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Kanetomo Sato

Publications and source records attributed to Kanetomo Sato.

12 recordsLinked to original sources
arXiv2024

Torsion birational motives of surfaces and unramified cohomology

Let $S$ and $T$ be smooth projective varieties over an algebraically closed field. Suppose that $S$ is a surface admitting a decomposition of the diagonal. We show that, away from the characteristic of $k$, if an algebraic correspondence $T \to S$ acts trivially on the unramified cohomology, then it acts trivially on any normalized, birational, and motivic functor. This generalizes Kahn's result on the torsion order of $S$. We also exhibit an example of $S$ over $\mathbb{C}$ for which $S \times S$ violates the integral Hodge conjecture.

math.AG↗
arXiv2019

Chern class and Riemann-Roch theorem for cohomology theory without homotopy invariance

In this paper, we formulate axioms of certain graded cohomology theory for which Chern class maps from higher K-theory are defined, following the method of Gillet [Gi1]. We will not include homotopy invariance nor purity in our axioms. It will turn out that the Grothendieck-Riemann-Roch theorem and the Riemann-Roch theorem without denominators hold for our higher Chern classes. We will also give two applications of our Riemann-Roch results.

math.AG↗
arXiv2019

On p-adic vanishing cycles of log smooth families

In this paper, we will show that the sheaf of p-adic vanishing cycles of a log smooth family over a DVR of mixed characteristic is generated by Milnor symboles. A key ingredient is a computation (due to K. Kato) on the graded quotients of a multi-indexed filtration on the sheaf concerned, which has been used in several papers of the first author.

math.NT↗
arXiv2014

Zero-cycles on varieties over p-adic fields and Brauer groups

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.

math.AG↗
arXiv2012

Cycle classes for p-adic étale Tate twists and the image of p-adic regulators

In this paper, we construct Chern class maps and cycle class maps with values in p-adic étale Tate twists [S2]. We also relate the p-adic étale Tate twists with the finite part of Bloch-Kato. As an application, we prove that the integral part of p-adic regulator maps has values in the finite part of Galois cohomology under certain assumptions.

math.AG↗
arXiv2012

Étale duality for constructible sheaves on arithmetic schemes

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).

math.AG↗
arXiv2010

Syntomic cohomology and Beilinson's Tate conjecture for $K_2$

In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic K3 surface $X$ over a p-adic field for which the torsion part of the Chow group $CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface $X$ over a p-adic field whose geometric genus is non-zero and for which the torsion part of $CH_0(X)$ is finite.

math.AG↗
arXiv2010

A Finiteness theorem for zero-cycles over $p$-adic fields

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.

math.AG↗
arXiv2009

A p-adic regulator map and finiteness results for arithmetic schemes

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.

math.AG↗
arXiv2007

\ell-adic class field theory for regular local rings

In this paper, we prove the $\ell$-adic abelian class field theory for henselian regular local rings of equi-characteristic assuming the surjectivity of Galois symbol maps, which is a $\ell$-adic variant of a result of Matsumi [13].

math.NT↗