arXiv subjects

Yuta Kusakabe

Publications and source records attributed to Yuta Kusakabe.

15 recordsLinked to original sources

arXiv2026

Oka Geometry of Complex Six-Spheres

Alpöge constructed a complex structure on $S^6$ through a torus fibration with exactly three special fibres. We prove that every member of the associated admissible period family is Oka and that the family projection over each relatively compact parameter disc is an Oka map. The Oka property survives the deletion or simultaneous blow-up of any finite set of distinct points in any member. Blowing up a single point therefore produces an Oka complex manifold diffeomorphic, but not biholomorphic, to $\mathbb{P}^3$. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.

math.CV↗

arXiv2026

Meromorphic approximation of holomorphic maps into algebraically elliptic projective varieties

We show that holomorphic maps from a neighborhood of a compact set in a Stein space to an algebraically elliptic smooth projective variety can be approximated by meromorphic maps, provided that the map extends continuously to the whole space and the compact set is convex with respect to a submonoid of the Picard group. The proof uses algebraic sprays and their polynomial lifts to affine cones, which we call conic lifts. Rational sprays, which generalize algebraic sprays, extend the result to maps avoiding the exceptional set of a rational spray. We also show that, for smooth projective varieties, the existence of an algebraic spray which is dominating at some point is a birational invariant.

math.CV↗

arXiv2026

On the fundamental groups of subelliptic varieties

We show that the fundamental group of any smooth subelliptic variety is finite. Moreover, it is also proved that every finite group can be realized as the fundamental group of a smooth subelliptic variety. As a consequence, it follows that there exists a smooth subelliptic variety homotopy equivalent to the $n$-sphere if and only if $n>1$. This result can be considered as a negative answer to the algebraic version of Gromov's problem on the homotopy types of Oka manifolds.

math.AG↗

arXiv2026

Every Gromov elliptic domain is meromorphically pseudoconvex

We prove that every Gromov elliptic domain $Ω$ in a complex manifold has the following property, which we call meromorphic pseudoconvexity: each boundary point has a neighborhood $V$ such that $Ω\cap V$ becomes Stein after removing a set which is, locally in $Ω\cap V$, the union of the polar sets of finitely many meromorphic functions on $Ω\cap V$. Since meromorphic functions on Hartogs figures extend to their hulls, it follows that, for a domain $Ω=\operatorname{int}\overlineΩ$ in a Stein manifold, Gromov ellipticity is equivalent to being Oka and Stein, and also to being Oka and admitting a complete Kähler metric. This complements a theorem of Matsumura and the author that every weakly pseudoconvex Oka manifold admitting a positive line bundle is elliptic. As applications, we show that the Oka manifolds $\mathbb C^n\setminus\overline{\mathbb B^n}$ and $\mathbb C^n\setminus\mathbb R^n$, $n\geq2$, are not elliptic, and that no proper domain with Lipschitz boundary in $\mathbb P^n$, $n\geq2$, is elliptic.

math.CV↗

arXiv2026

Oka realizations, fundamental groups, and holomorphic cofibrations

The homotopy types of Oka manifolds are exactly those of countable finite-dimensional CW complexes. We construct realizations as open subsets of complex Euclidean spaces with explicit dimension bounds. Other realizations have good open covers by sets biholomorphic to $\C^N$, with prescribed finite nerves, or are unions of Fatou--Bieberbach domains whose nonempty finite intersections are again Fatou--Bieberbach domains. Every nonempty Stein manifold admits a proper holomorphic embedding into an Oka open subset of some $\C^N$, and this embedding is a homotopy equivalence. We obtain holomorphic representatives of ordinary and generalized cohomology classes, and realize every countable group as the fundamental group of an Oka domain in $\C^5$. The fundamental groups of algebraically elliptic manifolds are precisely the finite groups, and every virtually abelian Kähler group is the fundamental group of a projective Oka manifold. Finally, we classify holomorphic cofibrations in Lárusson's intermediate model structures and establish factorizations over aspherical targets into proper holomorphic embeddings and holomorphic fibre bundles with Oka fibre.

math.CV↗

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↗

arXiv2024

Oka tubes in holomorphic line bundles

Let $(E,h)$ be a semipositive hermitian holomorphic line bundle on a compact complex manifold $X$ with $\dim X>1$. Assume that for each point $x\in X$ there exists a divisor $D\in |E|$ in the complete linear system determined by $E$ whose complement $X\setminus D$ is a Stein neighbourhood of $x$ with the density property. Then, the disc bundle $Δ_h(E)=\{e\in E:|e|_h<1\}$ is an Oka manifold while $D_h(E)=\{e\in E:|e|_h>1\}$ is a Kobayashi hyperbolic domain. In particular, the zero section of $E$ admits a basis of Oka neighbourhoods $\{|e|_h 0$. We show that this holds if $X$ is a rational homogeneous manifold of dimension $>1$. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

math.CV↗

arXiv2023

Oka properties of complements of holomorphically convex sets

Our main theorem states that the complement of a compact holomorphically convex set in a Stein manifold with the density property is an Oka manifold. This gives a positive answer to the well-known long-standing problem in Oka theory whether the complement of a compact polynomially convex set in $\mathbb{C}^{n}$ $(n>1)$ is Oka. Furthermore, we obtain new examples of nonelliptic Oka manifolds which negatively answer Gromov's question. The relative version of the main theorem is also proved. As an application, we show that the complement $\mathbb{C}^{n}\setminus\mathbb{R}^{k}$ of a totally real affine subspace is Oka if $n>1$ and $(n,k)\neq(2,1),(2,2),(3,3)$.

math.CV↗

arXiv2022

Surjective morphisms onto subelliptic varieties

We prove that every smooth subelliptic variety admits a surjective morphism from an affine space. This result gives partial answers to the questions of Arzhantsev and Forstnerič. As an application, we characterize open images of morphisms between affine spaces. We also obtain the jet interpolation theorem for morphisms from zero-dimensional subschemes of affine varieties to smooth subelliptic varieties.

math.AG↗

arXiv2020

Thom's jet transversality theorem for regular maps

We establish Thom's jet transversality theorem for regular maps from an affine algebraic manifold to an algebraic manifold satisfying a suitable flexibility condition. It can be considered as the algebraic version of Forstnerič's jet transversality theorem for holomorphic maps from a Stein manifold to an Oka manifold. Our jet transversality theorem implies genericity theorems for regular maps of maximal ranks. As an application, it follows that every connected compact locally flexible manifold is the image of a holomorphic submersion from an affine space. We also show that every algebraically degenerate subvariety of codimension at least two in a locally flexible manifold has an Oka complement.

math.AG↗

arXiv2020

An implicit function theorem for sprays and applications to Oka theory

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson's elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov's condition $\mathrm{Ell}_{1}$, and the equivariant localization principle is also given.

math.CV↗

arXiv2020

Elliptic characterization and unification of Oka maps

We generalize our elliptic characterization of Oka manifolds to Oka maps. The generalized characterization can be considered as an affirmative answer to the relative version of Gromov's conjecture. As an application, we unify previously known Oka principles for submersions; namely the Gromov type Oka principle for subelliptic submersions and the Forstnerič type Oka principle for holomorphic fiber bundles with CAP fibers. We also establish the localization principle for Oka maps which gives new examples of Oka maps.

math.CV↗

arXiv2019

Oka complements of countable sets and non-elliptic Oka manifolds

We study the Oka properties of complements of closed countable sets in $\mathbb{C}^{n}\ (n>1)$ which are not necessarily discrete. Our main result states that every tame closed countable set in $\mathbb{C}^{n}\ (n>1)$ with a discrete derived set has an Oka complement. As an application, we obtain non-elliptic Oka manifolds which negatively answer a long-standing question of Gromov. Moreover, we show that these examples are not even weakly subelliptic. It is also proved that every finite set in a Hopf manifold has an Oka complement and an Oka blowup.

math.CV↗

arXiv2018

Elliptic characterization and localization of Oka manifolds

We prove that Gromov's ellipticity condition $\mathrm{Ell}_1$ characterizes Oka manifolds. This characterization gives another proof of the fact that subellipticity implies the Oka property, and affirmative answers to Gromov's conjectures. As another application, we establish the localization principle for Oka manifolds, which gives new examples of Oka manifolds. In the appendix, it is also shown that the Oka property is not a bimeromorphic invariant.

math.CV↗

arXiv2017

Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps

We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain $Ω\Subset\mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(Ω,Y)$ contains a dense holomorphic disc. Our second result states that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$. In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain $Ω\Subset\mathbb{C}^n$, any fixed-point-free automorphism of $Ω$ and any connected complex manifold $Y$, there exists a universal map $Ω\to Y$. We also characterize Oka manifolds by the existence of universal maps.

math.CV↗