kernel asymptotics and Morse inequalities on CR manifolds

We consider an abstract compact orientable Cauchy-Riemann manifold endowed with a Cauchy-Riemann complex line bundle. We assume that the manifold satisfies condition Y(q) everywhere. In this paper we obtain a scaling upper-bound for the Szeg\"o kernel on (0, q)-forms with values in the high tensor powers of the line bundle. This gives after integration weak Morse inequalities, analogues of the holomorphic Morse inequalities of Demailly. By a refined spectral analysis we obtain also strong Morse inequalities which we apply to the embedding of some convex-concave manifolds.Comment: 40 pages, the constants in Theorems 1.1-1.8 have been modified by a multiplicative constant 1/2 ; v.2 is a final updat

Positivity of relative canonical bundles and applications

Given a family $f:\mathcal X \to S$ of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle $\mathcal K_{\mathcal X/S}$. We use a global elliptic equation to show that this metric is strictly positive on $\mathcal X$, unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil-Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space $\mathcal M_{\text{can}}$ of canonically polarized varieties follows. The direct images $R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal X/S}^{\otimes m})$, $m > 0$, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms $S^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal X/S}$ that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of $\mathcal M_{\text{can}}$.Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in Invent. mat

The impact of inflammatory bowel disease on sexual health in men: A scoping review.

AIMS AND OBJECTIVES: To review the literature on the impact of inflammatory bowel disease on the sexual health of men, and make recommendations for nursing practice and research. BACKGROUND: Inflammatory bowel disease is a chronic condition of the gastrointestinal tract, causing symptoms that may impact upon sexual health. Specialist nurses are well positioned to assess and manage sexual health, but there is a lack of clinical guidance, especially in relation to men. DESIGN: A systematic scoping review following the Arksey and O'Malley (2005) framework reported in line with the PRISMA-ScR checklist (Tricco et al. 2018). METHODS: OVID MEDLINE ALL [R], OVID EMBASE [R], OVID PsychINFO, EBSCO CINAHL Complete, The Cochrane Library and ProQuest were searched. Inclusion and exclusion criteria were applied independently by two reviewers. Data was extracted, charted and summarised from eligible studies. RESULTS: Thirty-one studies met the inclusion criteria. These were synthesised under three categories: mediators, moderators, and descriptors of sexual health. Depression, disease activity and surgery were the most commonly cited disease-related factors to affect sexual health in men. The most commonly used assessment tool was The International Index of Erectile Function. Descriptors of function included; frequency of intercourse, libido and the ability to maintain a desired sexual role. CONCLUSIONS: The effect of inflammatory bowel disease on sexual health in men involves a complex interaction of physical and psychosocial factors. Researchers must explore areas outside of erectile function to understand how the disease impacts sexuality, sexual well-being and masculinity. This can be achieved through qualitative exploration of patient, partner and health professional experiences. Relevance to clinical practice A holistic nursing assessment of men with inflammatory bowel disease should include sexual health. Developing understanding of how the disease influences sexual interaction and expression will facilitate support that is relevant, accessible and of value to men living with the disease

Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions