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Shunsuke Takagi

Publications and source records attributed to Shunsuke Takagi.

At least 19 recordsLinked to original sources
arXiv2026

On the uniform positivity of $F$-signature under reduction modulo $p$

Carvajal-Rojas, Schwede and Tucker asked whether the mod $p$ reductions of a complex klt type singularity have uniformly positive $F$-signature for almost all primes $p$. In this paper, we give an affirmative answer to this conjecture in the case of pure subrings of regular local rings--for example, reductive quotient singularities. We also show that the conjecture can be reduced to the Gorenstein case. Finally, we discuss the connection with $F$-alpha invariants--a characteristic $p$ analog of Tian's alpha invariants introduced by Pande--for log Fano pairs.

math.AG↗
arXiv2026

Quasi-$F$-splitting versus log canonicity

In this paper, we investigate the relationship between quasi-$F$-splitting and log canonicity. We show that if a numerically $\mathbb{Q}$-Gorenstein normal singularity is quasi-$F^e$-split for every $e\geq 1$, then it is numerically log canonical. In dimension two, we prove the converse under the condition that the Gorenstein index is not divisible by the characteristic $p$. We also classify two-dimensional quasi-$F$-split normal singularities.

math.AG↗
arXiv2025

On the behavior of adjoint ideals under pure morphisms

We characterize adjoint ideal sheaves via ultraproducts and, utilizing this characterization, study their behavior under pure morphisms. In particular, given a pure morphism $f:Y \to X$ between normal quasi-projective complex varieties, a reduced divisor $D$ and an effective $\mathbb{Q}$-Weil divisor $Γ$ on $X$ without common components, we have the following result: if the cycle-theoretic pullback $E:=f^{\natural}D$ is reduced and $(Y, E+f^*Γ)$ is of plt type along $E$, then $(X, D+Γ)$ is of plt type along $D$. This provides an affirmative answer to a question posed by Z. Zhuang.

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arXiv2024

A Gorenstein criterion for strongly $F$-regular and log terminal singularities

A conjecture of Hirose, Watanabe, and Yoshida offers a characterization of when a standard graded strongly $F$-regular ring is Gorenstein, in terms of an $F$-pure threshold. We prove this conjecture under the additional hypothesis that the anti-canonical cover of the ring is Noetherian. Moreover, under this hypothesis on the anti-canonical cover, we give a similar criterion for when a normal $F$-pure (resp. log canonical) singularity is quasi-Gorenstein, in terms of an $F$-pure (resp. log canonical) threshold.

math.AC↗
arXiv2023

Weak Akizuki-Nakano vanishing theorem for globally $F$-split 3-folds

In this paper, we prove that a weak form of the Akizuki-Nakano vanishing theorem holds on globally $F$-split 3-folds. Making use of this vanishing theorem, we study deformations of globally $F$-split Fano 3-folds and the Kodaira vanishing theorem for thickenings of locally complete intersection globally $F$-regular 3-folds.

math.AG↗
arXiv2021

Arithmetic and geometric deformations of $F$-pure and $F$-regular singularities

Given a normal $\mathbb{Q}$-Gorenstein complex variety $X$, we prove that if one spreads it out to a normal $\mathbb{Q}$-Gorenstein scheme $\mathcal{X}$ of mixed characteristic whose reduction $\mathcal{X}_p$ modulo $p$ has normal $F$-pure singularities for a single prime $p$, then $X$ has log canonical singularities. In addition, we show its analog for log terminal singularities, without assuming that $\mathcal{X}$ is $\mathbb{Q}$-Gorenstein, which is a generalization of a result of Ma-Schwede. We also prove that two-dimensional strongly $F$-regular singularities are stable under equal characteristic deformations. Our results give an affirmative answer to a conjecture of Liedtke-Martin-Matsumoto on deformations of linearly reductive quotient singularities.

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arXiv2017

General hyperplane sections of threefolds in positive characteristic

In this paper, we study the singularities of a general hyperplane section $H$ of a three-dimensional quasi-projective variety $X$ over an algebraically closed field of characteristic $p>0$. We prove that if $X$ has only canonical singularities, then $H$ has only rational double points. We also prove, under the assumption that $p>5$, that if $X$ has only klt singularities, then so does $H$.

math.AG↗
arXiv2016

The weak ordinarity conjecture and $F$-singularities

Recently M. Mustata and V. Srinivas related a natural conjecture about the Frobenius action on the cohomology of the structure sheaf after reduction to characteristic $p > 0$ with another conjecture connecting multiplier ideals and test ideals. We generalize this relation to the case of singular ambient varieties. Additionally, we connect these results to a conjecture relating $F$-injective and Du Bois singularities. Finally, using an unpublished result of Gabber, we also show that $F$-injective and Du Bois singularities have a common definition in terms of smooth hypercovers.

math.AG↗
arXiv2015

On the relationship between depth and cohomological dimension

Let $(S, m)$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $I$ be an ideal of $S$. We prove that if $\text{depth} S/I \ge 3$, then the cohomological dimension $\mathrm{cd}(S, I)$ of $I$ is less than or equal to $n-3$. We also show, under the assumption that $S$ has an algebraically closed residue field of characteristic zero, that if $\text{depth} S/I \ge 4$, then $\mathrm{cd}(S, I) \le n-4$ if and only if the local Picard group of the completion $\widehat{S/I}$ is torsion. We give a number of applications, including sharp bounds on cohomological dimension of ideals whose quotients satisfy good depth conditions such as Serre's conditions $(S_i)$.

math.AC↗
arXiv2015

Nilpotence of Frobenius action and the Hodge filtration on local cohomology

An $F$-nilpotent local ring is a local ring $(R, \mathfrak{m})$ of prime characteristic defined by the nilpotence of the Frobenius action on its local cohomology modules $H^i_{\mathfrak{m}}(R)$. A singularity in characteristic zero is said to be of $F$-nilpotent type if its modulo $p$ reduction is $F$-nilpotent for almost all $p$. In this paper, we give a Hodge-theoretic interpretation of three-dimensional normal isolated singularities of $F$-nilpotent type. In the graded case, this yields a characterization of these singularities in terms of divisor class groups and Brauer groups.

math.AG↗
arXiv2015

Comparing multiplier ideals to test ideals on numerically Q-Gorenstein varieties

We show that the reduction to positive characteristic of the multiplier ideal in the sense of de Fernex and Hacon agrees with the test ideal for infinitely many primes, assuming that the variety is numerically Q-Gorenstein. It follows, in particular, that this reduction property holds in dimension 2 for all normal surfaces.

math.AG↗
arXiv2013

Adjoint ideals and a correspondence between log canonicity and F-purity

This paper presents three results on F-singularities. First, we give a new proof of Eisenstein's restriction theorem for adjoint ideal sheaves, using the theory of F-singularities. Second, we show that a conjecture of Mustaţă and Srinivas implies a conjectural correspondence of F-purity and log canonicity. Finally, we prove this correspondence when the defining equations of the variety are very general.

math.AG↗