arXiv subjects

Shin-ichi Matsumura

Publications and source records attributed to Shin-ichi Matsumura.

At least 37 records · Page 2Linked to original sources

arXiv2023

Structure theorem for projective klt pairs with nef anti-canonical divisor

In this paper, we establish a structure theorem for projective klt pairs $(X,Δ)$ with nef anti-log canonical divisor; specifically, we prove that, up to replacing $X$ with a finite quasi-étale cover, $X$ admits a locally trivial rationally connected fibration onto a projective klt variety with numerically trivial canonical divisor. This structure theorem generalizes previous works for smooth projective varieties and reduces several structure problems to the singular Beauville-Bogomolov decomposition for Calabi-Yau varieties. As an application, projective varieties of klt Calabi-Yau type, which naturally appear as an outcome of the Log Minimal Model Program, are decomposed into building block varieties: rationally connected varieties and Calabi-Yau varieties.

math.AG↗

arXiv2022

Open problems on structure of positively curved projective varieties

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic sectional curvature, pseudo-effective tangent bundle, and nef anti-canonical divisor.

math.AG↗

arXiv2022

Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use $L^{2}$-harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

math.CV↗

arXiv2022

On the image of MRC fibrations of projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we pose several conjectures on structures and images of maximal rationally connected fibrations of smooth projective varieties admitting semi-positive holomorphic sectional curvature. Toward these conjectures, we prove that the canonical bundle of images of such fibrations is not big. Our proof gives a generalization of Yang's solution using RC positivity for Yau's conjecture. As an application, we show that any compact Kähler surface with semi-positive holomorphic sectional curvature is rationally connected, or a complex torus, or a ruled surface over an elliptic curve.

math.DG↗

arXiv2021

Strictly nef divisors on K-trivial fourfolds

In this paper, we prove the ampleness conjecture and Serrano's conjecture for strictly nef divisors on K-trivial fourfolds. Specifically, we show that any strictly nef divisors on projective fourfolds with trivial canonical bundle and vanishing irregularity are ample.

math.AG↗

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.

math.AG↗

arXiv2019

Projective klt pairs with nef anti-canonical divisor

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+Δ_{X_y})$ on a general fiber $X_y$ of $ψ: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness, which gives a sharper estimate than the question posed by Hacon-$\mathrm{M^{c}}$Kernan. Moreover, in the case of $X$ being smooth, we show that a maximally rationally connected fibration $ψ: X \to Y$ can be chosen to be a morphism to a smooth projective variety $Y$ with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair $(X, Δ)$, which can be seen as a generalization of Cao-Höring's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair $(X, Δ)$ with $-(K_X +Δ)$ nef, and obtain a structure theorem for projective orbifold surfaces.

math.AG↗

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.

math.AG↗

arXiv2018

Variation of numerical dimension of singular hermitian line bundles

The purpose of this paper is to give two supplements for vanishing theorems: One is a relative version of the Kawamata-Viehweg-Nadel type vanishing theorem, which is obtained from an observation for the variation of the numerical dimension of singular hermitian line bundles. The other is an analytic injectivity theorem for log canonical pairs on surfaces, which can be seen as a partial answer for Fujino's conjecture.

math.AG↗

arXiv2018

On projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $ϕ:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite étale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with the theory of holomorphic foliations. Moreover, we show that the universal cover of $X$ is biholomorphic and isometric to the product $\mathbb{C}^m \times F$ of the universal cover $\mathbb{C}^m$ of $Y$ with a flat metric and the rationally connected fiber $F$ with a Kähler metric whose holomorphic sectional curvature is semi-positive.

math.DG↗

arXiv2018

On morphisms of compact Kähler manifolds with semi-positive holomorphic sectional curvature

In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $ϕ: X \to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $ϕ$ is always smooth (that is, a submersion), the image $Y$ admits a finite etale cover $T \to Y$ by a complex torus $T$, and further that all the fibers are isomorphic when $X$ is projective. Moreover, by applying a modified method to maximal rationally connected fibrations, we show that $X$ is rationally connected, if $X$ is projective and $X$ has no truly flat tangent vectors at some point (which is satisfied when the holomorphic sectional curvature is quasi-positive). This result gives a generalization of Yau's conjecture. As a further application, we obtain a uniformization theorem for compact Kähler surfaces with semi-positive holomorphic sectional curvature.

math.DG↗

arXiv2018

A vanishing theorem of Koll'ar-Ohsawa type

For proper surjective holomorphic maps from K"ahler manifolds to analytic spaces, we give a decomposition theorem for the cohomology groups of the canonical bundle twisted by Nakano semi-positive vector bundles by means of the higher direct image sheaves, by using the theory of harmonic integrals developed by Takegoshi. As an application, we prove a vanishing theorem of Koll'ar-Ohsawa type by combining the L^2-method for the dbar-equation.

math.CV↗

arXiv2018

Injectivity theorems with multiplier ideal sheaves for higher direct images under K"ahler morphisms

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for K"ahler morphisms and singular hermitian metrics with non-algebraic singularities.

math.CV↗

arXiv2016

A transcendental approach to injectivity theorem for log canonical pairs

In this paper, we study transcendental aspects of the cohomology groups of adjoint bundles of log canonical pairs, aiming to establish an analytic theory for log canonical singularities. As a result, in the case of purely log terminal pairs, we give an analytic proof of the injectivity theorem originally proved by the Hodge theory. Our method is based on the theory of harmonic integrals and the L2-method for the dbar-equation, and it enables us to generalize the injectivity theorem to the complex analytic setting.

math.CV↗

arXiv2016

An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L2-estimates of solutions of the dbar-equation by using the de Rham-Weil isomorphism between the dbar-cohomology and the check{C}ech cohomology.

math.CV↗

arXiv2015

A Nadel vanishing theorem for metrics with minimal singularities on big line bundles

The purpose of this paper is to establish a Nadel vanishing theorem for big line bundles with multiplier ideal sheaves of singular metrics admitting an analytic Zariski decomposition (such as, metrics with minimal singularities and Siu's metrics). For this purpose, we apply the theory of harmonic integrals and generalize Enoki's proof of Koll'ar's injectivity theorem. Moreover we investigate the asymptotic behavior of harmonic forms with respect to a family of regularized metrics.

math.CV↗

arXiv2015

Injectivity theorems with multiplier ideal sheaves and their applications

The purpose of this survey is to present analytic versions of the injectivity theorem and their applications. The proof of our injectivity theorems is based on a combination of the L^2-method for the dbar-equation and the theory of harmonic integrals. As applications, we obtain Nadel type vanishing theorems and extension theorems for pluri-canonical sections of log pairs. Moreover, we give some results on semi-ampleness related to the abundance conjecture in birational geometry (the minimal model program).

math.CV↗

arXiv2014

Versions of injectivity and extension theorems

We give an analytic version of the injectivity theorem by using multiplier ideal sheaves, and prove some extension theorems for the adjoint bundle of dlt pairs. Moreover, by combining techniques of the minimal model program, we obtain some results for semi-ampleness related to the abundance conjecture in birational geometry and the Strominger-Yau-Zaslow conjecture for hyperKähler manifolds.

math.AG↗