arXiv subjects
Publications and source records attributed to Spencer Bloch.
The appearance of multiple zeta values in anomalous dimensions and $β$-functions of renormalizable quantum field theories has given evidence towards a motivic interpretation of these renormalization group functions. In this paper we start to hunt the motive, restricting our attention to a subclass of graphs in four dimensional scalar field theory which give scheme independent contributions to the above functions.
We consider correspondences on smooth quasiprojective varieties $U$. An algebraic cycle inducing the Künneth projector onto $H^1(U)$ is constructed. Assuming normal crossings at infinity, the existence of relative motivic cohomology is shown to imply the independence of $\ell$ for traces of open correspondences.
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.
This is the last version of AG/0111277. Here the old abstract: We define $ε$-factors in the de Rham setting and calculate the determinant of the Gauß-Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection.
Let $X\subset ¶^n$ 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.
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.
We define an additive version of the Bloch group of a field, together with an additive dilogarithm and an Artin-Schreier realization.
The cosimplicial scheme $$Delta^bullet = Δ^0 smallmatrix \to smallmatrix Δ^1 smallmatrix to smallmatrix ...;\quad Δ^n :=\Spec\Big(k[t_0,...c,t_n]/(\sum t_i -t)\Big)$$ was used in B to define higher Chow groups. In this note, we let t tend to 0 and replace Δ^\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 $(n-1)$-Kähler forms $Ω^{n-1}_k$. An analogous degeneration on the level of de Rham cohomology associated to ``constant modulus'' degenerations of varieties in various contexts is discussed.
see the old abstract and the comments here.
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.
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 $f^*Ω^2_{k(S)}$. Then the Gauß-Manin sheaf $R^if_*(Ω^*_{X/S}({\rm log} Y)\otimes E)$ carries a Gauß-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 $Ω^1_{X/S}({\rm log}Y)$.
We give an explicit formula for the determinant of the Gauß-Manin connection for an irregular connection on a Zariski open set of the projective line $¶^1_K$ over a function field $K$ over a field $k$ of characteristic zero.
Gauß-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 Gauß-Manin determinant are discussed.
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.
We study the character of the infinite wedge projective representation of the algebra of differential operators on the circle. We prove quasi-modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions. These correlation functions are sums of determinants built from genus 1 theta functions and their derivatives.
An analog of Chern-Simons theory is developed in an algebro-geometric setting.