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Teppei Takamatsu

Publications and source records attributed to Teppei Takamatsu.

31 records · Page 2Linked to original sources
arXiv2025

Arithmetic monodromy of hyper-Kähler varieties over $p$-adic fields

In this paper, we study the $p$-adic and $\ell$-adic monodromy operators associated with hyper-Kähler varieties over $p$-adic fields, in connection with Looijenga-Lunts-Verbitsky Lie algebras. We investigate a conjectural relation between the nilpotency indices of these monodromy operators on higher-degree cohomology groups and on the second cohomology, which may be viewed as an arithmetic analogue of Nagai's conjecture for degenerations of hyper-Kähler manifolds over a disk. We verify this arithmetic version of Nagai's conjecture for hyper-Kähler varieties over $p$-adic fields, assuming they belong to one of the four known deformation types. As part of our approach, we introduce a new method to analyze the $p$-adic cohomology of hyper-Kähler varieties via Sen's theory.

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arXiv2025

Reduction of bielliptic surfaces

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic surfaces over a field of characteristic different from $2$ and $3$, in order to prove the Shafarevich conjecture for bielliptic surfaces with rational points. Furthermore, we demonstrate that the Shafarevich conjecture generally fails for bielliptic surfaces without rational points. In particular, this paper completes the study of the Shafarevich conjecture for minimal surfaces of Kodaira dimension $0$. In the second part of this paper, we study a Néron model of a bielliptic surface. We establish the potential existence of a Néron model for a bielliptic surface when the residual characteristic is not equal to $2$ or $3$.

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arXiv2025

On the finiteness of twists of irreducible symplectic varieties

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the finiteness of twists of irreducible symplectic varieties via a fixed finite field extension of characteristic $0$. The main ingredient of the proof is the cone conjecture for irreducible symplectic varieties, which was proved by Markman and Amerik--Verbitsky. As byproducts, we also discuss the cone conjecture over non-closed fields by Bright--Logan--van Luijk's method. We also give an application to the finiteness of derived equivalent twists. Moreover, we discuss the case of K3 surfaces or Enriques surfaces over fields of positive characteristic.

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arXiv2024

Quasi-F-splittings in birational geometry

We develop the theory of quasi-$F$-splittings in the context of birational geometry. Amongst other things, we obtain results on liftability of sections and establish a criterion for whether a scheme is quasi-$F$-split employing the higher Cartier operator. As one of the applications of our theory, we prove that three-dimensional klt singularities in large characteristic are quasi-$F$-split, and so, in particular, they lift modulo $p^2$.

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arXiv2024

Quasi-F-splittings in birational geometry II

Over an algebraically closed field of characteristic $p>41$, we prove that three-dimensional $\mathbb Q$-factorial affine klt varieties are quasi-$F$-split. Furthermore, we show that the bound on the characteristic is optimal.

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arXiv2024

On Frobenius liftability of surface singularities

We show that a plt surface singularity $(P\in X,B)$ is $F$-liftable if and only if it is $F$-pure and is not a rational double point of type $E_8^1$ in characteristic $p=5$. As a consequence, we prove the logarithmic extension theorem for $F$-pure surface pairs and Bogomolov-Sommese vanishing for globally $F$-split surface pairs. These results were previously known to hold in characteristic $p>5$.

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arXiv2023

Minimal model program for semi-stable threefolds in mixed characteristic

In this paper, we study the minimal model theory for threefolds in mixed characteristic. As a generalization of a result of Kawamata, we show that the MMP holds for strictly semi-stable schemes over an excellent Dedekind scheme $V$ of relative dimension two without any assumption on the residue characteristics of $V$. We also prove that we can run a $(K_{X/V}+Δ)$-MMP over $Z$, where $π\colon X \to Z$ is a projective birational morphism of $\mathbb{Q}$-factorial quasi-projective $V$-schemes and $(X,Δ)$ is a three-dimensional dlt pair with $\mathrm{Exc}(π) \subset \lfloor Δ\rfloor $.

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arXiv2022

On the Shafarevich conjecture for irreducible symplectic varieties

Irreducible symplectic varieties are higher-dimensional analogues of K3 surfaces. In this paper, we prove the Shafarevich conjecture for irreducible symplectic varieties of fixed deformation class. We also observe that the second cohomological generalization of the Shafarevich conjecture does not hold in general, and discuss another formulation of a cohomological generalization.

math.NT↗
arXiv2022

On behavior of conductors, Picard schemes, and Jacobian numbers of varieties over imperfect fields

Let $X$ be a regular geometrically integral variety over an imperfect field $K$. Unlike the case of characteristic $0$, $X':=X\times_{\mathrm{Spec}\,K}\mathrm{Spec}\,K'$ may have singular points for a (necessarily inseparable) field extension $K'/K$. In this paper, we define new invariants of the local rings of codimension $1$ points of $X'$, and use these invariants for the calculation of $δ$-invariants (, which relate to genus changes,) and conductors of such points. As a corollary, we give refinements of Tate's genus change theorem and the Patakfalvi-Waldron Theorem. Moreover, when $X$ is a curve, we show that the Jacobian number of $X$ is $2p/(p-1)$ times of the genus change by using the above calculation. In this case, we also relate the structure of the Picard scheme of $X$ with invariants of singular points of $X$. To prove such a relation, we give a characterization of the geometrical normality of algebras over fields of positive characteristic.

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arXiv2020

On a cohomological generalization of the Shafarevich conjecture for K3 surfaces

The Shafarevich conjecture for K3 surfaces asserts the finiteness of isomorphism classes of K3 surfaces over a fixed number field admitting good reduction away from a fixed finite set of finite places. André proved this conjecture for polarized K3 surfaces of fixed degree, and recently She proved it for polarized K3 surfaces of unspecified degree. In this paper, we prove a certain generalization of their results, which is stated by the unramifiedness of l-adic étale cohomology groups for K3 surfaces over finitely generated fields of characteristic 0. As a corollary, we get the original Shafarevich conjecture for K3 surfaces without assuming the extendability of polarization, which is stronger than the results of André and She. Moreover, as an application, we get the finiteness of twists of K3 surfaces via a finite extension of characteristic 0 fields.

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arXiv2019

On the Shafarevich conjecture for Enriques surfaces

Enriques surfaces are minimal surfaces of Kodaira dimension $0$ with $b_{2}=10$. If we work with a field of characteristic away from $2$, Enriques surfaces admit double covers which are K3 surfaces. In this paper, we prove the Shafarevich conjecture for Enriques surfaces by reducing the problem to the case of K3 surfaces. In our formulation of the Shafarevich conjecture, we use the notion "admitting a cohomological good K3 cover", which includes not only good reduction but also flower pot reduction.

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arXiv2019

The Shafarevich conjecture and some extension theorems for proper hyperbolic polycurves

In this paper, we prove the Shafarevich conjecture for proper hyperbolic polycurves, which is a higher dimensional analogue of that for proper hyperbolic curves. First, we study theories of proper hyperbolic polycurves over regular schemes. For example, we generalize the moduli theory of Kodaira fibrations due to Jost and Yau. We also show the Neron property of proper smooth models of proper hyperbolic polycurves over Dedekind schemes under an assumption on residual characteristics. We then apply these extension theories to the proof of the Shafarevich conjecture for proper hyperbolic polycurves.

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