Affine Simplices in Oka Manifolds

Finnur Lárusson

Received: June 8, 2009 Revised: July 27, 2009

Communicated by Thomas Peternell

Abstract. We show that the homotopy type of a complex manifold X satisfying the Oka property is captured by holomorphic maps from the affine spaces ℂ^{n}, n ≥ 0, into X. Among such X are all complex Lie groups and their homogeneous spaces. We present generalisations of this result, one of which states that the homotopy type of the space of continuous maps from any smooth manifold to X is given by a simplicial set whose simplices are holomorphic maps into X.

2000 Mathematics Subject Classification: Primary 32Q55. Secondary 18G30, 32C18, 55U10. Keywords and Phrases: Complex manifold, Stein manifold, Oka manifold, Oka property, simplicial set, singular set, affine simplex, homotopy type, weak equivalence.

1. Introduction

Motivated by Gromov’s comments in his seminal paper [12], Sec. 3.5.G and 3.5.G’, we prove in Sec. 2 that the homotopy type of an Oka manifold X (as a topological space) is captured by holomorphic maps from the affine spaces ℂ^{n}, n ≥ 0, into X. In Sec. 3 we present generalisations of this result. We start with a very brief review of some background material. The concept of an Oka manifold has evolved from Gromov’s paper and subsequent work, mainly due to Forstnerič, see in particular [4] and [5]. By a Stein inclusion we mean the inclusion into a reduced Stein space S (or a Stein manifold: the choice is immaterial) of a closed analytic subvariety T. A complex manifold X has the basic Oka property with interpolation (BOPI) with respect to T ↪ S if every continuous map h : S → X with h|T holomorphic can be deformed to a holomorphic map S → X with h|T fixed. Also, X has the interpolation property with respect to T ↪ S if every holomorphic map h : T → X extends to a holomorphic map S → X. The following are equivalent (see [15]) and define what it means for X to be Oka:

(1) X has BOPI with respect to every Stein inclusion.

(2) X has the interpolation property, or equivalently BOPI, with respect to every Stein inclusion T ↪ ℂ^{n}, n ≥ 1, where T is contractible (holomorphically or topologically: the choice is immaterial).

The Oka property has several other equivalent formulations. Each of these has a parametric version, where instead of a single map h as above we have a family of maps depending continuously on a parameter. The parametric Oka properties are all equivalent [4], and are equivalent to the Oka property [7]. A holomorphic map f : X → Y has the parametric Oka property with interpolation (POPI) if for every Stein inclusion T ↪ S, every finite polyhedron P with a subpolyhedron Q, and every continuous map g : S × P → X such that the restriction g|S × Q is holomorphic along S (meaning that g(·, q) : S → X is holomorphic for each q ∈ Q), the restriction g|T × P is holomorphic along T, and the composition f ∘ g is holomorphic along S, there is a continuous map G : S × P × I → X, where I = [0, 1], such that:

(1) G(·, ·, 0) = g,
(2) G(·, ·, 1) : S × P → X is holomorphic along S,
(3) G(·, ·, t) = g on S × Q and on T × P for all t ∈ I,
(4) f ∘ G(·, ·, t) = f ∘ g on S × P for all t ∈ I.

Equivalently, Q ↪ P may be taken to be any cofibration between cofibrant topological spaces, such as the inclusion of a subcomplex in a CW-complex, and the existence of G can be replaced by the stronger statement that the inclusion into the space, with the compact-open topology, of continuous maps h : S × P → X with h = g on S × Q and on T × P and f ∘ h = f ∘ g on S × P of the subspace of maps that are holomorphic along S is acyclic, that is, a weak homotopy equivalence (see [14], §16). Taking P to be a point and Q empty defines BOPI for f. A complex manifold X is Oka if and only if the constant map from X to a point satisfies BOPI or, equivalently, POPI. For maps in general, it is not known whether BOPI implies POPI. The notion of a holomorphic submersion being subelliptic was defined by Forstnerič [2], generalising the concept of ellipticity due to Gromov [12]. Subellipticity is the weakest currently-known sufficient geometric condition for a holomorphic map to satisfy POPI (see Forstnerič’s recently-proved parametric Oka principle for liftings [6]) and for a complex manifold to be Oka. By the influential work of Grauert in [9] and [10], the primary examples of Oka manifolds, to which our results apply, are complex Lie groups and their homogeneous spaces, that is, complex manifolds on which a complex Lie group acts holomorphically and transitively. Among other known examples are ℂ^{n}\A, where A is an algebraic or a tame analytic subvariety of codimension at least 2, ℙ^{n} \ A, where A is a subvariety of codimension at least 2, Hopf manifolds, Hirzebruch surfaces, and the complement of a finite set in a complex torus of dimension at least 2 (see [3] and [5]).

2. Oka manifolds are homotopically elliptic

Our results are naturally formulated in the language of simplicial sets. Simplicial sets are combinatorial objects that have a homotopy theory equivalent to that of topological spaces, but tend to be more useful or at least more convenient than topological spaces for various homotopy-theoretic purposes. For an introduction to simplicial sets, we refer the reader to [8] or [16]. We denote by Δ the category of finite ordinals and order-preserving maps. The objects of Δ are the sets n = {0, 1, 2, …, n}, n ∈ ℕ, with the usual order, and a morphism θ : n → m is a map such that θ(i) ≤ θ(j) whenever 0 ≤ i ≤ j ≤ n. A cosimplicial object in a category C is a functor Δ → C. A simplicial object in C is a functor from the opposite category Δ^{op} to C. In particular, a simplicial set is a functor from Δ^{op} to the category Set of sets. The category of simplicial objects in C is denoted sC. A cosimplicial object A_{•} in C induces a functor h_{A_{•}} : C → sSet, X ↦ hom_{C}(A_{•}, X). We call the simplicial set hom_{C}(A_{•}, X) the homotopy type of X with respect to A_{•}. The standard n-simplex T_{n}, n ≥ 0, is the subset

T_{n} = {(t_{0}, …, t_{n}) ∈ ℝ^{n+1} : t_{0} + ⋯ + t_{n} = 1, t_{0}, …, t_{n} ≥ 0}

of ℝ^{n+1} with the subspace topology. An order-preserving map θ : n → m induces a continuous map θ_{∗} : T_{n} → T_{m} defined by the formula θ_{∗}(t_{0}, …, t_{n}) = (s_{0}, …, s_{m}), where

s_{i} = ∑_{j∈θ^{−1}(i)} t_{j}

(the sum is interpreted as zero if θ^{−1}(i) is empty). It is easy to check that this defines a cosimplicial object T_{•} in the category of topological spaces. The homotopy type sX = 𝒞 (T_{•}, X) of a topological space X with respect to T_{•} is the usual homotopy type of X. Here, for each n ≥ 0, 𝒞 (T_{n}, X) denotes the set of continuous maps T_{n} → X. The simplicial set sX is called the singular set of X. It is a fibrant simplicial set, that is, a Kan complex. The affine n-simplex A_{n}, n ≥ 0, is the affine subspace

A_{n} = {(t_{0}, …, t_{n}) ∈ ℂ^{n+1} : t_{0} + ⋯ + t_{n} = 1}

of ℂ^{n+1}, viewed as a complex manifold biholomorphic to ℂ^{n}. An orderpreserving map θ : n → m induces a holomorphic map θ_{∗} : A_{n} → A_{m} defined by the same formula as above, and we have a cosimplicial object A_{•} in the category of complex manifolds. We call the homotopy type eX = 𝒪(A_{•}, X) of a complex manifold X with respect to A_{•} the affine homotopy type of X. Here, for each n ≥ 0, 𝒪(A_{n}, X) denotes the set of holomorphic maps A_{n} → X. We also call the simplicial set eX the affine singular set of X. A holomorphic map A_{n} → X is determined by its restriction to T_{n} ⊂ A_{n}, so we have a monomorphism, that is, a cofibration eX ↪ sX. The following lemma comes from basic homotopy theory.

Lemma. For a complex manifold X, the following are equivalent.

(a) The affine singular set eX is fibrant and the cofibration eX ↪ sX is a weak equivalence of simplicial sets.

(b) The cofibration eX ↪ sX is the inclusion of a strong deformation retract.

Proof. (a) ⇒ (b) by [13], Prop. 7.6.11.

(b) ⇒ (a) by [13], Prop. 7.8.3, and since a retract of a fibrant object is fibrant. □

We say that X is homotopically elliptic if conditions (a) and (b) are satisfied. Then the usual homotopy type of X as a topological space is represented by the affine singular set eX of X. If X is connected and homotopically elliptic, then X is ℂ-connected, meaning that any two points in X can be joined by an entire curve. In fact, any finite subset of X lies in a holomorphic image of ℂ. On the other hand, if X is Brody hyperbolic, then eX is discrete.

Theorem 1. An Oka manifold is homotopically elliptic.

Proof. Let Z_{n} = {(z_{1}, …, z_{n}) ∈ ℂ^{n} : z_{j} = 0 for some j} be the union of the coordinate hyperplanes in ℂ^{n}, n ≥ 2. If X is an Oka manifold, every holomorphic map Z_{n} → X extends to a holomorphic map ℂ^{n} → X, but this is precisely what it means for eX to be fibrant. The homotopy groups π_{m}(K, ∗), m ≥ 1, of a Kan complex K with respect to a base point ∗ ∈ K_{0} may be simply described as follows:

π_{m}(K, ∗) = {a ∈ K_{m} : d_{j}a = ∗ for j = 0, …, m}/ ∼,

where d_{j} : K_{m} → K_{m−1} is the face map that in the case of sX and eX acts by precomposition by the map

δ_{j} : (t_{0}, …, t_{m−1}) ↦ (t_{0}, …, t_{j−1}, 0, t_{j}, …, t_{m−1}),

and ∼ is the equivalence relation with a ∼ b for a, b ∈ K_{m} with all faces ∗ if there is c ∈ K_{m+1} such that d_{j}c = a for some j, d_{j}c = b for another j, and d_{j}c = ∗ for the remaining values of j. Identifying vertices a, b ∈ K_{0} if there is c ∈ K_{1} with d_{0}c = a and d_{1}c = b (this is an equivalence relation) gives the set π_{0}(K) of path components of K. (See e.g. [1], Th. 2.4, or [17], Sec. 8.2—homotopy groups of non-fibrant simplicial sets are not so easily dealt with.) Since X is Oka, two points in the same path component of X can be joined by a holomorphic image of ℂ. Thus the inclusion eX ↪ sX induces a bijection π_{0}(eX) → π_{0}(sX). By induction over m we obtain continuous retractions ρ_{m} : A_{m} → T_{m}, m ≥ 0, such that ρ_{m+1} ∘ δ_{j} = δ_{j} ∘ ρ_{m} for j = 0, …, m, so ρ_{m} retracts each face of A_{m} onto the corresponding face of T_{m}. The continuous surjection σ_{m} : T_{m} × I → T_{m+1},

(t_{0}, …, t_{m}, s) ↦ (t_{0}(1 − s), t_{1}, …, t_{m}, t_{0}s), m ≥ 1, collapses each segment {x} × I, where x belongs to the face of T_{m} with t_{0} = 0, and makes no other identifications. Let m ≥ 1 and choose a base point ∗ ∈ X. To prove surjectivity of the induced map π_{m}(eX, ∗) → π_{m}(sX, ∗), we need to show that if a ∈ s_{m}X has all faces ∗, then there is b ∈ e_{m}X with all faces ∗ that is equivalent to a by some c ∈ s_{m+1}X. Now a_{0} = a ∘ ρ_{m} : A_{m} → X is continuous with all faces ∗, so since X is Oka, there is a continuous deformation a_{t}, t ∈ I, of a_{0}, such that a_{1} is holomorphic and a_{t} has all faces ∗ for all t ∈ I. The restriction to T_{m} × I of the deformation factors through σ_{m} by a map T_{m+1} → X, which is continuous since σ_{m} is a quotient map, and which is the desired c. To prove injectivity of the induced map π_{m}(eX, ∗) → π_{m}(sX, ∗), we need to show that if a, b ∈ e_{m}X with all faces ∗ are equivalent by c ∈ s_{m+1}X, say dc = (a, b, ∗, …, ∗), then a and b are also equivalent by some c′ ∈ e_{m+1}X. Continuously extend c to T_{m+1} ∪ W_{m+1}, where W_{m+1} = {(t_{0}, …, t_{m+1}) ∈ A_{m+1} : t_{j} = 0 for some j}, such that dc is still (a, b, ∗, …, ∗). Use the acyclic cofibration T_{m+1} ∪ W_{m+1} ↪ A_{m+1} to further extend c to a continuous map c : A_{m+1} → X. Since X is Oka, c may be deformed to c′ ∈ e_{m+1}X with dc′ = dc. □

The author has tried to directly construct a strong deformation retraction from sX onto eX, but without success. The proof shows that a complex manifold is homotopically elliptic if and only if it satisfies the interpolation property with respect to the Stein inclusions Z_{n} ↪ ℂ^{n}, n ≥ 2, and a weak version of BOPI with respect to the Stein inclusions W_{n} ↪ A_{n} ≅ ℂ^{n}, n ≥ 1.

3. Generalisations

Theorem 1 is a special case of a more general result. Let f : X → Y be a holomorphic map between complex manifolds and T ↪ S be a Stein inclusion. Let

\xymatrix{T \ar[r] \ar[d] & X \ar[d] \\ S \ar[r] & Y}

be a commuting square of holomorphic maps. Let L_{𝒪} be the space, with the compact-open topology, of holomorphic liftings in the square, and L_{𝒞} be the space of continuous liftings. Let eL_{𝒪} be the simplicial set whose n-simplices, n ≥ 0, are the holomorphic maps λ : S ×A_{n} → X such that λ(·, t) is a lifting in the square for every t ∈ A_{n}, and whose maps taking m-simplices to n-simplices are given by precomposing in the second variable by the holomorphic maps θ_{∗} : A_{n} → A_{m} described above. There are inclusions

eL_{𝒪} ↪^{i′} sL_{𝒪} ↪^{i′′} sL_{𝒞} .

If f satisfies POPI, then i′′ is a weak equivalence (see [14], §16). Also, the proof of the Theorem is easily generalised to show that if f satisfies BOPI, then eL_{𝒪} is fibrant and i′′ ∘ i′ is a weak equivalence. Thus, if f satisfies POPI, i′ is a weak equivalence of Kan complexes. Theorem 1 is the case when T is empty and S and Y are points. A less special case is when T is empty and Y is a point. Then liftings in the square are simply maps S → X, so we write e𝒪(S, X) for eL_{𝒪} and conclude that if X is Oka, then the inclusions e𝒪(S, X) ↪ s𝒪(S, X) ↪ s𝒞 (S, X) are weak equivalences of Kan complexes. Generalising this in a different direction, we can represent the the homotopy type of the space 𝒞 (M, X) of continuous maps from any smooth manifold M to an Oka manifold X by a simplicial set whose simplices are holomorphic maps into X. Namely, assuming as we may that M is real-analytic, by a wellknown result of Grauert [11], M can be real-analytically embedded into a Stein manifold S such that M is a strong deformation retract of S. Then, if X is Oka, the homotopy type of 𝒞 (M, X) is given by the Kan complex e𝒪(S, X). For ease of reference, we summarise the above as a theorem.

Theorem 2. Let X be an Oka manifold.

(1) For every Stein manifold S, the inclusions

e𝒪(S, X) ↪ s𝒪(S, X) ↪ s𝒞 (S, X)

are weak equivalences of Kan complexes.

(2) For every smooth manifold M, there is a Stein manifold S such that the homotopy type of 𝒞 (M, X) is given by the Kan complex e𝒪(S, X).

References

[1] E. B. Curtis. Simplicial homotopy theory. Adv. Math. 6 (1971) 107–209.

[2] F. Forstnerič. The Oka principle for sections of subelliptic submersions. Math. Z. 241 (2002) 527–551.

[3] F. Forstnerič. The homotopy principle in complex analysis: a survey. Explorations in complex and Riemannian geometry, 73–99, Contemp. Math., 332, Amer. Math. Soc., 2003.

[4] F. Forstnerič. Extending holomorphic mappings from subvarieties in Stein manifolds. Ann. Inst. Fourier (Grenoble) 55 (2005) 733–751.

[5] F. Forstnerič. Runge approximation on convex sets implies the Oka property. Ann. of Math. (2) 163 (2006) 689–707.

[6] F. Forstnerič. Invariance of the parametric Oka property. In Complex analysis: Several complex variables and connections with PDE theory and geometry, volume in honour of Linda P. Rothschild, Trends in Mathematics, Birkhäuser, to appear, arXiv:0901.4373.

[7] F. Forstnerič. Oka manifolds. C. R. Math. Acad. Sci. Paris 347 (2009) 1017–1020.

[8] P. G. Goerss and J. F. Jardine. Simplicial homotopy theory. Progress in Mathematics, 174. Birkhäuser Verlag, 1999.

[9] H. Grauert. Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. Math. Ann. 133 (1957) 450–472.

[10] H. Grauert. Analytische Faserungen über holomorph-vollständigen Räumen. Math. Ann. 135 (1958) 263–273.

[11] H. Grauert. On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math. (2) 68 (1958) 460–472.

[12] M. Gromov. Oka’s principle for holomorphic sections of elliptic bundles. J. Amer. Math. Soc. 2 (1989) 851–897.

[13] P. S. Hirschhorn. Model categories and their localizations. Mathematical Surveys and Monographs, 99. Amer. Math. Soc., 2003.

[14] F. Lárusson. Model structures and the Oka Principle. J. Pure Appl. Algebra 192 (2004) 203–223.

[15] F. Lárusson. Applications of a parametric Oka principle for liftings. In Complex analysis: Several complex variables and connections with PDE theory and geometry, volume in honour of Linda P. Rothschild, Trends in Mathematics, Birkhäuser, to appear, arXiv:0901.4388.

[16] J. P. May. Simplicial objects in algebraic topology. Van Nostrand Mathematical Studies, 11. D. Van Nostrand Co., 1967. Chicago Lectures in Mathematics. University of Chicago Press, 1992.

[17] P. Selick. Introduction to homotopy theory. Fields Institute Monographs, 9. Amer. Math. Soc., 1997.

Finnur Lárusson

School of Mathematical Sciences University of Adelaide Adelaide SA 5005 Australia. finnur.larusson@adelaide.edu.au 698

Documenta Mathematica 14 (2009)