Stein Manifolds, the Oka Principle, and Gromov Ellipticity¶
Job: develop the Stein characterization (holomorphic convexity / separation / embedding), the Oka–Grauert–Forster principle (topological = holomorphic on Stein), the Gromov Oka / dominating spray / ellipticity, and the Oka manifolds / flexibility vs hyperbolicity — with the Grauert Oka for the Picard group as the worked example.
Core¶
Stein manifolds and the characterization¶
A complex manifold \(X\) is Stein iff: (a) holomorphically convex (the holomorphic hull \(\hat K=\{x:|f(x)|\le\sup_K|f|\,\forall f\in\mathcal{O}(X)\}\) is compact for \(K\) compact); (b) holomorphic separation (the holomorphic functions separate points); (c) countable topology.
| Stein characterization | Description |
|---|---|
| Holomorphic convexity | \(\hat K\) compact for \(K\) compact |
| Holomorphic separation | \(\forall x\ne y\in X\), \(\exists f\in\mathcal{O}(X):f(x)\ne f(y)\) |
| Countable topology | \(X\) has a countable base |
| Embedding (Bishop–Narasimhan–Remmert) | every Stein \(n\)-fold embeds as a closed complex submanifold of \(\mathbb{C}^{2n+1}\) |
| Stein vs affine | every smooth affine variety is Stein (analytification); not every Stein is algebraic |
Theorem (Bishop–Narasimhan–Remmert embedding). Every Stein manifold of dimension \(n\) embeds as a closed complex submanifold of \(\mathbb{C}^{2n+1}\) (the holomorphic embedding, the Stein manifolds as the affine / non-compact complex manifolds). The Stein manifolds are the non-compact complex manifolds with the maximal holomorphic function theory.
The Oka principle (Oka–Grauert–Forster)¶
The Oka principle: on a Stein manifold, the topological obstructions to the holomorphic problems vanish — the holomorphic and the topological classifications coincide.
Theorem (Grauert Oka). The classification of holomorphic vector bundles on a Stein \(X\) coincides with the topological classification: \(H^1(X,\mathcal{O}^\times)\cong H^1(X,\mathcal{C}^\times)\) (the holomorphic vector bundles \(\leftrightarrow\) the topological vector bundles, via the exponential / the Oka–Cartan, the vanishing of the obstruction \(H^1(\mathcal{O})\to H^1(\mathcal{O}^\times)\to H^2(\mathbb{Z})\)).
Theorem (Forster–Ramspott). The holomorphic maps / the homotopy classes coincide on Stein (the Oka for maps, the topological-to-holomorphic: a continuous map from a Stein to a complex manifold is homotopic to a holomorphic map, under the Oka conditions on the target).
| Oka principle | Description |
|---|---|
| Grauert Oka (vector bundles) | \(H^1(X,\mathcal{O}^\times)\cong H^1(X,\mathcal{C}^\times)\) (the holomorphic = topological bundles) |
| Forster–Ramspott (maps) | the continuous maps \(\to\) holomorphic maps (the homotopy, the Oka for maps) |
| Topological-to-holomorphic | the topological obstructions vanish on Stein, the holomorphic = topological classification |
The Gromov Oka principle and the elliptic complex geometry¶
The Gromov Oka principle — the modern / homotopy-theoretic formulation: a complex manifold \(X\) is Oka iff it satisfies the Oka property (the approximation / the interpolation / the ellipticity), and for an Oka target \(X\), the continuous maps from a Stein source \(\to X\) are homotopic to holomorphic maps, and the homotopy can be chosen via the holomorphic maps.
The elliptic complex geometry (Gromov): the Oka property is equivalent to the existence of a dominating spray / the elliptic / the h-principle for the holomorphic maps. A spray on \(X\) is a holomorphic map \(s:E\to X\) from a vector bundle \(E\to X\) (the spray bundle), with \(s|_{\mathrm{zero}}=\mathrm{id}\); the spray is dominating if the derivative \(Ds|_{\mathrm{zero}}:E_x\to T_xX\) is surjective for all \(x\) (the ellipticity, the flexibility).
| Oka property | Description |
|---|---|
| Gromov Oka / dominating spray | the existence of a dominating spray \(\Leftrightarrow\) the Oka property (the ellipticity / the h-principle) |
| Oka manifolds (Forstnerič) | the class of complex manifolds satisfying the Oka property (the flexible / the elliptic) |
| Oka vs hyperbolic | the Oka (flexible) vs the Kobayashi hyperbolic (rigid, the Brody hyperbolicity) — the opposite extremes |
The Oka manifolds (Forstnerič): the class of complex manifolds satisfying the Oka property — the flexible / the elliptic manifolds, the opposite of the hyperbolic (the Kobayashi hyperbolic, the Brody). The Oka property is the flexibility / the h-principle: the holomorphic maps satisfy the h-principle (the continuous = the holomorphic up to homotopy), the opposite of the rigidity (the Gromov–Eliashberg rigidity vs flexibility).
The Oka-Grauert–Gromov for principal bundles and the h-principle¶
| Oka / h-principle | Description |
|---|---|
| Oka for principal \(G\)-bundles | the holomorphic principal bundles on Stein \(\leftrightarrow\) the topological principal bundles (the Grauert–Gromov) |
| h-principle for holomorphic maps | the Gromov h-principle: the Oka property as the flexibility / the h-principle (the contrast with the rigidity) |
| Flexibility / rigidity | the Oka as the flexible complex geometry (the h-principle), the Gromov–Eliashberg rigidity (the rigid, the opposite) |
See it¶
Grauert Oka theorem for the Picard group of a Stein manifold. For a Stein manifold \(X\), the Picard group \(\mathrm{Pic}(X)=H^1(X,\mathcal{O}^\times)\) (the holomorphic line bundles). The exponential sequence $$ 0\to\mathbb{Z}\to\mathcal{O}\xrightarrow{\exp}\mathcal{O}^\times\to 0 $$ (the sheaves on \(X\), the \(\exp(f)=e^{2\pi if}\)) gives the long exact sequence $$ H^1(X,\mathcal{O})\to H^1(X,\mathcal{O}^\times)\to H^2(X,\mathbb{Z})\to H^2(X,\mathcal{O}). $$ By the Cartan B (the \(H^1(X,\mathcal{O})=H^2(X,\mathcal{O})=0\) on Stein), the map \(H^1(X,\mathcal{O}^\times)\to H^2(X,\mathbb{Z})\) is an isomorphism: the holomorphic line bundles are classified by the topological \(H^2(X,\mathbb{Z})\) (the first Chern class \(c_1\), the Oka–Grauert theorem for the line bundles). The key point: the Cartan B vanishing collapses the exponential sequence, and the holomorphic line bundles are the topological ones (classified by the Chern class), the Oka principle as the vanishing of the holomorphic obstruction. The same argument for the vector bundles of rank \(r\): the Grauert Oka \(H^1(X,\mathrm{GL}_r(\mathcal{O}))\cong H^1(X,\mathrm{GL}_r(\mathcal{C}))\) (the holomorphic = topological rank-\(r\) bundles).
Check¶
Show that the Cartan B vanishing \(H^1(X,\mathcal{O})=H^2(X,\mathcal{O})=0\) on a Stein \(X\) collapses the exponential sequence, and identify the Picard group.
The exponential sequence \(0\to\mathbb{Z}\to\mathcal{O}\xrightarrow{\exp}\mathcal{O}^\times\to 0\) gives \(H^1(\mathcal{O})\to H^1(\mathcal{O}^\times)\to H^2(\mathbb{Z})\to H^2(\mathcal{O})\). On a Stein \(X\), the Cartan B gives \(H^1(\mathcal{O})=H^2(\mathcal{O})=0\) (the vanishing of the coherent cohomology in degrees \(\ge1\)). So the long exact sequence collapses to \(0\to H^1(\mathcal{O}^\times)\xrightarrow{\sim} H^2(\mathbb{Z})\to 0\) (the isomorphism), i.e. \(\mathrm{Pic}(X)=H^1(\mathcal{O}^\times)\cong H^2(\mathbb{Z})\) (the Picard group is the second integral cohomology, the first Chern class). The holomorphic line bundles are classified by the topological \(H^2(\mathbb{Z})\) — the Oka principle for the line bundles.
Explain why the Oka property (the dominating spray) is the flexibility, and contrast with the Kobayashi hyperbolicity.
The Oka property (the dominating spray \(s:E\to X\) with \(Ds\) surjective) means the holomorphic maps from a Stein source to \(X\) satisfy the h-principle: every continuous map is homotopic to a holomorphic one (the flexibility, the absence of holomorphic rigidity). The Kobayashi hyperbolicity (the Kobayashi pseudo-distance is a true distance, the Brody hyperbolicity for compact) is the opposite: the holomorphic maps from \(\mathbb{C}\) (the Stein source) to \(X\) are constant (the rigidity, the absence of non-constant holomorphic maps from the flexible source). The Oka (flexible) and the hyperbolic (rigid) are the opposite extremes of the complex geometry: the Oka manifolds have the maximal holomorphic flexibility (the h-principle), the hyperbolic have the maximal rigidity (no non-constant maps from \(\mathbb{C}\)).
Pitfalls¶
- The Oka principle requires the Stein source (not every complex manifold is Oka — the hyperbolic / the rigid manifolds fail the Oka property). The Oka is for the target (the Oka target), the Stein for the source (the Stein source).
- The Gromov Oka / the dominating spray is the modern formulation (not the classical Oka–Grauert for the specific problems — the Gromov is the general homotopy-theoretic / the h-principle, the Grauert the vector bundles).
- The Oka manifolds (Forstnerič) are the flexible / elliptic complex manifolds, the opposite of the hyperbolic — the two are not exhaustive (there are complex manifolds that are neither Oka nor hyperbolic, the intermediate / the non-classified).
- The Stein vs affine: every smooth affine variety is Stein (the analytification), but not every Stein is algebraic (the non-algebraic Stein manifolds, the non-Stein affine — the analytification is the one direction).
- The embedding \(\mathbb{C}^{2n+1}\) is the Stein embedding (the holomorphic, not the algebraic — the algebraic embedding may require a larger ambient space, the algebraic vs analytic distinction).
- The h-principle for holomorphic maps (the Gromov) is the flexibility, the contrast with the Gromov–Eliashberg rigidity (the symplectic / the contact rigidity, the opposite of the Oka flexibility).
Refs¶
- Grauert, "Analytische Faserungen über holomorphvollständigen Räumen" (Math. Ann. 1957); Forster–Ramspott, "Okasche Princip und Injectivität"
- Gromov, "Oka's principle for holomorphic sections of elliptic bundles" (JAMS 1989); Forstnerič, Stein Manifolds and Holomorphic Mappings (2011)
- Bishop, "Analytic functions in several variables" (1961); Narasimhan, "Imbedding of holomorphically complete complex spaces" (1960)
- Kobayashi, Hyperbolic Complex Spaces (1998); Brody, "Compact manifolds and hyperbolicity" (1978)
Next¶
SCV index · Foundations · \(\bar\partial\)-Neumann & Kohn Estimates · CR Geometry & \(\bar\partial_b\) · Analytic Methods & \(L^2\) · Complex geometry · Mirror symmetry