arXiv subjects
Publications and source records attributed to Shou Yoshikawa.
In this paper, we apply Fedder-type criteria for quasi-$F$-splitting to provide explicit computations of quasi-$F$-split heights for Calabi-Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. We also find interesting phenomena concerned with inversion of adjunction, fiber products, Fano varieties, and general fibers of fibrations.
In this paper, we introduce the notion of quasi-$F$-splitting for rings in mixed characteristic. By comparing quasi-$F$-splitting with perfectoid purity, we obtain a new inversion of adjunction-type result. Furthermore, we study the possible values of the perfectoid pure threshold of $\mathrm{div}(p)$ and construct new examples of perfectoid pure rings.
We study quasi-$F^e$-split and quasi-$F$-regular singularities, which generalize Yobuko's quasi-$F$-splitting. We establish Fedder type criteria that characterize these properties for hypersurfaces. These criteria offer explicit tools for computation and verification. As an application, we construct a counterexample to the inversion of adjunction for quasi-$F$-regularity and compute the quasi-$F$-split threshold of the cone over the ordinary cusp.
Let $k$ be a complete non-archimedean non-trivial valued field. In this paper, we investigate whether every $k$-algebra homomorphism between $k$-affinoid algebras is automatically bounded. We show that this property holds if and only if either $|k^\times|^\mathbb{Q} = \mathbb{R}_{>0}$ holds, or $k$ has positive characteristic and is $F$-finite.
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 $p^2$.
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.
In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an étale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization of toric varieties. We further prove that, without assumption of global $F$-splitting, a smooth projective separably rationally connected variety of arbitrary characteristic with numerically flat log cotangent bundle is a toric variety.
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 $.
In this paper, we discuss a generalization of log canonical singularities in the non-$\mathbb{Q}$-Gorenstein setting. We prove that if a normal complex projective variety has a non-invertible polarized endomorphism, then it has log canonical singularities in our sense. As a corollary, we give an affirmative answer to a conjecture of Broustet and Höring.
In this paper, we obtain a characterization of toric varieties via int-amplified endomorphisms. We prove that if $f \colon X \to X$ is an int-amplified endomorphism of a smooth complex projective variety $X$, then $X$ is toric if and only if $f_*L$ is a direct sum of line bundles on $X$ for every line bundle $L$.
In this paper, we study the structure of Fano fibrations of varieties admitting an int-amplified endomorphism. We prove that if a normal $\mathbb{Q}$-factorial klt projective variety $X$ has an int-amplified endomorphism, then there exists an étale in codimension one finite morphism $\widetilde{X} \to X$ such that $\widetilde{X}$ is of Fano type over its albanese variety. As a corollary, if we further assume that $X$ is smooth and rationally connected, then $X$ is of Fano type.
In this paper, we study the global $F$-splitting of varieties admitting an int-amplified endomoprhism. We prove that surfaces admitting an int-amplified endomorphism are of dense globally $F$-split type and, in particular, of Calabi-Yau type.
We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.