arXiv subjects
Publications and source records attributed to Shuji Saito.
In this paper, we study the Brauer-Manin pairing of smooth proper varieties over local fields, and determine the $p$-adic part of the kernel of one side. We also compute the $A_0$ of a potentially rational surface which splits over a wildly ramified extension.
In this paper we study weight homology of singular schemes. Weight homology is an invariant of a singular scheme defined in terms of hypercoverings of resolution of singularities. Our main result is McKay principle for weight homology of quotient singularities, i.e. we describe weight homology of a quotient scheme in terms of weight homology of an equivariant scheme. Our method is to reduce the geometric McKay principle for weight homology to Kato's cohomological Hasse principle for arithmetic schemes. The McKay principle for weight homology implies McKay principle for the homotopy type of the dual complex of the exceptional divisors of a resolution of a quotient singularity. As a consequence we show that the dual complex is contractible for isolated quotient singularities.
We prove the existence of resolution of singularities for arbitrary (not necessarily reduced or irreducible) excellent two-dimensional schemes, via permissible blow-ups. The resolution is canonical, and functorial with respect to automorphisms or etale or Zariski localizations. We treat the embedded case as well as the non-embedded case, with or without a boundary, and we relate the diferent versions. In the non-embedded case, a boundary is a collection of locally principal closed subschemes. Our main tools are the stratifications by Hilbert-Samuel functions and the characteristic polyhedra introduced by H. Hironaka. In an appendix we show that the standard method used in characteristic zero - the theory of maximal contact - does not work for surfaces in positive characteristic (the counterexamples are hypersurfaces in affine threespace and work over any field of positive characteristic). In this new version, we treat the case of locally noetherian but not necessarily noetherian schemes in an appropriate way. Here one does not have a finite resolution sequence, but still a canonical resolution morphism by glueing. The same techniques allow to treat algebraic spaces and stacks.
In this note we relate three topics for arithmetic schemes: a general duality for étale constructible torsion sheaves, an étale homology theory, and a Gersten-Bloch-Ogus-Kato complex. The results in this paper have been used in other papers of the authors ([JS], [Sa], [SaH] in the list of references).
In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.
In this paper we prove a finiteness result concerning the Chow group of zero-cycles for varieties over $p$-adic local fields. In this final version, there are several corrections concerning mathematical symbols and reference to related known results.
We show the existence of good hyperplane sections for schemes over discrete valuation rings with good or (quasi) semistable reduction, and the existence of good Lefschetz pencils for schemes with good reduction or ordinary quadratic reduction. As an application we prove that the reciprocity map introduced for smooth projective varieties over local fields by Bloch, Kato and Saito is an isomorphism after profinite completion, if the variety has good reduction or 'almost good' reduction.
For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective smooth over a finite field. We give an application to finiteness questions for some motivic cohomology groups over finite fields.
A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension 2 of ${\mathscr X}$ to a new $p$-adic cohomology of ${\mathscr X}$ introduced by the second author, which is a candidate of the conjectural étale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.
Let $X$ be a projective algebraic manifold and let $CH^r(X)$ be the Chow group of algebraic cycles of codimension $r$ on $X$, modulo rational equivalence. Working with a candidate Bloch-Beilinson filtration $\{F^ν\}_{ν\geq 0}$ on $CH^r(X)\otimes {\Bbb Q}$ due to the second author, we construct a space of arithmetic Hodge theoretic invariants $\nabla J^{r,ν}(X)$ and corresponding map $ϕ_{X}^{r,ν} : Gr_{F}^νCH^r(X)\otimes {\Bbb Q} \to \nabla J^{r,ν}(X)$, and determine conditions on $X$ for which the kernel and image of $ϕ_{X}^{r,ν}$ are ``uncountably large''.
We show an example of Chow group of 0-cycles on surface over a p-adic field which has infinite torsion subgroup.
The contents is changed.
In this paper, we develop the theory of Jacobian rings of open complete intersections, which mean a pair $(X,Z)$ where $X$ is a smooth complete intersection in the projective space and and $Z$ is a simple normal crossing divisor in $X$ whose irreducible components are smooth hypersurface sections on $X$. Our Jacobian rings give an algebraic description of the cohomology of the open complement $X-Z$ and it is a natural generalization of the Poincaré residue representation of the cohomology of a hypersurface originally invented by Griffiths. The main results generalize the Macaulay's duality theorem and the Donagi's symmetrizer lemma for usual Jacobian rings for hypersurfaces. A feature that distinguishes our generalized Jacobian rings from usual ones is that there are instances where duality fails to be perfect while the defect can be controlled explicitly by using the defining equations of $Z$ in $X$. Two applications of the main results are given: One is the infinitesimal Torelli problem for open complete intersections. Another is an explicit bound for Nori's connectivity in case of complete intersections. The results have been applied also to study of algebraic cycles in several other works.
We study the higher Chow groups $CH^2(X,1)$ and $CH^3(X,2)$ of smooth, projective algebraic surfaces over a field of char 0. We develop a theoretical framework to study them by using so-called higher normal functions and higher infinitesimal invariants when the cycles are supported on normal crossing divisors. Then we investigate their structure on hypersurfaces in projective 3-space. This leads to vanishing or decomposbility results. On the other hand we find examples of these groups in degrees 4 and 5 where the result is a highly indecomposably (hence big) abelian group. Part of the examples are due to Alberto Collino.