Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic family.Comment: 19 pages, This is the final version with significant corrections and improvements. To appear in Arkiv f\"or matemati

Effective algebraic degeneracy

We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}

A Schumpeterian approach to examine the development boundary of casino tourism

In The Theory of Economic Development, Joseph Schumpeter ([1934] 1983, Chapter II) explicates how the dynamic role of innovative mindset and ability of entrepreneurs, through introducing new methods of production and new economic commodities, leads up to the expansion of the boundary for economic/industrial development. By following Schumpeter’s approach, we explore the development boundary of casino tourism. Through a post facto analysis of the hard evidences derived across the world (e.g., Las Vegas, Macao and Singapore) since the 1980s, the dimensions of innovation to the development boundary of casino tourism are scrutinized and explicated. Besides, we uncovered that the expanding capacity and changing institutions of casino tourism generates positive feedbacks which further stimulates the progressive changes in the innovative process. Nevertheless, different from the industrial world being examined by Schumpeter, casino tourism supplies leisure and related hospitality services by incorporating a unique component of casino gaming. In practice, innovations in casino gaming may promote gambling behavior by various communities which is not necessarily a socially desired consequence. This actually generates a new form of negative externality that may as well set a limit to the industry’s development boundary at both the regional and global levels

Local syzygies of multiplier ideals

In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties involving higher syzygies. It follows that in dimensions three and higher, multiplier ideals are very special among all integrally closed ideals.Comment: 8 page

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

On the cohomology of pseudoeffective line bundles

The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

Geometric Aspects of the Moduli Space of Riemann Surfaces

This is a survey of our recent results on the geometry of moduli spaces and Teichmuller spaces of Riemann surfaces appeared in math.DG/0403068 and math.DG/0409220. We introduce new metrics on the moduli and the Teichmuller spaces of Riemann surfaces with very good properties, study their curvatures and boundary behaviors in great detail. Based on the careful analysis of these new metrics, we have a good understanding of the Kahler-Einstein metric from which we prove that the logarithmic cotangent bundle of the moduli space is stable. Another corolary is a proof of the equivalences of all of the known classical complete metrics to the new metrics, in particular Yau's conjectures in the early 80s on the equivalences of the Kahler-Einstein metric to the Teichmuller and the Bergman metric.Comment: Survey article of our recent results on the subject. Typoes corrrecte

Long-term cardiovascular outcomes after pregnancy in women with heart disease

BACKGROUND: Women with heart disease are at risk for pregnancy complications, but their long-term cardiovascular outcomes after pregnancy are not known. METHODS AND RESULTS: We examined long-term cardiovascular outcomes after pregnancy in 1014 consecutive women with heart disease and a matched group of 2028 women without heart disease. The primary outcome was a composite of mortality, heart failure, atrial fibrillation, stroke, myocardial infarction, or arrhythmia. Secondary outcomes included cardiac procedures and new hypertension or diabetes mellitus. We compared the rates of these outcomes between women with and without heart disease and adjusted for maternal and pregnancy characteristics. We also determined if pregnancy risk prediction tools (CARPREG [Canadian Cardiac Disease in Pregnancy] and World Health Organization) could stratify long-term risks. At 20-year follow-up, a primary outcome occurred in 33.1% of women with heart disease, compared with 2.1% of women without heart disease. Thirty-one percent of women with heart disease required a cardiac procedure. The primary outcome (adjusted hazard ratio, 19.6; 95% CI, 13.8–29.0; P\u3c0.0001) and new hypertension or diabetes mellitus (adjusted hazard ratio, 1.6; 95% CI, 1.4–2.0; P\u3c0.0001) were more frequent in women with heart disease compared with those without. Pregnancy risk prediction tools further stratified the late cardiovascular risks in women with heart disease, a primary outcome occurring in up to 54% of women in the highest pregnancy risk category. CONCLUSIONS: Following pregnancy, women with heart disease are at high risk for adverse long-term cardiovascular outcomes. Current pregnancy risk prediction tools can identify women at highest risk for long-term cardiovascular events

5-Phenyl-2-(4-pyrid­yl)pyrimidine

The title compound, C15H11N3, crystallizes with two independent mol­ecules in the asymmetric unit. The dihedral angles between the phenyl and pyridine rings in each mol­ecule are 53.48 (5) and 50.80 (5)°. In the crystal structure, weak inter­molecular C—H⋯N hydrogen bonds connect mol­ecules into one-dimensional chains. In addition, the crystal structure is stabilized by weak C—H⋯π(arene) inter­actions

Cohomology of bundles on homological Hopf manifold

We discuss the properties of complex manifolds having rational homology of $S^1 \times S^{2n-1}$ including those constructed by Hopf, Kodaira and Brieskorn-van de Ven. We extend certain previously known vanishing properties of cohomology of bundles on such manifolds.As an application we consider degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex variables and Complex Geometry. Xiamen. Chin