The K\"ahler-Ricci flow on surfaces of positive Kodaira dimension

The existence of K\"ahler-Einstein metrics on a compact K\"ahler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the K\"ahler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized K\"ahler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metric classification for K\"aher surfaces with a numerical effective canonical line bundle by the K\"ahler-Ricci flow. In general, we propose a program of finding canonical metrics on canonical models of projective varieties of positive Kodaira dimension

Forward Symplectic Integrators and the Long Time Phase Error in Periodic Motions

We show that when time-reversible symplectic algorithms are used to solve periodic motions, the energy error after one period is generally two orders higher than that of the algorithm. By use of correctable algorithms, we show that the phase error can also be eliminated two orders higher than that of the integrator. The use of fourth order forward time step integrators can result in sixth order accuracy for the phase error and eighth accuracy in the periodic energy. We study the 1-D harmonic oscillator and the 2-D Kepler problem in great details, and compare the effectiveness of some recent fourth order algorithms.Comment: Submitted to Phys. Rev. E, 29 Page

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

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

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.Comment: additional references, fixed some typos, minor additions including a questions and some remark

Arbuscular Mycorrhizal Symbiosis Leads to Differential Regulation of Drought-Responsive Genes in Tissue-Specific Root Cells of Common Bean

Arbuscular mycorrhizal fungi (AMF) colonization in plants promotes both local and systemic changes in the gene expression profiles of the host that might be relevant for drought-stress perception and response. Drought-tolerant common bean plants (cv. BAT 477), colonized by a mixture of AMF (Glomus clarum, Acaulospora scrobiculata, and Gigaspora rosea), were exposed to a water deprivation regime of 96 h during pre-flowering. Root transcriptomes were accessed through RNA-Seq revealing a set of 9,965 transcripts with significant differential regulation in inoculated plants during a water deficit event, and 10,569 in non-inoculated. These data include 1,589 transcripts that are exclusively regulated by AMF-inoculation, and 2,313 under non-inoculation conditions. Relative gene expression analyses of nine aquaporin-related transcripts were performed in roots and leaves of plants harvested at initial stages of treatment. Significant shifts in gene expression were detected in AM water deficit-treated roots, in relation to non-inoculated, between 48 and 72 h. Leaves also showed significant mycorrhizal influence in gene expression, especially after 96 h. Root cortical cells, harboring or not arbuscules, were collected from both inoculation treatments through a laser microdissection-based technique. This allowed the identification of transcripts, such as the aquaporin PvPIP2;3 and Glucan 1,3 β-Glucosidase, that are unique to arbuscule-containing cells. During the water deficit treatment, AMF colonization exerted a fine-tune regulation in the expression of genes in the host. That seemed to initiate in arbuscule-containing cells and, as the stressful condition persisted, propagated to the whole-plant through secondary signaling events. Collectively, these results demonstrate that arbuscular mycorrhization leads to shifts in common bean’s transcriptome that could potentially impact its adaptation capacity during water deficit events

Big Line Bundles over Arithmetic Varieties

We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution

Molecular characterization of fluoroquinolone-resistant Mycobacterium tuberculosis clinical isolates from Shanghai, China

China is one of the countries with the highest prevalence of fluoroquinolone-resistant (FQ r) Mycobacterium tuberculosis. Nevertheless, knowledge on the molecular characterization of the FQ r M. tuberculosis strains of this region remains very limited. This study was performed to investigate the frequencies and types of mutations present in FQ r M. tuberculosis clinical isolates collected in Shanghai, China. A total of 206 FQ r M. tuberculosis strains and 21 ofloxacin-sensitive (FQ s) M. tuberculosis strains were isolated from patients with pulmonary tuberculosis in Shanghai. The phenotypic drug susceptibilities were determined by the proportion method, and the mutations inside quinolone resistance-determining region (QRDR) of gyrA and gyrB genes were identified by DNA sequence analyses. Among 206 FQ r M. tuberculosis strains, 44% (90/206) were multidrug-resistant isolates and 39% (81/206) were extensively drug-resistant isolates. Only 9% (19/206) were monoresistant to ofloxacin. In total, 79.1% (163/206) of FQ r isolates harboured mutations in either gyrA or gyrB QRDR. Mutations in gyrA QRDR were found in 75.7% (156/206) of FQ r clinical isolates. Among those gyrA mutants, a majority (75.6%) harboured mutations at amino acid position 94, with D94G being the most frequent amino acid substitution. Mutations in gyrA QRDR showed 100% positive predictive value for FQ r M. tuberculosis in China. Mutations in gyrB were observed in 15.5% (32/206) of FQ r clinical isolates. Ten novel mutations were identified in gyrB. However, most of them also harboured mutations in gyrA, limiting their contribution to FQ r resistance in M. tuberculosis. Our findings indicated that, similar to other geographic regions, mutations in gyrA were shown to be the major mechanism of FQ r resistance in M. tuberculosis isolates. The mutations in gyrA QRDR can be a good molecular surrogate marker for detecting FQ r M. tuberculosis in China. © 2012 Elsevier Inc.postprin