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Shin-ichi Matsumura

Publications and source records attributed to Shin-ichi Matsumura.

At least 19 recordsLinked to original sources
arXiv2026

Atiyah classes and ellipticity of Oka manifolds

We prove that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic in the sense of Gromov, thereby proving Gromov's ellipticity conjecture. The proof establishes a cohomological construction of local sprays. Given a bundle morphism $φ\colon E\to T_X$, we introduce quadratic $φ$-vector fields on $E$ whose flows yield local sprays with fibre derivative $φ$. Their existence is characterized by the vanishing of the symmetrized $φ$-Atiyah class of $E$. This gives a local dominating spray on every weakly pseudoconvex manifold admitting a positive line bundle, without any Oka assumption. The positivity hypothesis is essential even for local existence: blow-ups at a single point of complex tori of algebraic dimension zero, and Kummer surfaces of algebraic dimension zero, admit no local dominating spray. The torus examples give compact Kähler Oka manifolds that are not elliptic and show that ellipticity is not preserved under blowing up, and that it is neither open nor closed in holomorphic families of compact Oka manifolds. Combined with the work of Xie and Zhao, the Kummer examples yield Oka K3 surfaces that are not elliptic. Under the additional Oka assumption, we globalize the local dominating sprays constructed above. The proof combines Oka approximation with a gluing argument based on weighted $L^2$ estimates for the $\bar\partial$-equation.

math.CV↗
arXiv2026

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.

math.AG↗
arXiv2026

Beauville--Bogomolov--Yau decomposition for Kähler generalized pairs

In this paper, we establish the Beauville--Bogomolov--Yau decomposition for generalized pairs in the Kähler setting, thereby extending this structure theorem to the natural framework of the Kähler generalized Minimal Model Program. More precisely, we prove that, after passing to a finite quasi-étale cover, a Kähler generalized klt pair $(X,Δ,\boldsymbolβ)$ of Calabi--Yau type admits a locally constant fibration $(X,Δ)\to Y$ over a Calabi--Yau variety $Y$ whose fiber $(F,Δ|_{F})$ is rationally connected. Equivalently, after base change to the universal cover of $Y$, the fibration becomes a product, and its global structure is determined by a monodromy action preserving the pair $(F,Δ|_{F})$. As a principal application, when the nef b-part $\boldsymbolβ$ vanishes, we show that the monodromy can be eliminated after a further finite quasi-étale cover, yielding a product decomposition into a rationally connected pair, strict Calabi--Yau varieties, irreducible holomorphic symplectic varieties, and complex tori. The proof introduces new analytic methods involving relative projectivity, localized positivity and flatness of direct image sheaves, generalized pairs in the analytic Minimal Model Program, and foliations on singular Kähler varieties.

math.AG↗
arXiv2026

Positivity of compact Kähler varieties admitting an int-amplified endomorphism

We study compact Kähler varieties admitting int-amplified endomorphisms from the viewpoint of positivity of tangent sheaves. Our main result shows that the tangent sheaf of such a variety is weakly positively curved; in particular, it is pseudo-effective in a strong sense. This establishes a new link between complex dynamics and the positivity theory of tangent sheaves, and provides an alternative to equivariant MMP techniques in the compact Kähler setting. As an application, we prove a Kähler analogue of Yoshikawa's structure theorem: for a compact Kähler klt variety admitting an int-amplified endomorphism, after an equivariant quasi-étale cover, it admits an equivariant flat MRC fibration with irreducible fibres onto a complex torus; in the smooth case, the fibration is smooth whose periodic fibres are of Fano type. To this end, we prove the existence of minimal models in the case of numerical dimension zero canonical divisors for non-projective Kähler varieties. We further study rationally connected manifolds with pseudo-effective tangent bundle, focusing on their relation to Fano-type properties, almost homogeneity, and fundamental groups. In the proof, we also show that every surjective endomorphism of a compact klt Kähler variety lifts to a suitable maximally quasi-étale cover.

math.AG↗
arXiv2026

A note on compact Kähler varieties with pseudo-effective tangent sheaf

In this paper, we discuss recent developments and open problems concerning the geometry of compact Kähler varieties whose tangent sheaves are pseudo-effective in a strong sense. As our main result, we prove that, after passing to a finite quasi-étale cover, any compact Kähler variety with quotient singularities and pseudo-effective tangent sheaf admits a flat fibration onto a complex torus, with rationally connected fibers. We also obtain an analogous structure theorem for klt compact Kähler varieties, assuming the existence of minimal models in numerical dimension zero; consequently, the result holds unconditionally in the projective setting and for low-dimensional compact Kähler varieties.

math.AG↗
arXiv2026

A flatness criterion for pseudo-effective sheaves on compact Kähler spaces

In this paper, we prove that if $E$ is a pseudo-effective sheaf with vanishing first Chern class on a klt compact Kähler space $X$, then, after passing to a finite quasi-étale cover, the reflexive pullback of $E$ is locally free and flat. This extends the flatness criterion of Höring--Peternell, originally established for projective varieties, to the Kähler setting. The proof relies on two main ingredients, both of which are new even in the projective case. The first is a flatness theorem for stable sheaves: we show that a slope-stable pseudo-effective sheaf with vanishing first Chern class is Hermitian flat. This is obtained by combining Hermitian--Einstein theory with the subharmonicity properties of direct image sheaves. The second is a singular Kähler analogue of Simpson's flatness theorem for extensions of locally free Hermitian flat sheaves.

math.AG↗
arXiv2026

Injectivity theorems for higher direct images under proper Kähler morphisms on snc spaces

Let $X$ be a complex manifold, and let $Y$ and $D$ be two reduced simple-normal-crossing (snc) divisors on $X$ with no common irreducible components. Given a proper locally Kähler morphism $π\colon X \to Δ$ from $X$ to a complex analytic space $Δ$, we prove Fujino's conjecture on the injectivity theorem in the relative setting in a generalized form. Specifically, we establish an injectivity result for the higher direct images under $π$ for the lc pairs $(X, D)$ as well as $(Y, D_Y)$, where $D_Y := D \cap Y$. As an application, this result immediately implies the injectivity theorem on holomorphically convex Kähler manifolds with reduced snc divisors. The main technique in the proof consists of the theory of harmonic integrals together with residue formulae associated with adjoint ideal sheaves, which are developed from our previous work for the absolute case (where $Δ$ is a point and $X$ is compact). Additionally, we make use of the Takegoshi harmonic forms to deal with the non-compactness of $X$.

math.CV↗
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.

math.AG↗
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.

math.AG↗
arXiv2025

On compact Kähler manifolds with pseudo-effective tangent bundle

In this paper, we prove that a compact Kähler manifold $X$ with pseudo-effective (resp. singular positively curved) tangent bundle admits a smooth (resp. locally constant) rationally connected fibration $ϕ\colon X \to Y$ onto a finite étale quotient $Y$ of a compact complex torus. This result extends the structure theorem previously established for smooth projective varieties to compact Kähler manifolds.

math.AG↗
arXiv2025

Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature

In this paper, we prove that a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature admits a locally trivial fibration $ϕ\colon X \to Y$, where the fiber $F$ is a rationally connected projective manifold and the base $Y$ is a finite étale quotient of a torus. This result extends the structure theorem, previously established for projective manifolds, to compact Kähler manifolds. A key part of the proof involves analyzing the foliation generated by truly flat tangent vectors and showing the abelianness of the topological fundamental group $π_{1}(X)$, with a focus on varieties of special type.

math.DG↗
arXiv2025

Compact Kähler three-folds with nef anti-canonical bundle

In this paper, we prove that a non-projective compact Kähler three-fold with nef anti-canonical bundle is, up to a finite étale cover, one of the following: a manifold with vanishing first Chern class; the product of a K3 surface and the projective line; or a projective space bundle over a $2$-dimensional torus. This result extends Cao-Höring's structure theorem for projective manifolds to compact Kähler manifolds in dimension $3$. For the proof, we investigate the Minimal Model Program for compact Kähler three-folds with nef anti-canonical bundles by using the positivity of direct image sheaves, $\mathbb{Q}$-conic bundles, and orbifold vector bundles.

math.AG↗
arXiv2024

An injectivity theorem on snc compact Kähler spaces: an application of the theory of harmonic integrals on log-canonical centers via adjoint ideal sheaves

Let $(X,D)$ be a log-canonical (lc) pair, in which $X$ is a compact Kähler manifold and $D$ is a reduced snc divisor, and let $F$ be a holomorphic line bundle on $X$ equipped with a smooth metric $h_F = e^{-φ_F}$. Via the use of the adjoint ideal sheaves (constructed from $φ_F$ and $D$) and the associated residue morphisms, sections of $K_D \otimes \left. F\right|_D$ on $D$ (as well as those of $K_X \otimes D \otimes F$ on $X$) can be related to the $F$-valued holomorphic top-forms on each lc center of $(X,D)$ by an inductive use of a certain residue exact sequence derived from the adjoint ideal sheaves. The theory of harmonic integrals is valid on each lc center (which is compact Kähler), so this provides a pathway to apply the techniques in harmonic theory to the possibly singular Kähler space $D$. To illustrate the use of such apparatus in problems concerning lc pairs, we prove a Kollár-type injectivity theorem for the cohomology on $D$ when $F$ is semi-positive. This in turn also solves the conjecture by Fujino on the injectivity theorem for the compact Kähler lc pair $(X,D)$, providing an alternative proof of a recent result by Cao and Păun.

math.CV↗
arXiv2024

Singular Nakano positivity of direct image sheaves of adjoint bundles

In this paper, we consider a proper Kähler fibration $f \colon X \to Y$ and a singular Hermitian line bundle $(L, h)$ on $X$ with semi-positive curvature. We prove that the direct image sheaf $f_{*}(\mathcal{O}_{X}(K_{X/Y}+L) \otimes \mathcal{I}(h))$, equipped with the Narasimhan-Simha metric, is singular Nakano semi-positive in the sense that the $\overline{\partial}$-equation can be solved with optimal $L^{2}$-estimate. Our proof does not rely on the theory of Griffiths positivity for the direct image sheaf.

math.AG↗
arXiv2024

Nakano positivity of singular Hermitian metrics: Approximations and applications

This paper studies the approximation of singular Hermitian metrics on vector bundles using smooth Hermitian metrics with Nakano semi-positive curvature on Zariski open sets. We show that singular Hermitian metrics capable of this approximation satisfy Nakano semi-positivity as defined through the $\overline{\partial} $-equation with optimal $L^2$-estimates. Furthermore, for a projective fibration $f \colon X \to Y$ with a line bundle $L$ on $X$, we provide a specific condition under which the Narasimhan-Simha metric on the direct image sheaf $f_{*}\mathcal{O}_{X}(K_{X/Y}+L)$ admits this approximation. As an application, we establish several vanishing theorems.

math.CV↗