Henri Gillet's Preprints

Preprints are generally in pdf format.

Title (linked to pdf file) Abstract

Differential Algebra - a scheme theory approach Two results in Differential Algebra, Kolchin's irreducibility theorem, and a result on
descent of projective varieties (due to Buium) are proved using methods of ``modern''
or ``Grothendieck style'' algebraic geometry

An Explicit Proof of the Generalized Gauss-Bonnet Formula
(With Fatih Unlu)
In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior algebra of E, which gives a "twisted resolution" of the structure sheaf of Z. The "generalized super-trace" of A^{2r}/r!, where r is the rank of E, is an explicit map of complexes from the twisted resolution to the Dolbeault complex of X, which represents [Z]. One may then read off the Gauss-Bonnet formula from this map of complexes.

Volumes of symmetric spaces via lattice points. (with Daniel R. Grayson) In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases.
We also present a proof of the closely related result that the Tamagawa number of Sl_k Q is 1 that is somewhat simpler and more arithmetic than Weil's. His proof proceeds by induction on k and appeals to the Poisson summation formula, whereas the proof here brings to the forefront local versions of the formula, one for each prime p, which help to illuminate the appearance of values of zeta functions in formulas for volumes.

K-Theory and Intersection Theory
Chapter for the forthcoming handbook of algebraic K-theory

Title (linked to pdf file)	Abstract
Differential Algebra - a scheme theory approach	Two results in Differential Algebra, Kolchin's irreducibility theorem, and a result on descent of projective varieties (due to Buium) are proved using methods of ``modern'' or ``Grothendieck style'' algebraic geometry
An Explicit Proof of the Generalized Gauss-Bonnet Formula (With Fatih Unlu)	In this paper we construct an explicit representative for the Grothendieck fundamental class [Z] of a complex submanifold Z of a complex manifold X, under the assumption that Z is the zero locus of a real analytic section of a holomorphic vector bundle E. To this data we associate a super-connection A on the exterior algebra of E, which gives a "twisted resolution" of the structure sheaf of Z. The "generalized super-trace" of A^{2r}/r!, where r is the rank of E, is an explicit map of complexes from the twisted resolution to the Dolbeault complex of X, which represents [Z]. One may then read off the Gauss-Bonnet formula from this map of complexes.
Volumes of symmetric spaces via lattice points. (with Daniel R. Grayson)	In this paper we show how to use elementary methods to prove that the volume of Sl_k R / Sl_k Z is zeta(2) * zeta(3) * ... * zeta(k) / k. Using a version of reduction theory presented in this paper, we can compute the volumes of certain unbounded regions in Euclidean space by counting lattice points and then appeal to the machinery of Dirichlet series to get estimates of the growth rate of the number of lattice points appearing in the region as the lattice spacing decreases. We also present a proof of the closely related result that the Tamagawa number of Sl_k Q is 1 that is somewhat simpler and more arithmetic than Weil's. His proof proceeds by induction on k and appeals to the Poisson summation formula, whereas the proof here brings to the forefront local versions of the formula, one for each prime p, which help to illuminate the appearance of values of zeta functions in formulas for volumes.
K-Theory and Intersection Theory	Chapter for the forthcoming handbook of algebraic K-theory