On surfaces of general type with maximal Albanese dimension

Given a minimal surface equipped with a generically finite map to an Abelian variety, we give an optimal bound on the canonical degree of a rational or an elliptic curve. As a corollary, we obtain the finiteness of rational and elliptic curves on any surface of general type with two linearly independent regular one forms.Comment: 13 page

A refined Kodaira dimension and its canonical fibration

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of $f^*K_Y$, where $K_Y$ is the canonical sheaf of $Y$. We define the Kodaira dimension of the orbifold base $(Y,f)$ by that of $L_f$ and call the fibration to be of general type if this dimension is the maximal \dime Y. We call $X$ special if it does not have a general type fibration with positive dimensional base and call $f$ special if its general fibers are. We note that Iitaka and rationally connected fibrations are special. Our main theorem is that any compact complex $X$ has a special fibration of general type which dominates any fibration of general type and factors through any special fibration of general type from $X$ in the birational category, thus unique in this category. Also, the fibration has positive dimensional fibers if $X$ is not of general type thus resolving a problem in Mori's program for varieties with negative Kodaira dimension. Our main theorem is stated in the full orbifold context of log pairs as in Mori's program via which we give an application to the Albanese map of projective manifolds with nef anticanonical bundle a part of which was derived using positive charateristic techniques by Qi Zhang.Comment: Some additional theorems and remarks added and corrections mad

Maser mechanism of optical pulsations from anomalous X-ray pulsar 4U0142+61

A maser curvature emission mechanism in the presence of curvature drift is used to explain the optical pulsations from anomalous X-ray pulsars. For the source of AXP0142+61,the optical pulsation occurs at the radial distance $R(\nu_M)\sim 4.75\times 10^9$ cm to the neutron star. The corresponding curvature maser frequency is about $\nu_M\approx1.39\times 10^{14}$ Hz. The result is consistent with the observation of the optical pulsations from the anomalous X-ray pulsar 4U0142+61.Comment: 18pages,LaTeX2e, accetpetd for publication in MNRA

Positivity criteria for log canonical divisors and hyperbolicity

Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the log minimal model program, natural to our setting, we obtain a geometric criterion for K_X+D to be numerically effective as well as a geometric version of the cone theorem, generalizing to the context of log pairs these results of Mori. A criterion for K_X+D to be pseudo-effective with mild hypothesis on D follows. We also obtain, assuming the abundance conjecture and the existence of rational curves on Calabi-Yau manifolds, an optimal geometric sharpening of the Nakai-Moishezon criterion for the ampleness of a divisor of the form K_X+D, a criterion verified under a canonical hyperbolicity assumption on (X,D). Without these conjectures, we verify this ampleness criterion with assumptions on the number of ample and non ample components of D.Comment: Journal f\"ur die reine und angewandte Mathematik (to appear); final version; this arXiv version contains the proof of Remark 4.9 (2-dimensional case for dlt pairs) as Appendi

Algebraic Surfaces Holomorphically Dominable by C2

An n-dimensional complex manifold M is said to be (holomorphically) dominable by \CC^n if there is a map F:\CC^n \ra M which is holomorphic such that the Jacobian determinant $\det(DF)$ is not identically zero. Such a map F is called a dominating map. In this paper, we attempt to classify algebraic surfaces X which are dominable by \CC^2 using a combination of techniques from algebraic topology, complex geometry and analysis. One of the key tools in the study of algebraic surfaces is the notion of Kodaira dimension (defined in section 2). By Kodaira's pioneering work and its extensions, an algebraic surface which is dominable by \CC^2 must have Kodaira dimension less than two. Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by \CC^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is rational without any logarithmic 1-form. More specifically, in the case when X is compact (namely projective), we need to exclude only the case when X is birationally equivalent to a K3 surface (a simply connected compact complex surface which admits a globally non-vanishing holomorphic 2-form) that is neither elliptic nor Kummer (see sections 3 and 4 for the definition of these types of surfaces).Comment: 39 page

On the nature of the flares from three candidate tidal disruption events

The X-ray flares of NGC 5905, RX J1242.6-1119A, and RX J1624.9+7554 observed by Chandra in 2001 and 2002 have been suggested as the candidate tidal disruption events. The distinct features observed from these events may be used to determine the type of a star tidally disrupted by a massive black hole. We investigate these three events, focusing on the differences for the tidal disruption of a giant star and a main sequence, resulted from their different relation between the mass and the radius. We argue that their X-ray flare properties could be modeled by the partial stripping of the outer layers of a solar type star. The tidal disruption of a giant star is excluded completely. This result may be useful for understanding the growth of a supermassive black hole by capturing stars, versus the growth mode through continuous mass accretion.Comment: 3 pages, Conference proceeding to appear in "The Central Engine of Active Galactic Nuclei", ed. L. C. Ho and J.-M. Wang (San Francisco: ASP

Regularity criterion and classification for algebras of Jordan type

We show that Artin-Schelter regularity of a $\mathbb{Z}$-graded algebra can be examined by its associated $\mathbb{Z}^r$-graded algebra. We prove that there is exactly one class of four-dimensional Artin-Schelter regular algebras with two generators of degree one in the Jordan case. This class is strongly noetherian, Auslander regular, and Cohen-Macaulay. Their automorphisms and point modules are described.Comment: 24 page

The structure of connected (graded) Hopf algebras

In this paper, we establish a structure theorem for connected graded Hopf algebras over a field of characteristic $0$ by claiming the existence of a family of homogeneous generators and a total order on the index set that satisfy some desirable conditions. The approach to the structure theorem is constructive, based on the combinatorial properties of Lyndon words and the standard bracketing on words. As a surprising consequence of the structure theorem, we show that connected graded Hopf algebras of finite Gelfand-Kirillov dimension over a field of characteristic $0$ are all iterated Hopf Ore extensions of the base field. In addition, some keystone facts of connected Hopf algebras over a field of characteristic $0$ are observed as corollaries of the structure theorem, without the assumptions of having finite Gelfand-Kirillov dimension (or affineness) on Hopf algebras or of that the base field is algebraically closed.Comment: 25 pages, add more details on some proof

Nonrelativistic phase in gamma-ray burst afterglows

The discovery of multiband afterglows definitely shows that most $\gamma$-ray bursts are of cosmological origin. $\gamma$-ray bursts are found to be one of the most violent explosive phenomena in the Universe, in which astonishing ultra-relativistic motions are involved. In this article, the multiband observational characteristics of $\gamma$-ray bursts and their afterglows are briefly reviewed. The standard model of $\gamma$-ray bursts, i.e. the fireball model, is described. Emphasis is then put on the importance of the nonrelativistic phase of afterglows. The concept of deep Newtonian phase is elaborated. A generic dynamical model that is applicable in both the relativistic and nonrelativistic phases is introduced. Based on these elaborations, the overall afterglow behaviors, from the very early stages to the very late stages, can be conveniently calculated.Comment: A review paper accepted for publication in a Chinese journal of: Progress in Natural Science. 6 figures, 21 page

Nakayama automorphisms of twisted tensor products

In this paper, we study homological properties of twisted tensor products of connected graded algebras. We focus on the Ext-algebras of twisted tensor products with a certain form of twisting maps firstly. We show those Ext-algebras are also twisted tensor products, and depict the twisting maps for such Ext-algebras in-depth. With those preparations, we describe Nakayama automorphisms of twisted tensor products of noetherian Artin-Schelter regular algebras