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Tatsuro Kawakami

Publications and source records attributed to Tatsuro Kawakami.

At least 19 recordsLinked to original sources
arXiv2026

Liftability and vanishing theorems for Fano threefolds in positive characteristic I

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results for the case when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$.

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arXiv2026

The Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails

We show that the Frobenius--stable version of the Grauert--Riemenschneider vanishing theorem fails for threefolds in any positive characteristic, and for terminal 3-folds in characteristic $p \in \{2, 3, 5\}$. To prove this, we introduce the notion of $\mathbb{F}_p$-rationality for singularities in positive characteristic and we show that klt singularities in dimension at most 4 are $\mathbb{F}_p$-rational. We apply this to prove a Frobenius--stable version of the Kawamata--Viehweg vanishing theorem on $K$-trivial 3-folds.

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arXiv2026

Local vanishing for F-pure threefolds

We establish Grauert--Riemenschneider vanishing for $F$-pure threefolds over a perfect field $k$ of characteristic $p>5$. We apply this to prove Steenbrink vanishing for three-dimensional sharply $F$-pure pairs in characteristic $p>5$. As a consequence, we obtain the logarithmic extension for one-forms in this setting.

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arXiv2026

Mumford vanishing for threefolds in positive characteristic

Let $X$ be a projective klt threefold in characteristic $p>5$ and let $L$ be a nef Cartier divisor on $X$. We show that $H^1(X, -L)=0$ for the following two cases: (1) $K_X$ is not big and $L$ is big; (2) $-K_X$ is nef and $L$ is of numerical dimension two.

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arXiv2026

Higher F-rational singularities

We introduce higher $F$-rationality generalising $F$-rationality. We prove that a normal variety over a field of characteristic zero is $m$-rational if and only if it is $m$-$F$-rational after reduction modulo a sufficiently large prime $p$. Additionally, we establish new results on the logarithmic extension of forms.

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arXiv2026

Extending one-forms on $F$-regular singularities

We prove the logarithmic extension theorem for one-forms on strongly $F$-regular singularities. Additionally, we establish the logarithmic extension theorem for one-forms on three-dimensional klt singularities in characteristic $p>41$. To this end, we reduce the problem to the logarithmic extension theorem for two-dimensional klt singularities with imperfect residue fields using a technique based on Cartier operators.

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arXiv2025

Fedder type criteria for quasi-$F$-splitting I

Yobuko recently introduced the notion of quasi-$F$-splitting and quasi-$F$-split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that quasi-$F$-split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. In this paper, we prove Fedder type criteria for quasi-$F$-splittings of complete intersections, and in particular, obtain a simple formula to compute Artin-Mazur heights of Calabi-Yau hypersurfaces. As one of its applications, we prove that there exist Calabi-Yau varieties of arbitrarily high Artin-Mazur height over $\mathbb{F}_2$. We also give explicit defining equations of quartic K3 surfaces over $\mathbb{F}_{3}$ realizing all the possible Artin-Mazur heights.

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arXiv2025

Fedder type criteria for quasi-$F$-splitting II

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.

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arXiv2025

Frobenius liftable hypersurfaces

Let $D$ be a reduced divisor in $\mathbb P^n_k$ for an algebraically closed field $k$ of positive characteristic $p > 0$. We prove that if $(\mathbb P^n_k, D)$ is Frobenius liftable modulo $p^2$, then $D$ is a toric divisor. As a corollary, we show that if there exists a finite surjective morphism $f\colon Y\to X$ onto a smooth projective complex variety $X$ of Picard rank $1$ such that $(Y, f^{-1}(D)_{\mathrm{red}})$ is a toric pair, then $X$ is the projective space and $D$ is a toric divisor.

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arXiv2025

On Grauert-Riemenschneider vanishing for Cohen-Macaulay schemes of klt type

Given a Cohen-Macaulay scheme of klt type $X$ and a resolution $π\colon Y\to X$, we show that $R^1π_*ω_Y=0$. We deduce that if $\mathrm{dim}(X)=3$, then $X$ satisfies Grauert-Riemenschneider vanishing and therefore has rational singularities. We also obtain that in arbitrary dimension, if $X$ is of finite type over a perfect field of characteristic $p>0$, then $X$ has $\mathbb{Q}_p$-rational singularities.

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arXiv2025

Liftability and vanishing theorems for Fano threefolds in positive characteristic II

In our series of papers, we prove that smooth Fano threefolds in positive characteristic lift to the ring of Witt vectors. Moreover, we show that they satisfy Akizuki-Nakano vanishing, $E_1$-degeneration of the Hodge to de Rham spectral sequence, and torsion-freeness of Crystalline cohomologies. In this paper, we establish these results except when $|-K_X|$ is very ample and the Picard group is generated by $ω_X$. To this end, we show that an arbitrary smooth Fano threefold is quasi-$F$-split when the Picard number or the Fano index is larger than one.

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arXiv2024

Higher F-injective singularities

We introduce the concept of higher $F$-injectivity, a generalisation of $F$-injectivity. We prove that an isolated singularity over a field of characteristic zero is $k$-Du Bois if it is $k$-$F$-injective after reductions modulo infinitely many primes $p$. Under the ordinarity conjecture, we also establish the converse. As an application, we study Frobenius liftable hypersurfaces.

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