Beatriz Graña Otero

University of Salamanca · Department of Mathematics

We briefly review an open conjecture about Higgs bundles that are semistable after pulling back to any curve, and prove it in the rank 2 case. We also prove some results in higher rank under suitable additional assumptions. Moreover, we establish a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton–Lazarsfeld ine...

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 satura...

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.

After reviewing some “fundamental group schemes” that can be attached to a variety by means of Tannaka duality, we consider the example of the Higgs fundamental group scheme, surveying its main properties and relations with the other fundamental groups, and giving some examples.

In this paper we study Higgs and co-Higgs $G$-bundles on compact K\"ahler manifolds $X$. Our main results are: (1) If $X$ is Calabi--Yau, and $(E,\,\theta)$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the principal $G$-bundle $E$ is semistable. In particular, there is a deformation retract of ${\mathcal M}_H(G)$ onto $\mathcal M(G)$,...

We generalize the Hitchin-Kobayashi correspondence between semistability and the existence of approximate Hermitian-Yang-Mills structures to the case of principal Higgs bundles. We prove that a principal Higgs bundle on a compact Kaehler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it adm...

We announce a result about the extension of the Hitchin-Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian-Yang-Mills structure.

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...

It is known that semistable sheaves V admit a filtration whose quotients are stable and have the same slope of V, named the Jordan–Hölder filtration. We give the analogous result for principal Higgs bundles on curves. Let G be a reductive algebraic group over C, if E=(E,ϕ) is a semistable principal Higgs G-bundle, there exists a parabolic subgroup...

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...

We give a Miyaoka-type semistability criterion for principal Higgs G-bundles E on complex projective manifolds of any dimension, i.e., we prove that E is semistable and the second Chern class of its adjoint bundle vanishes if and only if certain line bundles, obtained from the characters of the parabolic subgroups of $G$, are numerically effective....

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 quo...

This work provides a complete classification of the smooth three-folds in the Grassmann variety of lines in , 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 indep...

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...

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...