Motivic cohomology and unramified cohomology of quadrics

This is the last of a series of three papers where we compute the unramified cohomology of quadrics in degree up to 4. Complete results were obtained in the two previous papers for quadrics of dimension ≤4 and ≥11. Here we deal with the remaining dimensions between 5 and 10. We also prove that the unramified cohomology of Pfister quadrics with divisible coefficients always comes from the ground field, and that the same holds for their unramified Witt rings. We apply these results to real quadrics. For most of the paper we have to assume that the ground field has characteristic 0, because we use Voevodsky’s motivic cohomology. Received August 18, 1999 / final version received December 10, 1999?Published online April 19, 2000     

Product structures in motivic cohomology and higher Chow groups

It is shown that the product structures of motivic cohomology groups and of higher Chow groups are compatible under the comparison isomorphism of Voevodsky (2002) [11]. This extends the result of Weibel (1999) [14], where he used the comparison isomorphism which assumed that the base field admits resolution of singularities.The mod n motivic cohomology groups and product structures in motivic homotopy theory are defined, and it is shown that the product structures are compatible under the comparison isomorphisms.     

Failure of tameness for local cohomology

We give an example that shows that not all local cohomology modules are tame in the sense of Brodmann and Hellus.     

Motivic integral of K3 surfaces over a non-archimedean field

We prove a formula expressing the motivic integral (Loeser and Sebag, 2003) [34] of a K3 surface over C((t)) with semi-stable reduction in terms of the associated limit mixed Hodge structure. Secondly, for every smooth variety over a complete discrete valuation field we define an analogue of the monodromy pairing, constructed by Grothendieck in the case of abelian varieties, and prove that our monodromy pairing is a birational invariant of the variety. Finally, we propose a conjectural formula for the motivic integral of maximally degenerate K3 surfaces over an arbitrary complete discrete valuation field and prove this conjecture for Kummer K3 surfaces.     

A bilinear form of dilogarithm and motivic regulator map

The Bloch-Wigner function D₂ is a single-valued version of a dilogarithm function and is used by Bloch to describe the Borel regulator map from K₃(C) into R explicitly (c.f. [Bloch, Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves, American Mathematical Society, Providence, RI, 2000]). We introduce a new way to formulate a single-valued dilogarithm function and use it to explicitly define a motivic regulator map for , defined in terms of the motivic complex of Goodwillie and Lichtenbaum. We also detect certain explicit nonzero elements in the motivic cohomology group. Throughout this paper, a path will be a C¹-function from the unit interval [0,1] into C-{0}.     

On the Shimura variety having Mumford’s fake projective plane as a connected component

We give an explicit construction of a unitary Shimura surface that has Mumford’s fake projective plane as one of its connected components. Moreover, as a byproduct of the construction, we show that Mumford’s fake projective place has a model defined over the 7th cyclotomic field.     

Morphic cohomology and singular cohomology of motives over the complex numbers

Morphic cohomology and singular cohomology of motives over the complex numbers are defined via the triangulated category of motives. Regarding morphic cohomology as functors defined on the triangulated category of motives, natural transformations of morphic cohomology are studied.     

Fibred sites and stack cohomology

The usual notion of the site associated to a stack is expanded to a definition to a site fibred over a presheaf of categories A on a site . If the presheaf of categories is a presheaf of groupoids G, then the associated homotopy theory is Quillen equivalant to the homotopy theory of simplicial presheaves over BG, and so the homotopy theory for the fibred site is an invariant of the homotopy type of G. Similar homotopy invariance results obtain for presheaves of spectra and presheaves of symmetric spectra on . In particular, stack cohomology can be calculated on the fibred site for any representing presheaf of groupoids within a fixed homotopy type.     

Brauer groups and cohomology

We explain in an elementary way an example showing that the Brauer group of a scheme X does not always coincide with the torsion of Received: 22 June 2004     

Motivic equivalence of quadratic forms. II

Let F be a field of characteristic ≠2 and φ be a quadratic form over F. By X _φ we denote the projective variety given by the equation φ=0. For each positive even integer d≥8 (except for d=12) we construct a field F and a pair φ, ψ of anisotropic d-dimensional forms over F such that the Chow motives of X _φ and X _ψ coincide but . For a pair of anisotropic (2 ^{n -1})-dimensional quadrics X and Y, we prove that existence of a rational morphism Y→X is equivalent to existence of a rational morphism Y→X. Received: 27 September 1999 / Revised version: 27 December 1999     

Deformation spaces of one-dimensional formal modules and their cohomology

Let Mm be the formal scheme which represents the functor of deformations of a one-dimensional formal module over equipped with a level-m-structure. By work of Boyer (in equal characteristic) and Harris and Taylor, the ?-adic étale cohomology of the generic fibre Mm of Mm realizes simultaneously the local Langlands and Jacquet-Langlands correspondences. The proofs given so far use Drinfeld modular varieties or Shimura varieties to derive this local result. In this paper we show without the use of global moduli spaces that the Jacquet-Langlands correspondence is realized by the Euler-Poincaré characteristic of the cohomology. Under a certain finiteness assumption on the cohomology groups, it is shown that the correspondence is realized in only one degree. One main ingredient of the proof consists in analyzing the boundary of the deformation spaces and in studying larger spaces which can be considered as compactifications of the spaces Mm.     

Computing Gauss-Manin systems for complete intersection singularitiesS μ

The Gauss-Manin systems with coefficients having logarithmic poles along the discriminant sets of the principal deformations of complete intersection quasihomogeneous singularitiesS are calculated. Their solutions in the form of generalized hypergeometric functions are presented.     

Functional equations of the dilogarithm in motivic cohomology

We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH²(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields.     

Degree of the divisor of solutions of a differential equation on a projective variety

Using the data schemes from [1] we give a rigorous definition of algebraic differential equations on the complex projective space Pⁿ. For an algebraic subvariety S?Pⁿ, we present an explicit formula for the degree of the divisor of solutions of a differential equation on S and give some examples of applications. We extend the technique and result to the real case.     

On Certain Subcovers of the Hermitian Curve

We present a simple construction that gives explicit equations for certain subcovers of the Hermitian curve. We show that maximal curves with a certain type of defining equations are covered by the Hermitian curve.     

Multiplication maps and vanishing theorems for Toric varieties

We use multiplication maps to give a characteristic-free approach to vanishing theorems on toric varieties. Our approach is very elementary but is enough powerful to prove vanishing theorems.