We embed the category of complex manifolds into the simplicial category of prestacks on the simplicial site of Stein manifolds, a prestack being a contravariant simplicial functor from the site to the category of simplicial sets. The category of prestacks carries model structures, one of them defined for the first time here, which allow us to develop "holomorphic homotopy theory". More specifically, we use homotopical algebra to study lifting and extension properties of holomorphic maps, such as those given by the Oka Principle. We prove that holomorphic maps satisfy certain versions of the Oka Principle if and only if they are fibrations in suitable model structures. We are naturally led to a simplicial, rather than a topological, approach, which is a novelty in analysis.
A parametric Oka principle for liftings, recently proved by Forstneric, provides many examples of holomorphic maps that are fibrations in a model structure introduced in previous work of ours. We use this to show that the basic Oka property is equivalent to the parametric Oka property for a large class of manifolds. We introduce new versions of the basic and parametric Oka properties and show, for example, that a complex manifold $X$ has the basic Oka property if and only if every holomorphic map to $X$ from a contractible submanifold of $\mathbb C^n$ extends holomorphically to $\mathbb C^n$.
We apply concepts and tools from abstract homotopy theory to complex analysis and geometry, continuing our development of the idea that the Oka Principle is about fibrancy in suitable model structures. We explicitly factor a holomorphic map between Stein manifolds through mapping cylinders in three different model structures and use these factorizations to prove implications between ostensibly different Oka properties of complex manifolds and holomorphic maps. We show that for Stein manifolds, several Oka properties coincide and are characterized by the geometric condition of ellipticity. Goin…
A complex manifold $X$ satisfies the Oka-Grauert property if the inclusion $\Cal O(S,X) \hookrightarrow \Cal C(S,X)$ is a weak equivalence for every Stein manifold $S$, where the spaces of holomorphic and continuous maps from $S$ to $X$ are given the compact-open topology. Gromov's Oka principle states that if $X$ has a spray, then it has the Oka-Grauert property. The purpose of this paper is to investigate the Oka-Grauert property using homotopical algebra. We embed the category of complex manifolds into the model category of simplicial sheaves on the site of Stein manifolds. Our main result …
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.
We prove a homotopy theorem for sheaves. Its application shortens and simplifies the proof of many Oka principles such as Gromov's Oka principle for elliptic submersions.
The name of Oka principle, or Oka-Grauert principle, is traditionally used to refer to the holomorphic incarnation of the homotopy principle: on a Stein space, every problem that can be solved in the continuous category, can be solved in the holomorphic category as well. In this note, we begin the study of the same kind of questions on a Levi-flat manifold; more precisely, we try to obtain a classification of CR-bundles on a semiholomorphic foliation of type (n, 1). Our investigation should only be considered a preliminary exploration, as it deals only with some particular cases, either in ter…
This paper presents a proof of the generalized Oka-Grauert principle for 1-convex manifolds: Every continuous mapping from a 1-convex manifold X to a complex manifold Y which is already holomorphic on a neighborhood of the exceptional set is homotopic to a holomorphic one provided that either Y satisfies CAP or we are free to change the complex structure on X.
The paper is related to the author's talk at the Hayama Symposium in Complex Analysis in December 2000. In section 1 we survey results on the Oka principle for sections of holomorphic submersions over Stein manifolds. In section 2 we apply these results to the lifting problem for holomorphic mappings from Stein manifolds, and in section 3 we apply them to the problem of removability of intersections of holomorphic maps from Stein manifolds with closed complex subvarieties of the target space.
In this paper we establish a basic version of the Oka principle for multivalued sections of ramified holomorphic maps h from a complex manifold Z onto a Stein manifold X. If the ramification locus of h projects into a closed complex subvariety X' of X and if h admits a fiber dominating spray over a small neighborhood of any point in X\X' then any multivalued continuous section of h which is holomorphic in a neighborhood of X' and unramified in X\X' can be homotopically deformed to a global holomorphic multivalued section. The corresponding results for sections of unramified submersions were es…
Let X and Y be complex manifolds. One says that maps from X to Y satisfy the Oka principle if the inclusion of the space of holomorphic maps from X to Y into the space of continuous maps is a weak homotopy equivalence. In 1957 H. Grauert proved the Oka principle for maps from Stein manifolds to complex Lie groups and homogeneous spaces, as well as for sections of fiber bundles with homogeneous fibers over a Stein base. In 1989 M. Gromov extended Grauert's result to sections of submersions over a Stein base which admit dominating sprays over small open sets in the base; for proof see [F. Forstn…
Let $X$ be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps $X \to (\mathbb{C}^*)^2$ by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on $X$ outside a certain set. This generalises and strengthens a recent result of Alarcon and Lopez. We also give a Forstneric-Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into $(\mathbb{C}^*)^2$.
We prove a theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, 851-897, 1989) to the effect that sections of certain holomorphic submersions h from a complex manifold Z onto a Stein manifold X satisfy the Oka principle, meaning that the inclusion of the space of holomorphic sections into the space of continuous sections is a weak homotopy equivalence. The Oka principle holds if the submersion admits a fiber-dominating spray over a small neighborhood of any point in X. This extends a classical result of Grauert (Holomorphe Funktionen mit We…
The classical Grauert and Ramspott theorems constitute the foundation of the Oka principle on Stein spaces. In this paper we establish analogous results on the maximal ideal space $M(H^\infty)$ of the Banach algebra $H^\infty$ of bounded holomorphic functions on the open unit disk $\mathbb D\subset\mathbb C$. We illustrate our results by some examples and applications to the theory of operator-valued $H^\infty$ functions.
Gromov, in his seminal 1989 paper on the Oka principle, introduced the notion of an elliptic manifold and proved that every continuous map from a Stein manifold to an elliptic manifold is homotopic to a holomorphic map. We show that a much stronger Oka principle holds in the special case of maps from certain open Riemann surfaces called circular domains into CxC*, namely that every continuous map is homotopic to a proper holomorphic embedding. An important ingredient is a generalisation to CxC* of recent results of Wold and Forstneric on the long-standing problem of properly embedding open Rie…
We give a proof of the following theorem of M. Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc., 2 (1989), 851-897). Let Z be a holomorphic fiber bundle over a Stein manifold. If the fiber of Z admits a dominating holomorphic spray then the inclusion of the space of global holomorphic sections of Z into the space of continuous sections is a weak homotopy equivalence. In the classical case when the fiber is a complex Lie group or a homogeneous space this was proved by H. Grauert (Math. Ann., 135 (1958), 263--273). For further results in this direction se…
We prove a parametric Oka principle for equivariant sections of a holomorphic fibre bundle $E$ with a structure group bundle $\mathscr G$ on a reduced Stein space $X$, such that the fibre of $E$ is a homogeneous space of the fibre of $\mathscr G$, with the complexification $K^\mathbb C$ of a compact real Lie group $K$ acting on $X$, $\mathscr G$, and $E$. Our main result is that the inclusion of the space of $K^\mathbb C$-equivariant holomorphic sections of $E$ over $X$ into the space of $K$-equivariant continuous sections is a weak homotopy equivalence. The result has a wide scope; we describ…
We survey recent work, published since 2015, on equivariant Oka theory. The main results described in the survey are as follows. Homotopy principles for equivariant isomorphisms of Stein manifolds on which a reductive complex Lie group $G$ acts. Applications to the linearisation problem. A parametric Oka principle for sections of a bundle $E$ of homogeneous spaces for a group bundle $\mathscr G$, all over a reduced Stein space $X$ with compatible actions of a reductive complex group on $E$, $\mathscr G$, and $X$. Application to the classification of generalised principal bundles with a group a…
It is an elementary fact that the action by holomorphic automorphisms on C^n is infinitely transitive, i.e., m-transitive for any m in N. The same holds on any Stein manifold with the holomorphic density property X. We study a parametrized case: we consider m points on X parametrized by a Stein manifold W, and seek a family of automorphisms of X, parametrized by W, putting them into a standard form which does not depend on the parameter. This general transitivity is shown to enjoy an Oka principle, to the effect that the obstruction to a holomorphic solution is of a purely topological nature. …
This is a survey on the homotopy principle in complex analysis on Stein manifolds, also called the Oka principle in this context. We concentrate on the following topics: the Oka-Grauert principle (classification of holomorphic vector bundles); the homotopy principle for holomorphic mappings from Stein manifolds and, more generally, for sections of holomorphic submersions with sprays; question on removability of intersections of holomorphic mappings with complex subvarieties; embeddings and immersions of Stein manifolds in affine spaces of minimal dimension; embeddings of open Riemann surfaces …
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.
A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, ma…
Oka theory has its roots in the classical Oka-Grauert principle whose main result is Grauert's classification of principal holomorphic fiber bundles over Stein spaces. Modern Oka theory concerns holomorphic maps from Stein manifolds and Stein spaces to Oka manifolds. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. In this expository paper we discuss Oka manifolds and Oka maps. We describe equivalent characterizations of Oka manifolds, the functorial properties of this class, and geometric sufficient conditions for …
Let $G$ be a reductive complex Lie group acting holomorphically on Stein manifolds $X$ and $Y$. Let $p_X\colon X\to Q_X$ and $p_Y\colon Y\to Q_Y$ be the quotient mappings. Assume that we have a biholomorphism $Q:= Q_X\to Q_Y$ and an open cover $\{U_i\}$ of $Q$ and $G$-biholomorphisms $Φ_i\colon p_X^{-1}(U_i)\to p_Y^{-1}(U_i)$ inducing the identity on $U_i$. There is a sheaf of groups $\mathcal A$ on $Q$ such that the isomorphism classes of all possible $Y$ is the cohomology set $H^1(Q,\mathcal A)$. The main question we address is to what extent $H^1(Q,\mathcal A)$ contains only topological inf…
Over the past decade, the class of Oka manifolds has emerged from Gromov's seminal work on the Oka principle. Roughly speaking, Oka manifolds are complex manifolds that are the target of "many" holomorphic maps from affine spaces. They are "dual" to Stein manifolds and "opposite" to Kobayashi-hyperbolic manifolds. The prototypical examples are complex homogeneous spaces, but there are many other examples: there are many ways to construct new Oka manifolds from old. The class of Oka manifolds has good formal properties, partly explained by a close connection with abstract homotopy theory. These…
Assume that E and B are complex manifolds and that pi is a holomorphic Serre fibration from E onto E such that E admits a finite dominating family of holomorphic fiber-sprays over a small neighborhood of any point in B. We show that the parametric Oka property (POP) of B implies POP of E; conversely, POP of E implies POP of B for contractible parameter spaces. This follows from a parametric Oka principle for holomorphic liftings which we establish in the paper.
Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that …
Oka theory has its roots in the classical Oka principle in complex analysis. It has emerged as a subfield of complex geometry in its own right since the appearance of a seminal paper of M. Gromov in 1989. Following a brief review of Stein manifolds, we discuss the recently introduced category of Oka manifolds and Oka maps. We consider geometric sufficient conditions for being Oka, the most important of which is ellipticity, introduced by Gromov. We explain how Oka manifolds and maps naturally fit into an abstract homotopy-theoretic framework. We describe recent applications and some key open p…
Let $G$ be a reductive complex Lie group acting holomorphically on normal Stein spaces $X$ and $Y$, which are locally $G$-biholomorphic over a common categorical quotient $Q$. When is there a global $G$-biholomorphism $X\to Y$? If the actions of $G$ on $X$ and $Y$ are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that $X$ and $Y$ are $G$-biholomorphic …
In this paper we present the main developments in Oka theory since the publication of my book Stein Manifolds and Holomorphic Mappings (The Homotopy Principle in Complex Analysis)}, Second Edition, Springer, 2017. We also give several new results, examples and constructions of Oka domains in Euclidean and projective spaces. Furthermore, we show that for $n>1$ the fibre $\mathbb C^n$ in a Stein family can degenerate to a non-Oka fibre, thereby answering a question of Takeo Ohsawa. Several open problems are discussed.
We give the following positive answer to Gromov's question (in "Oka's principle for holomorphic sections of elliptic bundles", J. Amer. Math. Soc. 2, 851-897 (1989), 3.4.(D), page 881). THEOREM: If every holomorphic map from a compact convex set in a complex Euclidean space C^n to a certain complex manifold Y is a uniform limit of entire maps of C^n to Y, then Y enjoys the parametric Oka property. In particular, for any reduced Stein space X the inclusion of the space of holomorphic maps of X to Y into the space of continuous maps is a weak homotopy equivalence. This shows that all Oka type pr…
The presented splitting lemma extends the techniques of Gromov and Forstnerič to glue local sections of a given analytic sheaf, a key step in the proof of all Oka principles. The novelty on which the proof depends is a lifting lemma for transition maps of coherent sheaves, which yields a reduction of the proof to the work of Forstnerič. As applications we get shortcuts in the proofs of Forster and Ramspott's Oka principle for admissible pairs and of the interpolation property of sections of elliptic submersions, an extension of Gromov's results obtained by Forstnerič and Prezelj.
The geometric notion of ellipticity for complex manifolds was introduced by Gromov in his seminal 1989 paper on the Oka principle, and is a sufficient condition for a manifold to be Oka. In the current paper we present contributions to three open questions involving elliptic and Oka manifolds. We show that quotients of C^n by discrete groups of affine transformations are elliptic. Combined with an example of Margulis, this yields new examples of elliptic manifolds with free fundamental groups and vanishing higher homotopy. Finally we show that every open Riemann surface embeds acyclically into…
The basic result of Oka theory, due to Gromov, states that every continuous map $f$ from a Stein manifold $S$ to an elliptic manifold $X$ can be deformed to a holomorphic map. It is natural to ask whether this can be done for all $f$ at once, in a way that depends continuously on $f$ and leaves $f$ fixed if it is holomorphic to begin with. In other words, is $\scrO(S,X)$ a deformation retract of $\scrC(S,X)$? We prove that it is if $S$ has a strictly plurisubharmonic Morse exhaustion with finitely many critical points; in particular, if $S$ is affine algebraic. The only property of $X$ used in…
Gromov, in his seminal 1989 paper on the Oka principle, proved that every continuous map from a Stein manifold into an elliptic manifold is homotopic to a holomorphic map. Previously we have shown that, given a continuous map $X \to \C\times\C^*$ from a finitely connected planar domain $X$ without isolated boundary points, a stronger Oka property holds, namely that the map is homotopic to a proper holomorphic embedding. Here we show that every continuous map from a finitely connected planar domain, possibly with punctures, into $\C\times\C^*$ is homotopic to a proper immersion that identifies …
We show that a complex projective manifold X which satisfies the Gromov's h-principle is `special', and raise some questions about the reverse implication, the extension to the quasi-K\" ahler case, and the relationships of these properties to the `Oka' property. The guiding principle is that the existence of many Stein manifolds which have degenerate Kobayashi pseudometric gives strong obstructions to the complex hyperbolicity of X satisfying the h-principle.
In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka-Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930's, has had an enormous impact on the development of the area of Several Complex Variables and the number of it…
Let $X$ be an elliptic curve and $\mathbb{P}$ the Riemann sphere. Since $X$ is compact, it is a deep theorem of Douady that the set $\mathcal{O}(X,\mathbb{P})$ consisting of holomorphic maps $X\to \mathbb{P}$ admits a complex structure. If $R_n$ denotes the set of maps of degree $n$, then Namba has shown for $n\geq2$ that $R_n$ is a $2n$-dimensional complex manifold. We study holomorphic flexibility properties of the spaces $R_2$ and $R_3$. Firstly, we show that $R_2$ is homogeneous and hence an Oka manifold. Secondly, we present our main theorem, that there is a $6$-sheeted branched covering …
For each natural number d, the space R_d of rational maps of degree d on the Riemann sphere has the structure of a complex manifold. The topology of these manifolds has been extensively studied. The recent development of Oka theory raises some new and interesting questions about their complex structure. We apply geometric invariant theory to the cases of degree 2 and 3, studying a double action of the Möbius group on R_d. The action on R_2 is transitive, implying that R_2 is an Oka manifold. The action on R_3 has C as a categorical quotient; we give an explicit formula for the quotient map and…
Let X be a Stein manifold and let Y be a complex manifold which admits a spray in the sense of Gromov (Oka's principle for holomorphic sections of elliptic bundles, J. Amer. Math. Soc. 2, pp. 851-897 (1989)). We prove that for every closed complex subvariety X_0 of X and for every continuous map f_0 from X to Y whose restriction to X_0 is holomorphic there exists a homotopy of maps f_t from X to Y whose restrictions to X_0 agree with f_0 and such that the map f_1 is holomorphic on X. We obtain analogous results for sections of holomorphic submersions with sprays over Stein manifolds or Stein s…
We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$ can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provide…
Let π:Z\to X be a holomorphic submersion of a complex manifold Z onto a complex manifold X and D\Subset X a 1-convex domain with strongly pseudoconvex boundary. We prove that under certain conditions there always exists a spray of π-sections over \bar{D} which has prescribed core, it fixes the exceptional set E of D, and is dominating on \bar{D}\setminus E. Each section in this spray is of class C^k(\bar{D}) and holomorphic on D. As a consequence we obtain several approximation results for π-sections. In particular, we prove that π-sections which are of class C^k(\bar{D}) and holomorphic on D …
In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold $X$ is homotopic to a holomorphic contact structure on a Stein domain $Ω\subset X$ which is diffeotopic to $X$. We also prove a parametric h-principle in this setting, analogous to Gromov's h-principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our methods also furnish a parametric h-princip…
