In this paper, we consider the weight $i$ de Rham--Gauss--Manin bundles on a smooth variety arising from a smooth projective morphism $f:X\_U\lrar U$ for $i\geq 0$. 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.
We calculate the parabolic Chern character of a bundle with locally abelian parabolic structure on a smooth strict normal crossings divisor, using the definition in terms of Deligne-Mumford stacks. We obtain explicit formulas for $ch_1$, $ch_2$ and $ch_3$, and verify that these correspond to the formulas given by Borne for $ch_1$ and Mochizuki for $ch_2$.
If $ D \in X$ is a curve with multiple points in a surface, a parabolic bundle defined on $(X;D)$ away from the singularities can be extended in several ways to a parabolic bundle on a resolution of singularities. We investigate the possible parabolic Chern classes for these extensions.
In this paper, we obtain an explicit formula for the Chern character of a locally abelian parabolic bundle in terms of its constituent bundles. Several features and variants of parabolic structures are discussed. Parabolic bundles arising from logarithmic connections form an important class of examples. As an application, we consider the situation when the local monodromies are semi-simple and are of finite order at infinity. In this case the parabolic Chern classes of the associated locally abelian parabolic bundle are deduced to be zero in the rational Deligne cohomology in degrees $\geq 2$.
Given a parabolic vector bundle, we construct for it a projectivization and tautological line bundle. These are analogs of the projectivization and tautological line bundle for an usual vector bundle. Using these we give a construction of the parabolic Chern classes.
We give a de Rham interpretation of Nahm's transform for certain parabolic harmonic bundles on the projective line and compare it to minimal Fourier--Laplace transform of $\mathcal{D}$-modules. We give an algebraic definition of a parabolic structure on the transformed bundle and show that it is compatible with the transformed harmonic metric.
Unitary representations of the fundamental group of a Kahler manifold correspond to polystable vector bundles (with vanishing Chern classes). Semisimple linear representations correspond to polystable Higgs bundles. In this paper we find the objects corresponding to affine representations: the linear part gives a Higgs bundle and the translation part corresponds to an element of a generalized de Rham cohomology.
We give a formula for the Chern character on the DG-category of global matrix factorizations on a smooth scheme $X$ with superpotential $w\in Γ(\mathcal{O}_X)$. Our formula takes values in a Cech model for Hochschild homology. Our methods may also be adapted to get an explicit formula for the Chern character for perfect complexes of sheaves on $X$ taking values in right derived global sections of the De-Rham algebra. Along the way we prove that the DG version of the Chern character coincides with the classical one for perfect complexes.
Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms c_{i}(E;\nabla) in Γ(X, \cal{A}^{2i}_{X}). Here \cal{A}^{.}_{X} is the sheaf Beilinson adeles and \nabla is an adelic connection. When X is smooth these adeles calculate the algebraic De Rham cohomology, and c_{i}(E) = [c_{i}(E;\nabla)] are the usual Chern classes. We include three applications of the construction: (1) existence of adelic secondary (Chern-Simons) characteristic classes on any smooth X and any vector bundle E; (2) proof of the Bott Residue Formula for a vector field action…
Motivated by the Chern-Weil theory, we prove that for a given vector bundle $E$ on a smooth scheme $X$ over a field $k$ of any characteristic, the Chern classes of $E$ in the Hodge cohomology can be recovered from the Atiyah class. Although this problem was solved by Illusie in \cite{i}, we present another proof by means of derived algebraic geometry. Also, for a scheme $X$ over a field $k$ of characteristic $p$ with a vector bundle $E$ we construct elements $c^{cris}_n (E, α(E)) \in H_{dR}^{2n} (X) $ using an obstruction $α(E)$ to a lifting of $F^* E$ to a crystal modulo $p^2$ and prove that …
Classical Chern-Weil theory proves existence of a ring homomorphism - the Chern character - from the Grothendieck ring of locally free finite rank A-modules to the algebraic de Rham cohomology of A. In this paper we prove existence of a ring homomorphism from the Grothendieck ring of locally free finite rank A-modules with an L-connection to the Lie-Rinehart cohomology of L, where L is a Lie-Rinehart algebra. As a corollary we get the classical Chern character.
We study natural families of d-bar operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.
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 interpret it as a local index theorem for the family of dbar-operators in associated parabolic endomorphism bundles. The formula consists of two terms: one standard (proportional to the canonical Kaehler 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 curvat…
This is a sequel to "Kodaira-Saito vanishing via Higgs bundles in positive characteristic" (arXiv:1611.09880). However, unlike the previous paper, all the arguments here are in characteristic zero. The main result is a Kodaira vanishing theorem for semistable parabolic Higgs bundles with trivial parabolic Chern classes. This implies a general semipositivity theorem. This also implies a Kodaira-Saito vanishing theorem for complex variations of Hodge structure.
We introduce a notion of admissible Hermitian metrics on parabolic bundles and define positivity properties for the same. We develop Chern-Weil theory for parabolic bundles and prove that our metric notions coincide with the already existing algebro-geometric versions of parabolic Chern classes. We also formulate a Griffiths conjecture in the parabolic setting and prove some results that provide evidence in its favour for certain kinds of parabolic bundles. For these kinds of parabolic structures, we prove that the conjecture holds on Riemann surfaces. We also prove that a Berndtsson-type resu…
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
In this paper we prove a categorification of the Grothendieck-Riemann-Roch theorem. Our result implies in particular a Grothendieck-Riemann-Roch theorem for Toën and Vezzosi's secondary Chern character. As a main application, we establish a comparison between the Toën-Vezzosi Chern character and the classical Chern character, and show that the categorified Chern character recovers the classical de Rham realization.
Let $M$ be the moduli space of rank 3 parabolic vector bundles over a Riemann surface with several punctures. By the Mehta-Seshadri correspondence, this is the space of rank 3 unitary representations of the fundamental group of the punctured surface with specified conjugacy classes of the images of each boundary component, up to conjugation by elements of the unitary group. For each puncture we consider the torus bundle on $M$ consisting of those representations where the image of the corresponding boundary component is a fixed element of the torus. We associate line bundles to these torus bun…
Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group $π_1(X_K)$ of the generic fiber of $X$, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bund…
For an elliptic surface $q:Y \to Σ$, with prescribed singular fibres, Stefan Bauer proved directly via algebraic geometry that the stable bundles over $Y$, whose chern classes are pull backs from $Σ$, correspond to the stable (V-)bundles over $Σ$. We show, via a short proof in differential geometry, a generalisation to stable parabolic bundles. This uses extensions of Donaldson's deep result, giving the existence of Hermitian-Yang-Mills (or anti-self-dual) connections on stable parabolic bundles. In our cases these connections are flat and hence, correspond to representations of certain fundam…
Torsion of extended Chern-Simons classes for canonical extensions of flat bundles
Institute for Mathematical Sciences (HBNI)（数学科学研究所（HBNI））；Max-Planck Institute for Mathematics（马克斯·普朗克数学研究所）；CNRS, Laboratoire J. A. Dieudonné, UMR 6621 Université Cöte d’Azur（法国国家科学研究中心，J.A.迪永实验室，UMR 6621蔚蓝海岸大学）
