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

Differential Forms on Log Canonical Spaces

The present paper is concerned with differential forms on log canonical varieties. It is shown that any p-form defined on the smooth locus of a variety with canonical or klt singularities extends regularly to any resolution of singularities. In fact, a much more general theorem for log canonical pairs is established. The proof relies on vanishing theorems for log canonical varieties and on methods of the minimal model program. In addition, a theory of differential forms on dlt pairs is developed. It is shown that many of the fundamental theorems and techniques known for sheaves of logarithmic differentials on smooth varieties also hold in the dlt setting. Immediate applications include the existence of a pull-back map for reflexive differentials, generalisations of Bogomolov-Sommese type vanishing results, and a positive answer to the Lipman-Zariski conjecture for klt spaces.Comment: 72 pages, 6 figures. A shortened version of this paper has appeared in Publications math\'ematiques de l'IH\'ES. The final publication is available at http://www.springerlink.co

Anti-Pluricanonical Systems On Q-Fano Threefolds

We investigate birationality of the anti-pluricanonical map $\phi_{-m}$, the rational map defined by the anti-pluricanonical system $|-mK|$, on $\mathbb{Q}$-Fano threefolds.Comment: 18 page

Finite Generation of Canonical Ring by Analytic Method

In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425)

Overdiagnosis and overtreatment of breast cancer: Is overdiagnosis an issue for radiologists?

Overdiagnosis is diagnosis of cancers that would not present within the life of the patient and is one of the downsides of screening. This applies to low-grade ductal carcinoma in situ and some small grade 1 invasive cancers. Radiologists are responsible for cancer diagnosis, but at the time of diagnosis they cannot determine whether a particular low-grade diagnosis is one to which the definition of overdiagnosis applies. Overdiagnosis is likely to be driven by technological developments, including digital mammography, computer-aided detection and improved biopsy techniques. It is also driven by the patient's fear that cancer will be missed and the doctor's fear of litigation. It is therefore an issue of importance for radiologists, presenting them with difficult fine-tuned decisions in every assessment clinic that are ultimately counted later by those who evaluate their screening

Carbonate-Templated Self-Assembly of an Alkylthiolate-Bridged Cadmium Macrocycle

In the presence of Cd(ClO4)2 and a base, a new mixed N,S-donor alkylthiolate ligand supported both carbonate formation from atmospheric CO2 and the self-assembly of a novel bicapped puckered (CdS)6 molecular wheel. The remarkable stability of the complex was demonstrated by slow intermolecular ligand exchange on the 2J(HH) and J(111/113Cd1H) time scales at elevated temperature. Both CO2 and the base were required to convert amorphous “CdLClO4” precipitated in the absence of air to the carbonate complex. The complex shares structural features with the ζ-carbonic anhydrase class associating cadmium(II) with the biogeochemical cycling of carbon and is the first structurally characterized carbonate complex of any metal involving an alkylthiolate ligand

Magnetic resonance mammography in the evaluation of recurrence at the prior lumpectomy site after conservative surgery and radiotherapy

INTRODUCTION: The aim was to assess the value of magnetic resonance mammography (MRM) in the detection of recurrent breast cancer on the prior lumpectomy site in patients with previous conservative surgery and radiotherapy. METHODS: Between April 1999 and July 2003, 93 consecutive patients with breast cancer treated with conservative surgery and radiotherapy underwent MRM, when a malignant lesion on the site of lumpectomy was suspected by ultrasound and/or mammography. MRM scans were evaluated by morphological and dynamic characteristics. MRM diagnosis was compared with histology or with a 36-month imaging follow-up. Enhancing areas independent of the prior lumpectomy site, incidentally detected during the MRM, were also evaluated. RESULTS: MRM findings were compared with histology in 29 patients and with a 36-month follow-up in 64 patients. MRM showed 90% sensitivity, 91.6% specificity, 56.3% positive predictive value and 98.7% negative predictive value for detection of recurrence on the surgical scar. MRM detected 13 lesions remote from the scar. The overall sensitivity, specificity, positive predictive value and negative predictive value of MRM for detection of breast malignancy were 93.8%, 90%, 62.5% and 98.8%, respectively. CONCLUSION: MRM is a sensitive method to differentiate recurrence from post-treatment changes at the prior lumpectomy site after conservative surgery and radiation therapy. The high negative predictive value of this technique can avoid unnecessary biopsies or surgical treatments

Breast MRI: guidelines from the European Society of Breast Imaging

The aim of breast MRI is to obtain a reliable evaluation of any lesion within the breast. It is currently always used as an adjunct to the standard diagnostic procedures of the breast, i.e., clinical examination, mammography and ultrasound. Whereas the sensitivity of breast MRI is usually very high, specificity—as in all breast imaging modalities—depends on many factors such as reader expertise, use of adequate techniques and composition of the patient cohorts. Since breast MRI will always yield MR-only visible questionable lesions that require an MR-guided intervention for clarification, MRI should only be offered by institutions that can also offer a MRI-guided breast biopsy or that are in close contact with a site that can perform this type of biopsy for them. Radiologists involved in breast imaging should ensure that they have a thorough knowledge of the MRI techniques that are necessary for breast imaging, that they know how to evaluate a breast MRI using the ACR BI-RADS MRI lexicon, and most important, when to perform breast MRI. This manuscript provides guidelines on the current best practice for the use of breast MRI, and the methods to be used, from the European Society of Breast Imaging (EUSOBI)