Chern classes of tropical vector bundles

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in Torchiani, C., Line Bundles on Tropical Varieties, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern, 2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.     

Ample and spanned vector bundles of top Chern number two on smooth projective varieties

The purpose of this paper is to classify ample and spanned vector bundles of top Chern number two on smooth projective varieties of arbitrary dimension defined over an algebraically closed field of characteristic zero.

     

Chern numbers of ample vector bundles on toric surfaces

This article shows a number of strong inequalities that hold for the Chern numbers , of any ample vector bundle of rank on a smooth toric projective surface, , whose topological Euler characteristic is . One general lower bound for proven in this article has leading term . Using Bogomolov instability, strong lower bounds for are also given. Using the new inequalities, the exceptions to the lower bounds 4e(S)$"> and e(S)$"> are classified.

     

The first Chern form on moduli of parabolic bundles

For moduli space of stable parabolic bundles on a compact Riemann surface, we derive an explicit formula for the curvature of its canonical line bundle with respect to Quillen’s metric and interprete it as a local index theorem for the family of \(\overline\partial\)-operators in the associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kähler form on the moduli space), and one nonstandard, called a cuspidal defect, that is defined by means of special values of the Eisenstein–Maass series. The cuspidal defect is explicitly expressed through the curvature forms of certain natural line bundles on the moduli space related to the parabolic structure. We also compare our result with Witten’s volume computation.     

Monads and instanton bundles on smooth hyperquadrics

In this paper we study instanton bundles on quadric hypersurfaces from the algebraic and geometric point of view. We discuss examples and we give a cohomological characterization (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the connectedness of the first cohomology module for rank two vector bundles on projective space

We provide sufficient conditions for the connectedness of the Hartshorne-Rao module of integral subcanonical curves and of the first cohomology module for rank two vector bundles on projective space.

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Sunto Si danno condizioni sufficienti per la connessità del modulo di Hartshorne-Rao di una curva integra sottocanonica e del primo modulo di coomologia di un fibrato vettoriale di rango due sullo spazio proiettivo.

     

Chern classes of bundles on blown-up surfaces

Consider the blow upπ [Xtilde]→ Xof a complex surface Xat a point. Let [Etilde] be a holomorphic bundle on [Xtilde]whose restriction to the exceptional divisor is O(j)n ? O(-j), and define E+(π_* E*)^vv. Friedman and Morgan gave the following bounds for the second Chern classes j≤ c ₂([Etilde])- c ₂(E) ≤ j ²We show that these bounds are sharp.     

The Hochschild class of the Chern character for semifinite spectral triples

We provide a proof of Connes’ formula for a representative of the Hochschild class of the Chern character for (p,∞)-summable spectral triples. Our proof is valid for all semifinite von Neumann algebras, and all integral p?1. We employ the minimum possible hypotheses on the spectral triples.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

The Atiyah class of generalized holomorphic vector bundles

We introduce the notion of Atiyah class of a generalized holomorphic vector bundle, which captures the obstruction to the existence of generalized holomorphic connections on the bundle. As in the classical holomorphic case, this Atiyah class can be defined in three different ways: using Čech cohomology, using the first-jet short exact sequence, or adopting the Lie pair point of view.     

A relation between the parabolic Chern characters of the de Rham bundles

In this paper, we consider the weight i de Rham–Gauss–Manin bundles on a smooth variety arising from a smooth projective morphism for . We associate to each weight i de Rham bundle, a certain parabolic bundle on S and consider their parabolic Chern characters in the rational Chow groups, for a good compactification S of U. We show the triviality of the alternating sum of these parabolic bundles in the (positive degree) rational Chow groups. This removes the hypothesis of semistable reduction in the original result of this kind due to Esnault and Viehweg.     

Kähler-Ricci solitons on toric manifolds with positive first Chern class

In this paper we prove there exists a Kähler-Ricci soliton, unique up to holomorphic automorphisms, on any toric Kähler manifold with positive first Chern class, and the Kähler-Ricci soliton is a Kähler-Einstein metric if and only if the Futaki invariant vanishes.