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)

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}}

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

Roles of a novel splice variant of human IFI16 in innate immune response

Poster Presentation - Theme 4: Infection & immunityDNA from viral or bacterial pathogens activates innate immune response. The recognition of self-DNA would induce autoimmune diseases such as systemic lupus erythematosus (SLE). In human, AIM2 like receptors (ALRs) including AIM2, IFI16, IFIX and MNDA are DNA binding proteins implicated in DNA sensing. Most ALRs contain an N-terminal pyrin domain and C-terminal HIN200 domains. However, mouse SLE susceptibility locus p202 encodes only HIN200 domains. A human homolog of p202 was not found. Here, we identified and characterized a novel splice variant of human IFI16, which has a similar domain structure as mouse p202. We named it ...postprin

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

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

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

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