Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for $(p,q)$-forms on $n$-dimensional complex manifolds $X$ which admit a vector bundle of rank $n$ over $X \times X$, such that the diagonal of $X \times X$ has a defining section. We apply the method to \Pn and find weighted Koppelman formulas for $(p,q)$-forms with values in a line bundle over \Pn. As an application, we look at the cohomology groups of $(p,q)$-forms over \Pn with values in various line bundles, and find explicit solutions to the \dbar-equation in some of the trivial groups. We also look at cohomology groups of $(0,q)$-forms over \Pn \times \Pm with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.Comment: 25 page

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

A Joint Desalination and Power Plants for Water and Development: A Case study of the Sinai-Gaza

Desalination can be a cost-effective way to produce fresh water and possibly electricity. The Gaza Strip has had a complex hydro-political situation for many years. Gaza (enclosed area) is bordered by the Mediterranean in the west, by Israel in the north and east and by Egypt in the south. Water and electricity consumption in the Gaza Strip is expected to increase in the future due to the increasing population. In this paper, a solution for Sinai and the Gaza Strip is suggested involving the building of a joint power and desalination plant, located in Egypt close to the border of Gaza. Results of capital and unit costs have been derived from bench-mark studies of 18 different desalination projects mainly in the Middle East countries. The suggested joint Egypt-Palestine project would increases drinking water supply by 500,000 m3/d and the power supply by 500MW, whereof 2/3 is suggested to be used in Gaza and 1/3 in Sinai. The present lack of electricity and water in Gaza could be erased by such a project. But Egypt will probably gain more. More water and electricity will be available for the future development of Sinai; a significant value will be added to the sale of Egyptian natural gas used for water and power production in the project; more employment opportunities can be offered for people living in Sinai and Gaza; the domestic market for operation and maintenance of desalination plants can be boosted by the suggested project. Egypt may naturally and peacefully increase its cooperation with and presence in Gaza, which should lead to increased security around the border between Egypt and Gaza. This type of project could also get international support and can be a role-model for cooperation and trust-building between neighbours in the Middle East region. This study have also compared with more than five different alternatives

Human waste management in first phase response, protecting ground water and human health: a case study from Haiyan 2013

This briefing paper presents a case study of a Peepoo implementation in first phase humanitarian response. The case is taken from the Philippines, post typhoon Haiyan in 2013 and aims to demonstrate a safe way of handling of human waste without risking the contamination of water. This briefing paper will outline how the organisations, ACF, arche noVa, Oxfam and Red Cross implemented the Peepoo solution and lesson learnt so far. Some of the key lessons learnt in the projects were that; it is preferred to have the Peepoos stocked locally in order to have them available for phase one in humanitarian response; training implementing staff on Peepoo prior to the intervention increases the probability of success; and exit strategies must be set up in order to secure good sanitation practices after the implementing organisations stop distributing Peepoos

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

Pointwise estimates for the Bergman kernel of the weighted Fock space

We prove upper pointwise estimates for the Bergman kernel of the weighted Fock space of entire functions in $L^2(e^{-2\phi})$ where $\phi$ is a subharmonic function with $\Delta \phi$ a doubling measure. We derive estimates for the canonical solution operator to the inhomogeneous Cauchy-Riemann equation and we characterize the compactness of this operator in terms of $\Delta \phi$

PFAS in the Drinking Water Source: Analysis of the Contamination Levels, Origin and Emission Rates

Groundwater contamination caused by the use of the aqueous film-forming foam (AFFF) containing per- and polyfluoroalkyl substances (PFAS) was investigated in southern Sweden. sigma PFAS concentrations in groundwater ranged between 20 and 20,000 ng L-1; PFAS composition was primarily represented by PFOS and PFHxS. The PFAS chain length was suggested to have an impact on the contaminant distribution and transport in the groundwater. PFAS profiling showed that the use of PFSAs- and PFCAs/FTSAs-based PFAS-AFFF can be a contributor to PFAS contamination of the drinking water source (groundwater). PFAS emission was connected to PFAS-AFFF use during the fire-training and fire-fighting equipment tests at the studied location. PFAS emission per individual fire training was (semi-quantitatively) estimated as [1.4 < 11.5 +/- 5.7 < 43.7 kg] (n = 20,000). The annual emission estimates varied as [11 < 401 +/- 233 < 1125 kg yr(-1)] (n = 1005) considering possible [2 < 35 +/- 20 < 96] individual fire-training sessions per year

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

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