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One extends P. Deligne's notion of integrality over a finite field for a $\ell$-adic sheaf on a scheme of finite type over a local field with finite residue field. One shows that this integrality notion is preserved by $Rf_!$, as it is over a finite field after P. Deligne., Comment: 5 pages latex. See math.NT/0405318

We give a proof of Artin's vanishing theorem in characteristic zero, based on Deligne's Riemann-Hilbert correspondence. Just as a curiosity., Comment: 4 pages

If $V$ is a smooth projective variety defined over a local field $K$ with finite residue field, so that its \'etale cohomology over the algebraic closure $\bar{K}$ is supported in codimension 1, then the mod $p$ reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of $V$ over a large algebraically closed field is trivial, then the mod $p$ reduction of a projective regular model carries a rational point., Comment: 16 pages

We give a positive answer to the questions 4.4 on Hodge type and 4.2 on divisibility of the eigenvalues of Frobenius of the article by N. Fakhruddin and C. S. Rajan (NT/0402230)., Comment: 4 pages

Local Fourier trnasforms, analogous to the $\ell$-adic Fourier transforms, are constructed for connections over $k((t))$. Following a program of Katz, a meromorphic connection on a curve is shown to be rigid, i.e. determined by local data at the singularities, if and only if a certain cohomological condition is satisfied. We show that the index of rigidity is invariant under Fourier transform., Comment: 23 pages

If $X$ is an abelian variety over a field and $L$ is an invertible sheaf, we know that the degree of the 0-cycle $L^g$ is divisible by $g!$. As a 0-cycle, it is not, even over a field of cohomological dimension 1. But we show that over a finite field there is perhaps some hope., Comment: 7 pages

We show that Ax-Katz divisibility on the number of rational points of a variety defined over a finite field by equations of low degrees comes from divisibility of the eigenvalues of the Frobenius action of the $\ell$-adic cohomology with compact supports. This completes the motivc picture, as we knew already that in characteristic 0, the Hodge type was the expected one. [version 2 as the title contained a typo..., else no changes], Comment: 10 pages, latex

The eigenvalues of Frobenius acting on $\ell$-adic cohomology of a complete intersection over a finite field have the divisibility predicted by the theorem of Ax and Katz. We have corrected some unforgivable typos., Comment: 5 pages, latex

Let $X\subset \P^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz., Comment: 16 pages

This is meant to be a survey article for the Cubo Journal. We discuss the existence and number of rational points over a finite field, the Hodge type over the complex numbers, and the motivic conjectures which are controlling those invariants. We present a geometric version of it., Comment: 11 pages

We improve the known Hodge type bound for the exotic cohomology of complete intersections. In the revised version, we included a simplification of our original argument due to Pierre Deligne. The note appears in the C. R. de l'Aca. des Sc. de Paris., Comment: 7 pages

We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization., Comment: 28 pages

Smooth projective varieties $X$ over a finite field $k$ with $CH_0(X\otimes \bar{k(X)})=\mathbb Z$ have a rational point, in particular Fano varieties. We also refer to http://link.springer.de/link/service/journals/00222/tocs.htm where the last version, cleaned from some typos, should be published online., Comment: 6 pages

The purpose of this note is to give a survey on recent progress on characteristic classes of flat bundles, and how they behave in a family.

We show that the Chern character of a variation of polarized Hodge structures of weight one with nilpotent residues at $\infty$ dies up to torsion in the Chow ring, except in codimension 0., Comment: 19 pages

The cosimplicial scheme $$Delta^bullet = \Delta^0 smallmatrix \to smallmatrix \Delta^1 smallmatrix to smallmatrix ...;\quad \Delta^n :=\Spec\Big(k[t_0,...c,t_n]/(\sum t_i -t)\Big)$$ was used in B to define higher Chow groups. In this note, we let t tend to 0 and replace \Delta^\bullet by a degenerate version $$Q^\bullet = Q^0 smallmatrix to smallmatrix Q^1 smallmatrix to smallmatrix ...;\quad Q^n := \Spec\Big(k[t_0,...c,t_n]/(\sum t_i)\Big) $$ to define an additive version of the higher Chow groups. For a field k, we show the Chow group of 0-cycles on $Q^n$ in this theory is isomorphic to the absolute $(n-1)$-K\"ahler forms $\Omega^{n-1}_k$. An analogous degeneration on the level of de Rham cohomology associated to ``constant modulus'' degenerations of varieties in various contexts is discussed., Comment: 16 pages

This is the last version of AG/0111277. Here the old abstract: We define $\epsilon$-factors in the de Rham setting and calculate the determinant of the Gau\ss-Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection.

Bolibruch's examples of representations of pi_1(P^1-finitely many points) which are not realizable by Fuchsian differential systems are adapted to curves of higher genus., Comment: 10 pages, LaTeX

Homology with values in a connection with possibly irregular singular points on an algebraic curve is defined, generalizing homology with values in the underlying local system for a connection with regular singular points. Integration defines a perfect pairing between de Rham cohomology with values in the connection and homology with values in the dual connection., Comment: latex 16 pages, one picture

We give an explicit formula for the determinant of the Gau{\ss}-Manin connection for an irregular connection on a Zariski open set of the projective line $\P^1_K$ over a function field $K$ over a field $k$ of characteristic zero., Comment: Latex 2e, 37 pages

Gau{\ss}-Manin determinant connections associated to irregular connections on a curve are studied. The determinant of the Fourier transform of an irregular connection is calculated. The determinant of cohomology of the standard rank 2 Kloosterman sheaf is computed modulo 2 torsion. Periods associated to irregular connections are studied in the very basic $\exp(f)$ case, and analogies with the Gau{\ss}-Manin determinant are discussed., Comment: 32 pages, latex 2e

Let $f: X \to S$ be a smooth morphism in characteristic 0, and let $(E, \nabla_{X/S})$ be a relative regular connection. We define a cohomology of relative differential characters on $X$ which receives classes of $(E, \nabla_{X/S})$. It says in particular that the partial vanishing of the trace of the iterated Atiyah classs can be made canonical. While applied to a family of curves and $c_2$, the construction yields a connection of $f_*c_2(E)$. This one has been constructed analytically by A. Beilinson. We also relate our construction to the trace complex of Beilinson of Schechtman., Comment: Latex 2e, 25 pages

In Comm. Math. Physics 118 (1988), 651-701, A. Beilinson and V. Schechtman define on the total space of a smooth family of curves a so-called trace complex associated to a vector bundle, the 0-th relative cohomology of which is the Atiyah algebra of the determinant bundle. Their proof reduces the general case to the acyclic one. In particular, one needs a comparison of the image of the trace complex for a bundle, and its twist by an \'etale multisection. We analyse this and correct a point in the original proof., Comment: 6 pages, Latex 2e

We construct 0-cycles on the product of 2 elliptic curves, which are not detectable by Bloch's analytic motivic cohomology., Comment: Latex 2e, 17 pages

see the old abstract and the comments here., Comment: The article is already available on the server math.AG/9904088. Its main purpose is to compute a determinant formula for the determinant of Gau{\ss}-Manin cohomology of a rank one irregular connection on a family of curves. In the original version, we computed modulo torsion. To please the referee, we computed here the torsion as well. It is a 2-torsion term. We give for it two shapes. One version is to express it as the residue of some quadratic form stemming from the connection on the curve, and one as the discriminant of some Hessian on the generalized Picard variety. 51 pages

This note is an attempt to generalize Bolibruch's theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle., Comment: Latex 2e, 10 pages

Let $f: X \to S$ be flat morphism over an algebraically closed field $k$ with a relative normal crossings divisor $Y\subset X$, $(E, \nabla)$ be a bundle with a connection with log poles along $Y$ and curvature with values in $f^*\Omega^2_{k(S)}$. Then the Gau\ss-Manin sheaf $R^if_*(\Omega^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gau\ss-Manin connection $GM^i(\nabla)$. We establish a Riemann-Roch formula relating the algebraic Chern-Simons invariants of $\nabla$, $GM^i(\nabla)$ and the top Chern class of $\Omega^1_{X/S}({\rm log}Y)$., Comment: 46 pages, published version

This is a survey on the topic explained in the title, for the proceedings on the K-theory 1997 summer institute in Seattle., Comment: Latex 2e, 11 pages

We show that hypergeometric differential equations, unitary and Gauss-Manin connections give rise to de Rham cohomology sheaves which do not admit a Bloch-Ogus resolution. The latter is in contrast to Panin's theorem asserting that corresponding \'etale cohomology sheaves do fulfill Bloch-Ogus theory., Comment: latex 2e, 6 pages

We show the existence of a regular universal quotient as a smooth commutative algebraic group of the Chow group of 0-cycles on a projective reduced variety, and give over the field of complex numbers an analytic description of it. This generalizes the classical theory of the Albanese. The Chow group of 0-cycles is then isomorphic to this smooth algebraic group if and only if it is finite dimensional in the sense of Mumford. This generalizes the classical theorem of Roitman., Comment: Latex 2e

We propose an algebraic Riemann-Roch formula for moving flat bundles on contant families in characteristic zero with values in the ring of algebraic differential characters. This formula lifts Grothendieck-Riemann-Roch formula in the Chow groups and Bismut-Lott formula in the odd dimensional Betti cohomology with $\C/\Z$ coefficients over the field of complex numbers., Comment: Latex2e

We give a construction of algebraic differential characters, receiving classes of algebraic bundles with connection, lifitng the Chern-Simons invariants defined with S. Bloch, the classes in the Chow group and the analytic secondary invariants if the variety is defined over the field of complex numbers., Comment: Latex 2e

An analog of Chern-Simons theory is developed in an algebro-geometric setting., Comment: LaTeX