Escisión de fibrados en G(1,4) y sus variedades

La memoria se divide en dos partes diferenciadas. En la primera, correspondiente al capítulo uno, se clasifican los fibrados sin cohomología intermedia de la Grassamanniana G(1,4) de las rectas de P4. A diferencia de lo que ocurre en la Grassamanniana de rectas P3, se obtienen familias infinitas de fibrados. Como paso particular de la clasificación se caracterizan cohomológicamente las sumas directas de fibrados trivales y fibrados universales de la Grassamanniana, Q, S y S (y sus twists). La segunda parte, dividida en dos capítulos (2 y 3), consiste en la clasificación de las subvariedades lisas y de dimensión tres de G(1,4), llamadas congruencias, que además verifican que el fibrado universal cociente, Q, restringido a ellas escinde en suma directa de fibrados no lineales. La clasificación se hace interpretando geométricamente tanto el significado que tiene esta escisión, como el del número de secciones globales independientes que tienen los correspondientes fibrados lineale

Positivity for Higgs vector bundles: criteria and applications

Working in the category of smooth projective varieties over an algebraically closed field of characteristic 0, we review notions of ampleness and numerical nefness for Higgs bundles which "feel" the Higgs field and formulate criteria of the Barton-Kleiman type for these notions. We give an application to minimal surfaces of general type that saturate the Miyaoka-Yau inequality, showing that their cotangent bundle is ample. This will use results by Langer that imply that also for varieties over algebraically closed field of characteristic zero the so-called Simpson system is stable.Comment: 12 pages. v2: minor change in one exampl

Metrics on semistable and numerically effective Higgs bundles

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact K\"ahler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria for Higgs bundles on projective manifolds of any dimension.Comment: 25 pages. Changes in the expositio

Numerically flat Higgs vector bundles

After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.Comment: 11 page

On a conjecture about Higgs bundles for rank 2 and some inequalities

We briefly review an open conjecture about Higgs bundles that are semistable with after pulling back to any curve, and prove it in the rank 2 case. We also prove a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton-Lazarsfeld inequalities for numerically effective vector bundles.Comment: 13 pages. v2: 14 pages. Some results have been strengthened and the exposition has been reorganized. v3: minor changes, final version to appear in Mediterranean J. Mat

Semistable and numerically effective principal (Higgs) bundles

We study Miyaoka-type semistability criteria for principal Higgs G-bundles E on complex projective manifolds of any dimension. We prove that E has the property of being semistable after pullback to any projective curve if and only if certain line bundles, obtained from some characters of the parabolic subgroups of G, are numerically effective. One also proves that these conditions are met for semistable principal Higgs bundles whose adjoint bundle has vanishing second Chern class. In a second part of the paper, we introduce notions of numerical effectiveness and numerical flatness for principal (Higgs) bundles, discussing their main properties. For (non-Higgs) principal bundles, we show that a numerically flat principal bundle admits a reduction to a Levi factor which has a flat Hermitian-Yang-Mills connection, and, as a consequence, that the cohomology ring of a numerically flat principal bundle with real coefficients is trivial. To our knowledge this notion of numerical effectiveness is new even in the case of (non-Higgs) principal bundles.Comment: v1: 25 pages. This submission supersedes arXiv:0809.3936. v2: 28 pages, includes changes suggested by the referees. v3: Final version to appear in Advances in Mathematic

Nutrient supply does play a role on the structure of marine picophytoplankton communities

Conference communicationThe Margalef´s mandala (1978) is a simplified bottom-up control model that explains how mixing and nutrient concentration determine the composition of marine phytoplankton communities. Due to the difficulties of measuring turbulence in the field, previous attempts to verify this model have applied different proxies for nutrient supply, and very often used interchangeably the terms mixing and stratification. Moreover, because the mandala was conceived before the discovery of smaller phytoplankton groups (picoplankton <2 µm), it describes only the succession of vegetative phases of microplankton. In order to test the applicability of the classical mandala to picoplankton groups, we used a multidisciplinary approach including specifically designed field observations supported by remote sensing, database analyses, and modeling and laboratory chemostat experiments. Simultaneous estimates of nitrate diffusive fluxes, derived from microturbulence observations, and picoplankton abundance collected in more than 200 stations, spanning widely different hydrographic regimes, showed that the contribution of eukaryotes to picoautotrophic biomass increases with nutrient supply, whereas that of picocyanobacteria shows the opposite trend. These findings were supported by laboratory and modeling chemostat experiments that reproduced the competitive dynamics between picoeukaryote sand picocyanobacteria as a function of changing nutrient supply. Our results indicate that nutrient supply controls the distribution of picoplankton functional groups in the ocean, further supporting the model proposed by Margalef.Spanish Governmen

Congruences on G(1,4) with split universal quotient bundle

This work provides a complete classification of the smooth three-folds in the Grassmann variety of lines in P-4, for which the restriction of the universal quotient bundle is a direct sum of two line bundles. For this purpose we use the geometrical interpretation of the splitting of the quotient bundle as well as the meaning of the number of the independent global sections of each of its summands

Metrics on semistable and numerically effective Higgs bundles

We provide notions of numerical effectiveness and numerical flatness for Higgs vector bundles on compact K"ahler manifolds in terms of fibre metrics. We prove several properties of bundles satisfying such conditions and in particular we show that numerically flat Higgs bundles have vanishing Chern classes, and that they admit filtrations whose quotients are stable flat Higgs bundles. We compare these definitions with those previously given in the case of projective varieties. Finally we study the relations between numerically effectiveness and semistability, establishing semistability criteria for Higgs bundles on projective manifolds of any dimension