A homotopy theorem for Oka theory

Abstract

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.

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Acknowledgements

I would like to thank Frank Kutzschebauch for suggesting the topic and many helpful discussions. Moreover I would like to thank Finnur Lárusson and Gerald Schwarz for numerous valuable comments on a preprint. I am also very thankful for stimulating discussions with Jasna Prezelj and Franc Forstnerič. Moreover, I would like to thank the referee for valuable comments. The study was funded by Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (Grant no. 200021-178730).

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Communicated by Ngaiming Mok.

Appendix A. Applying Theorem 1 in Oka theory

Appendix A. Applying Theorem 1 in Oka theory

In this appendix we give references for the proofs of the assumptions of Theorem 1 in the settings of the Oka principles cited in the introduction. The intention is to give the reader a hint where the analytic challenges providing the assumptions of Theorem 1 are tackled in the original work. In some cases the cited work needs some adjustments, which will be pointed out. These adjustments were part of the author’s thesis [19], but cost too many lines to be included here.

A.1. Weak flexibility

The proof of the weak flexibility of a \({\mathcal {C}}\)-pair (AB) with respect to a given complex analytic sheaf is usually proved in two steps:

  1. (1)

    a parametric Runge approximation property, and

  2. (2)

    a gluing property.

The Oka principle for elliptic submersions: To show the weak flexibility assumptions of Theorem 1 in this setting one has to show that if \(h:Z \rightarrow X\) is a holomorphic submersion onto a reduced Stein space X and \(U\subset X\) is an open set such that the restriction \(h:h^{-1}(U) \rightarrow U\) admits a dominating spray, then every \({\mathcal {C}}\)-pair (AB) with \(B\subset U\) is weakly flexible for the sheaf of holomorphic sections of h. This was discovered by Gromov [12]. Detailed proofs of Gromov’s insight have been given by Forstnerič and Prezelj (see e.g. [9]). A convenient source is [8]: The weak flexibility of the pair (AB) follows from the Runge approximation property stated in Theorem 6.2.2, p. 284 and the gluing property stated in Proposition 6.7.2, p. 288.

The Oka principle for principal G-bundles: This Oka principle is a special case of the Oka principle for elliptic submersions. However, since the two remaining Oka principles build strongly on Cartan’s exposition of Grauert’s work [3], it makes sense to give references. The required Runge approximation property and gluing property which yield the weak flexibility of \({\mathcal {C}}\)-pairs follow in this case from a Runge approximation property and a splitting lemma in an associated sheaf of groups. These two key results are in Cartan’s exposition of Grauert’s work Proposition 1 and 2 (see [3], p. 109).

The Oka principle for admissible pairs of sheaves: This Oka principle builds on Cartan’s exposition of Grauert’s work. The necessary extensions of Cartan’s Proposition 1 and 2 from [3] are Lemma 2 and 3 from Forster and Ramspott’s work (see [4], p. 271 and p. 273).

The Oka principle for equivariant isomorphisms: This Oka principle builds likewise on Cartan’s text [3]. The necessary extensions of Cartan’s Proposition 1 and 2 in the work of Kutzschebauch, Lárusson and Schwarz are Proposition 10.2 and 10.3 (see [14], p. 7293).

The key results from the last three Oka principles, i.e. from those Oka principles which build on Cartan’s exposition of Grauert’s work, need some adjustments to yield complete proofs of the weak flexibility of \({\mathcal {C}}\)-pairs. These adjustments can be found in [19], Chapter 5 and 6.

A.2. Local weak homotopy equivalences

The difficulty of the proof that a given inclusion of sheaves \(\Phi \hookrightarrow \Psi \) is a local weak homotopy equivalence depends strongly on the setting.

The Oka principle for elliptic submersions: In this setting it suffices to show that if \(h:Z \rightarrow X\) is a holomorphic submersion onto a reduced complex space and \(\Phi \hookrightarrow \Psi \) is the inclusion of the sheaf of holomorphic sections to the sheaf of continuous sections of h, then \(\Phi \hookrightarrow \Psi \) is a local weak homotopy equivalence. Gromov seems to have taken this result for granted in [12]. In the more detailed work [9] local weak homotopy equivalences are not introduced. Instead, an analogue of our Proposition 2.9 is stated in the special case of holomorphic submersions, namely Proposition 4.7. The validity of Proposition 4.7 in [9] has been carefully checked in the thesis of Jasna Prezelj [18], which yields implicitly a proof of the fact that \(\Phi \hookrightarrow \Psi \) is a local weak homotopy equivalence in the mentioned case.

The Oka principle for principal G-bundles: This is a special case of the above.

The Oka principle for admissible pairs of sheaves: In the work of Forster and Ramspott [4] there is a slight weakening of \(\Phi \hookrightarrow \Psi \) being a local weak homotopy equivalence in the assumption, namely the homotopy property (PH) from Satz 1, p. 267. Using (PH), the fact that an inclusion of admissible pairs of sheaves \(\Phi \hookrightarrow \Psi \) in the sense of Forster and Ramspott is a local weak homotopy equivalence is a corollary to Lemma 1, p. 269 in [4].

The Oka principle for equivariant isomorphisms: In this setting it is hard to show that the given inclusion \(\Phi \hookrightarrow \Psi \) is a local weak homotopy equivalence. Theorem 1.3, p. 7253 in [14] reduces the proof to the case where we have \(X=Y\) for given Stein G-manifolds X and Y, where G is a complex reductive Lie group. Having this, one needs to extend some Lemmata from [14] in Section 3 and 5 to analogous parametric versions. These necessary adaptations are simple once Section 3 and 5 in [14] are understood.

For more details that the inclusions \(\Phi \hookrightarrow \Psi \) from the mentioned Oka principles are local weak homotopy equivalences see [19], Chapter 4.

A.3. Sheaves of topological spaces

Equipping all sets of local sections from the sheaves corresponding to the Oka principles from the introduction with the compact open topology turns these into sheaves of topological spaces. This is used in all the quoted work. To see that there is no pathological behavior when dealing with parametric sheaves, see Lemma 2.1, a basic fact which is usually taken for granted in Oka theory. Recall that (complete) metric sheaves are defined as those sheaves of topological spaces whose sets of local sections are equipped with a (complete) metric which induces the topology. That the complex analytic sheaves from the mentioned Oka principles are complete metric depends on the following two facts.

Fact 1

Let X be a space which admits an exhaustion \(K_1\subset K_2\subset K_3 \subset \cdots \) by compacts and (Yd) a complete metric space. Then

$$\begin{aligned} d_{C(X,Y)}(f,g)=\sum _{n\ge 1} \frac{1}{2^n} \frac{d_n(f,g)}{1+d_n(f,g)}, \ \ d_n(f,g)=\max _{x \in K_n} d(f(x),g(x)) \end{aligned}$$

for continuous maps \(f,g:X\rightarrow Y\) defines a complete metric on the space of continuous maps \(X \rightarrow Y\) which induces the compact open topology.

Fact 2

Let XY be two locally connected locally compact second countable metric spaces. Then \(d_H(f,g)=d_{C(X,Y)}(f,g)+d_{C(Y,X)}(f^{-1},g^{-1})\), where \(d_{C(X,Y)}\) and \(d_{C(Y,X)}\) are as in Fact 1, defines a complete metric on the set of homeomorphisms \(X \rightarrow Y\) which induces the compact open topology.

Fact 1 is well known, and it is easy to show that \(d_H\) from Fact 2 turns the set of homeomorphisms into a complete metric space. It is not obvious (and for locally disconnected topological spaces generally false) that the topology induced by \(d_H\) is not finer than the compact open topology. The proof depends on the following

Theorem

(Arens [2]) The homeomorphism group of a locally compact locally connected Hausdorff space equipped with the compact open topology is a topological group.

To show that \(d_H\) induces the compact open topology it suffices to show that \(d_H(\cdot , g)\) is continuous with respect to the compact open topology for a fixed homeomorphism \(g:X \rightarrow Y\); and this is the case if \(I:{{\,\mathrm{Homeo}\,}}(Y,Z) \rightarrow {{\,\mathrm{Homeo}\,}}(Z,Y), \ I(f)=f^{-1}\) is continuous. The latter follows from the fact that I is composition of the three continuous maps given by \(f \mapsto g^{-1} \circ f\), \(f \mapsto f^{-1}\) and \(f \mapsto f \circ g^{-1}\), where the continuity of the second factor is due to Arens result.

Fact 1 and 2 imply that the considered complex analytic sheaves from the mentioned Oka principles are complete metric in the following way: It is known that the sets of local sections of the complex analytic sheaves from the mentioned Oka principles are closed subspaces either of a space of continuous maps \(X \rightarrow Y\) for suitable X and Y or from the space of homeomorphisms \(X \rightarrow Y\) for suitable X and Y. In any case, Fact 1 and 2 imply that the sets of local sections are closed subsets of a complete metric space and hence complete if equipped with the suitable metric from Fact 1 resp. Fact 2.

A.4. Ordered flexibility

Ordered flexibility is—in the context of Oka theory—most of times rather easy to show. Proofs shall be given elsewhere. Instead, we would like to discuss the examples addressed in Remark 1.8. In most known Oka principles one looks at inclusions of sheaves \(\Phi \hookrightarrow \Psi \), where \(\Phi \) lives in the complex analytic category and \(\Psi \) lives in the category of topological spaces. However, in recent advances (see e.g. [14, 16]) one is forced to place \(\Psi \) in the smooth category instead. In the smooth category—opposed to the complex analytic and the topological category—completeness is more delicate. One can turn e.g. the space of smooth functions \(\mathbb {C}\rightarrow \mathbb {C}\) into a complete metric space by including all higher derivatives in the definition of the pseudometrics from Fact 1. However, the resulting topology is finer than the compact open topology. This is a disadvantage for proofs in Oka theory since Lemma 2.1 does not apply anymore. A similar example emerges from [6], where one is forced to look at a sheaf \(\Psi \) of continuous sections which are holomorphic in a neighborhood of some fixed subvariety. There seems to be no (natural) way to turn this sheaf into a complete metric sheaf without refining the compact open topology. In these examples one benefits from assumption (2) in Theorem 1 as an alternative to assumption (1), since in (2) no completeness is asked.

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Studer, L. A homotopy theorem for Oka theory. Math. Ann. 378, 1533–1553 (2020). https://doi.org/10.1007/s00208-019-01918-0

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