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Hiromu Tanaka

Publications and source records attributed to Hiromu Tanaka.

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 $ω_X$.

math.AG↗
arXiv2026

Relative semi-ampleness in positive characteristic

Given an invertible sheaf on a fibre space between projective varieties of positive characteristic, we show that fibrewise semi-ampleness implies relative semi-ampleness. The same statement fails in characteristic zero.

math.AG↗
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.

math.AG↗
arXiv2026

Prime Fano threefolds of genus 8 in positive characteristic

We prove that a prime Fano threefold of genus 8 over an algebraically closed field of positive characteristic is isomorphic to a linear section of the Grassmannian variety Gr(2, 6). As applications, it is shown that a prime Fano threefold of genus 8 is irrational and, if the characteristic is larger than two, then it is globally F-regular.

math.AG↗
arXiv2026

Fano threefolds in positive characteristic I

Over an algebraically closed field of positive characteristic, we classify smooth Fano threefolds of Picard number one whose anti-canonical linear systems are not very ample. Furthermore, we also prove that an anti-canonically embedded Fano threefold of genus at least five is an intersection of quadrics.

math.AG↗
arXiv2026

Precisely positioned generation of CsPbBr3 nano-light sources in a Cs4PbBr6 film by electron beam irradiation

Integration of high-quality photon emitters at specific locations within nanophotonic structures or optoelectronic devices is a key to innovating on-chip optical control and quantum technologies. Halide perovskite nanoparticles have great potential as single photon emitters with high quantum efficiency. To achieve their full potential, they must be embedded in a host material that ensures chemical stability and passivates surface defects. A previous experiment on a CsPbBr3-Cs4PbBr6 nanocomposite film suggested possibility that electron beam irradiation can be used to control positions of CsPbBr3 nano-light sources in the Cs4PbBr6 host although the effects of electron beam irradiation are not fully understood. Here, we fabricate a Cs4PbBr6-CsBr film, not containing the CsPbBr3 phase, and provide direct evidence that CsPbBr3 nanoparticles can be locally generated in the Cs4PbBr6 host by irradiation with a focused electron beam. We further demonstrate perovskite nano-light source arrays with submicron spacing using this method.

cond-mat.mes-hall↗
arXiv2025

Geometry and arithmetic of geometrically integral regular del Pezzo surfaces

We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration onto a curve over an algebraically closed field always admits a section. Moreover, we prove that the total space is rational if the base curve is rational and the anticanonical degree of a fibre is at least five.

math.AG↗
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.

math.AG↗
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 $p^2$.

math.AG↗
arXiv2024

Fano threefolds in positive characteristic II

Let $X$ be a smooth Fano threefold over an algebraically closed field of positive characteristic. Assume that $|-K_X|$ is very ample and each of the index and the Picard number is equal to one. We prove that $3 \leq g \leq 12$ and $g \neq 11$ for the genus $g$ of $X$. Moreover, we show that there exists no smooth curve on $X$ along which the blowup is Fano.

math.AG↗
arXiv2024

Discriminant divisors for conic bundles

We study some foundational properties on discriminant divisors for generically smooth conic bundles. In particular, we extend the formula $Δ_f \equiv -f_*K_{X/T}^2$ to arbitrary characteristics.

math.AG↗