Searcharxiv⌕ Search

arXiv subjects

Kazuya Kato

Publications and source records attributed to Kazuya Kato.

At least 19 recordsLinked to original sources
arXiv2025

Logarithmic geometry and Frobenius, II

Based on the strong analogy between the category of log mixed Hodge structures and the category ${\cal A}_X$ of $\ell$-adic nature, which we have introduced in the previous part and is closely related to the weight-monodromy conjecture, we prove the $\ell$-adic analogues of some theorems in Hodge theory related to the SL(2)-orbit theorem.

math.AG↗
arXiv2025

Logarithmic Tate conjectures over finite fields

We formulate an analogue of Tate conjecture on algebraic cycles, for the log geometry over a finite field. We show that the weight-monodromy conjecture follows from this conjecture and from the semi-simplicity of the Frobenius action. This conjecture suggests the existence of the monodromy cycle which gives the monodromy operator and an action of ${\frak{sl}}(2)$ on the cohomology, and which lives in the world of log motives.

math.AG↗
arXiv2024

Toroidal compactifications of the moduli spaces of Drinfeld modules

We construct toroidal compactifications of the moduli spaces of Drinfeld $\mathbb{F}_q[T]$-modules of rank $d$ with level $N$ structure as moduli spaces of log Drinfeld modules of rank $d$ with level $N$ structure. The toroidal compactifications are log regular schemes associated to rational cone decompositions, and there are regular ones among them. To construct these toroidal compactifications, we blow up the Satake compactification of Pink and employ the theory of formal moduli and a process of iterated Tate uniformization.

math.AG↗
arXiv2024

Logarithmic geometry and Frobenius

Based on the logarithmic algebraic geometry and the theory of Deligne systems, we define an abelian category of $\ell$-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of variations of mixed Hodge structure. We consider asymptotic behaviors and simple cases of higher direct images of objects of this category. This category is closely related to the monodromy-weight conjecture.

math.AG↗
arXiv2024

Upper Ramification Groups for Arbitrary Valuation Rings

T. Saito established a ramification theory for ring extensions locally of complete intersection. We show that for a Henselian valuation ring $A$ with field of fractions $K$ and for a finite Galois extension $L$ of $K$, the integral closure $B$ of $A$ in $L$ is a filtered union of subrings of $B$ which are of complete intersection over $A$. By this, we can obtain a ramification theory of Henselian valuation rings as the limit of the ramification theory of Saito. Our theory generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We study "defect extensions" which are not treated in these previous works.

math.NT↗
arXiv2023

Logarithmic Dieudonne theory

We develop the theory of logarithmic p-divisible groups and the theory of logarithmic finite locally free commutative group schemes.

math.AG↗
arXiv2022

Mixed objects are embedded into log pure objects

We prove that a variation of mixed Hodge structure is embedded in a logarithmic variation of pure Hodge structure, and a generalized version of this result. These results suggest some simple construction of the category of mixed motives by using log pure motives.

math.AG↗
arXiv2022

Deligne--Beilinson cohomology and log Hodge theory

We show that the description of Deligne--Beilinson cohomology is improved by using log Hodge theory. We consider the log relative version of it, and also present a fundamental conjecture in log Hodge theory.

math.AG↗
arXiv2020

Attempts on SGA for non-commutative rings

We define non-commutative schemes by using prime ideals of non-commutative rings, and discuss the étale cohomology, the Betti cohomology, and the fundamental groups of non-commutative schemes. For non-commutative schemes which are finite over centers, we prove the finiteness theorem for the higher direct images in étale cohomology theory, and the comparison theorem between étale cohomology and Betti cohomology. In Appendix, for non-commutative schemes over finite fields which are finite over centers and satisfy a certain condition, $L$-functions are expressed by using étale cohomology with compact supports.

math.AG↗