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Masataka Iwai

Publications and source records attributed to Masataka Iwai.

15 recordsLinked to original sources
arXiv2026

The Miyaoka-Yau inequality and the delta invariant for Fano varieties

We establish the following Miyaoka-Yau inequality for any $n$-dimensional klt Fano variety $X$, possibly K-unstable, in terms of its delta invariant: $$ \left(2(n+1)\widehat{c}_2(X)-n c_1(X)^2\right)\cdot c_1(X)^{n-2} \ge -n \left(1-\min\{1,δ(X)\}\right)^2 \cdot c_1(X)^n. $$ Furthermore, inspired by recent work of Inoue and Hallam-Lahdili, we formulate and prove an equivariant version of this inequality in the soliton setting, in which the Chern classes and the delta invariant are replaced by the equivariant Chern classes and the weighted delta invariant, respectively. As a consequence, we obtain the equivariant Miyaoka-Yau inequality for every smooth Fano manifold admitting a Kähler-Ricci soliton.

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arXiv2026

The Miyaoka-Yau inequality for singular varieties with big canonical or anticanonical divisors

We establish the Miyaoka-Yau inequality for $n$-dimensional projective klt varieties with big canonical divisor $K_X$: \[ (2(n+1)\widehat{c}_2(X) - n \widehat{c}_1(X)^2) \cdot \langle c_1(K_X)^{n-2} \rangle \ge 0. \] We also prove the Miyaoka-Yau inequality for K-semistable projective klt varieties with big anticanonical divisor $-K_X$. As part of our approach, we define the non-pluripolar product $\langle α_1 \cdots α_p \rangle$ on singular varieties, and establish the Bogomolov-Gieseker type inequality for $\langle α^{n-1} \rangle$-semistable Higgs sheaves with respect to a big class $α$.

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arXiv2026

Semipositivity of the orbifold second Chern class in Fujiki's class

We study inequalities for orbifold second Chern classes of compact normal analytic varieties in Fujiki's class. We prove Miyaoka's inequality for singular varieties in Fujiki's class with nef canonical divisor, as well as the semipositivity of the orbifold second Chern class for varieties with nef anti-canonical divisor. To prove these results, we establish generic nefness theorems for tangent and cotangent sheaves and an orbifold Bogomolov--Gieseker inequality for mixed polarizations.

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arXiv2025

Minimal projective varieties satisfying Miyaoka's equality

In this paper, we establish a structure theorem for minimal projective klt varieties $X$ that satisfiy Miyaoka's equality $3c_2(X) = c_1(X)^2$. Specifically, we prove that the canonical divisor $K_X$ is semi-ample and that the Kodaira dimension $κ(K_X)$ is either $0$, $1$, or $2$. Furthermore, based on this abundance result, we show that a maximally quasi-étale cover of $X$ is smooth, and we describe explicitly the structure of the Iitaka fibration. Additionally, we prove a similar result for projective klt varieties with a nef anti-canonical divisor.

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arXiv2025

Positivity of tangent sheaves of projective varieties -- the structure of MRC fibrations

In this paper, we extend the structure theorem for smooth projective varieties with nef tangent bundle to projective klt varieties whose tangent sheaf is either positively curved or almost nef. Specifically, we show that such a variety $X$, up to a finite quasi-étale cover, admits a rationally connected fibration $X \to A$ onto an abelian variety $A$. For the proof, we develop the theory of positivity of coherent sheaves on projective varieties. As applications, we establish some relations between the geometric properties and positivity of tangent sheaves.

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arXiv2023

Positivity of extensions of vector bundles

In this paper, we study when positivity conditions of vector bundles are preserved by extension. We prove that an extension of a big (resp. pseudo-effective) line bundle by an ample (resp. a nef) vector bundle is big (resp. pseudo-effective). We also show that an extension of an ample line bundle by a big line bundle is not necessarily pseudo-effective. In particular, this implies that an almost nef vector bundle is not necessarily pseudo-effective.

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arXiv2023

Miyaoka type inequality for terminal threefolds with nef anti-canonical divisors

In this paper, we study the Miyaoka type inequality on Chern classes of terminal projective $3$-folds with nef anti-canonical divisors. Let $X$ be a terminal projective $3$-fold such that $-K_X$ is nef. We show that if $c_1(X)\cdot c_2(X)\neq 0$, then $c_1(X)\cdot c_2(X)\geq \frac{1}{252}$; if further $X$ is not rationally connected, then $c_1(X)\cdot c_2(X)\geq \frac{4}{5}$ and this inequality is sharp. In order to prove this, we give a partial classification of such varieties along with many examples. We also study the nonvanishing of $c_1(X)^{\dim X-2}\cdot c_2(X)$ for terminal weak Fano varieties and prove a Miyaoka--Kawamata type inequality for terminal weak Fano $3$-folds.

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arXiv2022

Abundance theorem for minimal compact Kähler manifolds with vanishing second Chern class

In this paper, for compact Kähler manifolds with nef cotangent bundle, we study the abundance conjecture and the associated Iitaka fibrations. We show that, for a minimal compact Kähler manifold, the second Chern class vanishes if and only if the cotangent bundle is nef and the canonical bundle has the numerical dimension $0$ or $1$. Additionally, in this case, we prove that the canonical bundle is semi-ample. Furthermore, we give a relation between the variation of the fibers of the Iitaka fibration and a certain semipositivity of the cotangent bundle.

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arXiv2021

Characterization of pseudo-effective vector bundles by singular Hermitian metrics

In this paper, we give complex geometric descriptions of the notions of algebraic geometric positivity of vector bundles and torsion-free coherent sheaves, such as nef, big, pseudo-effective and weakly positive, by using singular Hermitian metrics. As an applications, we obtain a generalization of Mori's result. We also give a characterization of the augmented base locus by using singular Hermitian metrics on vector bundles and the Lelong numbers.

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arXiv2021

Almost nef regular foliations and Fujita's decomposition of reflexive sheaves

In this paper, we study almost nef regular foliations. We give a structure theorem of a smooth projective variety $X$ with an almost nef regular foliation $\mathcal{F}$: $X$ admits a smooth morphism $f: X \rightarrow Y$ with rationally connected fibers such that $\mathcal{F}$ is a pullback of a numerically flat regular foliation on $Y$. Moreover, $f$ is characterized as a relative MRC fibration of an algebraic part of $\mathcal{F}$. As a corollary, an almost nef tangent bundle of a rationally connected variety is generically ample. For the proof, we generalize Fujita's decomposition theorem. As a by-product, we show that a reflexive hull of $f_{*}(mK_{X/Y})$ is a direct sum of a hermitian flat vector bundle and a generically ample reflexive sheaf for any algebraic fiber space $f : X \rightarrow Y$. We also study foliations with nef anti-canonical bundles.

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arXiv2020

On asymptotic base loci of relative anti-canonical divisors of algebraic fiber spaces

In this paper, we study the relative anti-canonical divisor $-K_{X/Y}$ of an algebraic fiber space $ϕ: X \to Y$, and we reveal relations among positivity conditions of $-K_{X/Y}$, certain flatness of direct image sheaves, and variants of the base loci including the stable (augmented, restricted) base loci and upper level sets of Lelong numbers. This paper contains three main results: The first result says that all the above base loci are located in the horizontal direction unless they are empty. The second result is an algebraic proof for Campana--Cao--Matsumura's equality on Hacon--$\rm{M^c}$Kernan's question, whose original proof depends on analytics methods. The third result partially solves the question which asks whether algebraic fiber spaces with semi-ample relative anti-canonical divisor actually have a product structure via the base change by an appropriate finite étale cover of $Y$. Our proof is based on algebraic as well as analytic methods for positivity of direct image sheaves.

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arXiv2019

On projective manifolds with pseudo-effective tangent bundle

In this paper, we develop the theory of singular hermitian metrics on vector bundles. As an application, we give a structure theorem of a projective manifold $X$ with pseudo-effective tangent bundle: $X$ admits a smooth fibration $X \to Y$ to a flat projective manifold $Y$ such that its general fiber is rationally connected. Moreover, by applying this structure theorem, we classify all the minimal surfaces with pseudo-effective tangent bundle and study general non-minimal surfaces, which provide examples of (possibly singular) positively curved tangent bundles.

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arXiv2019

On the global generation of direct images of pluri-adjoint line bundles

We study the Fujita-type conjecture proposed by Popa and Schnell. We obtain an effective bound on the global generation of direct images of pluri-adjoint line bundles on the regular locus. We also obtain an effective bound on the generic global generation for a Kawamata log canonical $\mathbb{Q}$-pair. We use analytic methods such as $L^2$ estimates, $L^2$ extensions and injective theorems of cohomology groups.

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arXiv2018

Nadel-Nakano vanishing theorems of vector bundles with singular Hermitian metrics

We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type vanishing theorem of a vector bundle with a singular Hermitian metric. We do not use an approximation technique of a singular Hermitian metric. We apply these theorems to a singular Hermitian metric induced by holomorphic sections and a big vector bundle, and we obtain a generalization of Griffiths' vanishing theorem. Finally, we show a generalization of Ohsawa's vanishing theorem.

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