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

Abstract

In this paper, we consider the weight i de Rham-Gauss-Manin bundles on a smooth variety arising from a smooth projective morphism f : X U −→ U for i ≥ 0. 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.

Iterated destabilizing modifications for vector bundles with connection

“…These objects correspond to parabolic vector bundles with real parabolic weights and λ-connection ∇ respecting the parabolic filtration and inducing the appropriate multiple of the identity on each graded piece. If in addition the conjugacy classes are assumed to be of finite order, the parabolic weights should be rational, and our objects may then be viewed as lying on an orbicurve or Deligne-Mumford stack X ′ with ramification orders m i corresponding to the common denominators of the weights at x i [6] [12] [28]. Denote by…”

Local Systems on Proper Algebraic V -manifolds

“…However, when the Chern classes vanish then the condition no longer depends on a choice of polarization so we can expect that it gives a reasonable condition on a DM-stack too. Recall that Vistoli's theorem provides the notion of rational Chern classes on X, see [53]. Thus, the condition c i (E) = 0 in H 2i (|X|, Q) makes good sense.…”

“…Let Z → X be the root stack corresponding to denominators n i for the irreducible components D i of D. As in the original article of Seshadri [93], a vector bundle on Z corresponds to a parabolic bundle on (X, D) such that the weights along D i are in 1 n i Z. This correspondence has been used and studied by many authors, see for example Boden [18], Balaji et al [5], Biswas [11], Borne [20] [21] as well as [53] and [54].…”

“…Our discussion of nonabelian harmonic theory on stacks adds to a subject which has already been treated by several authors [12] [36] [45] [103], and for the case of root stacks it is closely related to harmonic theory for parabolic bundles [15] [29] [64] [74] [70] [71] [84] [100]. The relationship between local systems and ramified covers can be related to the Chern class calculations of Esnault and Viehweg in [41], going back also to [80], and related formulae involving parabolic and orbifold bundles were studied in [53] [54]. This subject also connects with Viehweg's recent works such as [109] [110], since Shimura varieties are best considered as DM-stacks and indeed symmetric spaces were a main part of the original motivation for the notion of V -manifold [89] [90] which appears in our title.…”

A Construction of Chern Classes of Parabolic Vector Bundles

“…In [13], Maruyama and Yokogawa introduced parabolic vector bundles on higher dimensional complex projective varieties. The notion of Chern classes of a vector bundle extends to the context of parabolic vector bundles [3,12,15].…”