EXCISION FOR SIMPLICIAL SHEAVES ON THE STEIN SITE AND GROMOV'S OKA PRINCIPLE Finnur Larusson University of Western Ontario December 2000 Abstract. A complex manifold X satisfies the Oka-Grauert property if the i* *nclusion O(S; X) ,! 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-G* *rauert 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 categor* *y of simplicial sheaves on the site of Stein manifolds. Our main result is that the Oka-Gr* *auert property is equivalent to X representing a finite homotopy sheaf on the Stein site. Th* *is expresses the Oka-Grauert property in purely holomorphic terms, without reference to con* *tinuous maps. 1. Introduction The purpose of this paper is to investigate Gromov's Oka principle using abstra* *ct homo- topy theory and to recast it in intrinsic, holomorphic terms, and thereby to in* *troduce some of the concepts and methods of homotopical algebra into complex geometry. The Oka principle is a vague maxim, supported by many results. It may be phr* *ased by saying that on a Stein manifold (complex submanifold of Euclidean space), an* *alytic problems of a cohomological nature have only topological obstructions. It has a* * long and venerable history, starting with the 1939 result of Oka stating, in modern lang* *uage, that a holomorphic line bundle on a domain of holomorphy is trivial if it is topologic* *ally trivial. Deep generalizations to vector bundles and certain other fiber bundles were obt* *ained by Grauert in the late 1950s. Another manifestation of the Oka principle is the 19* *75 result _____________ 2000 Mathematics Subject Classification. Primary: 32Q28; secondary: 18F10, 1* *8F20, 18G30, 18G55, 32E10, 32H02, 55U35. The author was supported in part by the Natural Sciences and Engineering Res* *earch Council of Canada. Typeset by AM S-* *TEX 1 of Cornalba and Griffiths that every rational cohomology class of degree 2k on * *a Stein manifold is a rational multiple of the fundamental class of a k-codimensional a* *nalytic subvariety. For a survey, see [Lei]. A major development appeared in Gromov's 1989 paper [Gro]. He discovered th* *at if a complex manifold X has a geometric structure called a spray, then the incl* *usion O(S; X) ,! 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* *. We refer to this as the Oka-Grauert property. It implies in particular that every * *continuous map from a Stein manifold to X can be deformed to a holomorphic map. A spray on X consists of holomorphic maps Cm ! X, 0 7! x, one for each x 2 X, submersive * *at 0, and varying holomorphically with x. For a detailed proof of Gromov's theore* *m and a thorough introduction, see [FP1]. A more general version for sections of subm* *ersions is contained in [FP2]. Gromov's Oka principle has been applied to the famous pr* *oblem of embedding Stein manifolds into Euclidean spaces of the smallest possible dim* *ension: Sch"urmann has used it to prove Forster's conjecture in higher dimensions [Sch]* *. Other applications may be found in [For1], [For2], and [FP3]. The Oka-Grauert property certainly has a homotopy-theoretic flavour. Our goa* *l is to turn this impression into a precise statement in an abstract homotopy-theoretic* * setting. At the same time, we will express the Oka-Grauert property in purely holomorphi* *c terms, without reference to continuous maps. Abstract homotopy theory, also known as homotopical algebra, is due to Quill* *en [DS, GJ, Hir, Hov, Qui]. Its fundamental notion is that of a model category: a cat* *egory satisfying certain axioms that allow us to develop an analogue of ordinary homo* *topy theory. There has been much activity in recent years in both the theory and app* *lications of homotopical algebra, perhaps the most notable example being Voevodsky's prog* *ramme to prove the Milnor conjecture [Vo1, Vo2]. As far as I know, the present paper * *is the first attempt to introduce homotopical algebra into analytic geometry. The first step, just as for schemes, is to embed the category of complex man* *ifolds into the model category of simplicial objects in a topos by a Yoneda embedding * *of some sort, where we can then do homotopy theory with them. Here, this is done in Se* *ction 4. Whereas in arithmetic geometry the focus is on generalized cohomology theori* *es and ultimately motives, on the analytic side it seems of more immediate interest to* * try to do algebraic topology with complex manifolds and holomorphic maps instead of topol* *ogical spaces and continuous maps, and then our attention is immediately drawn to Grom* *ov's Oka principle. For the purpose of studying the Oka principle, we associate to a* * complex manifold X the simplicial sheaf sO(.; X) on the site of all Stein manifolds. He* *re, spaces of holomorphic maps are given the compact-open topology; it is for technical re* *asons that we turn them into simplicial sets by applying the singular functor s. Using the* * compact- open topology allows us to work at a relatively simple technical level: we neit* *her have to localize nor stabilize to get something interesting. The main result of this paper is Theorem 2.1, later rephrased as Theorem 4.3* *. It 2 states that the Oka-Grauert property is equivalent to the simplicial sheaf sO(.* *; X) being a finite homotopy sheaf on the Stein site. This homotopy-theoretic property is * *also called excision. It gives rise to Mayer-Vietoris sequences of homotopy groups and is * *familiar from topology and appears nowadays in algebraic geometry: see e.g. [MV, x3.1.2]* *. I have tried to make the proof of Theorem 2.1 as understandable as possible to those u* *nfamiliar with homotopy theory. However, the very definition of excision requires the not* *ion of a homotopy limit (a deformation invariant approximate limit), for which I refer t* *he reader to [GJ, VIII.2] and [Hir, Ch. 19]. The proof uses the main result of Section 3,* * Theorem 3.4, whose crucial ingredient is a classical theorem of Brown and Gersten [BG, * *Thm. 1], foundational in the homotopy theory of simplicial sheaves. Section 3 is pure ho* *motopy theory and constitutes the bulk of the paper. The proof of Theorem 2.1 also use* *s Siu's theorem on the existence of Stein neighbourhoods of Stein subvarieties, and Whi* *tney's lemma on decomposing an open subset in Euclidean space into a union of cubes wi* *th special properties. Those familiar with the homotopy theory of simplicial sheaves would now ask * *if the Oka-Grauert property is actually equivalent to descent, making Gromov's Oka pri* *nci- ple analogous to such results as Brown-Gersten, Nisnevich, and Thomason descent* * in algebraic geometry [Mit]. Descent is a stronger property than excision: roughly* * speak- ing, it signifies weak homotopy equivalence with a fibrant simplicial sheaf, wh* *ich may be thought of as a resolution, analogous to a projective resolution of a module* * or a CW approximation of a topological space. I do not know the answer: the finiteness * *properties that make descent possible in algebra do not hold in analysis. There are indic* *ations, however, that the answer is affirmative in a somewhat different model category,* * which might provide an appropriate framework for analytic homotopy theory. I hope to * *address this in a future paper. In the meantime, Section 5 contains a partial descent t* *heorem of sorts for quasi-projective manifolds. A word about terminology: we take manifolds to be second countable by defini* *tion, but not necessarily connected. Acknowledgements. I am grateful for discussions with Paul Balmer, Dan Christen* *sen, Gaunce Lewis, Sergei Yagunov, and especially Rick Jardine, who has generously a* *nd patiently helped me understand homotopy theory. 2. The Oka-Grauert property is equivalent to excision We say that a complex manifold X (second countable but not necessarily connecte* *d) has the Oka-Grauert property if the inclusion map O(S; X) ,! C(S; X) is a weak equivalence for all Stein manifolds S, where the spaces of holomorphi* *c and continuous maps from S to X carry the compact-open topology. This means that t* *he inclusion induces isomorphisms of all homotopy groups for all base points, as w* *ell as a 3 bijection of path components. In particular, surjectivity on path components me* *ans that every continuous map S ! X is homotopic to a holomorphic map. Note that requiri* *ng S to be connected results in an equivalent condition. We say that X satisfies excision (or finite excision) if whenever {U1; : :;:* *Um } is a finite cover of a Stein manifold S by Stein open subsets, O(S; X) is not only the limi* *t but also the homotopy limit of the diagram whose objects are the spaces O(Ui1\ . .\.Uik;* * X) for 1 i1 < . .<.ik m and k = 1; : :;:m, and whose arrows are the restriction maps* *. This diagram forms an m-dimensional cube with one vertex missing; the limit or homot* *opy limit provides the missing vertex. For m = 2, this simply means that the square O(S; X) ----! O(U1; X) ?? ? y ?y O(U2; X) ----! O(U1 \ U2; X) is not only a pullback but also a homotopy pullback. Let us make clear what we mean by a homotopy limit. We view homotopy limits * *as determined only up to weak equivalence. We say that a topological space Y with * *a map to a diagram X of spaces over an index category J is the homotopy limit of X (o* *r more properly that the diagram Y ! X is a homotopy limit) if and only if the singul* *ar set sY of Y is the homotopy limit of the diagram sX of simplicial sets. This in tur* *n means that if A is a fibrant model for sX in the category of diagrams of simplicial s* *ets over J with the pointwise cofibration structure, then the natural map from sY to the * *limit of A is a weak equivalence [GJ, VIII.2.11]. It is my understanding that this defi* *nition is standard (up to weak equivalence): it agrees up to weak equivalence with the de* *finitions of Bousfield-Kan and Hirschhorn [Hir, Ch. 19]. The homotopy limit of a diagram* * of spaces or simplicial sets may also be described somewhat explicitly as the func* *tion space or complex of morphisms to the diagram from a certain diagram associated to the* * index category [GJ, VIII.2.3; Hir, 19.1.10]. We will make frequent use of the basic f* *act that if one of two weakly equivalent diagrams of spaces or fibrant simplicial sets is a* * homotopy limit, then so is the other one. A deep and important theorem of Gromov's, proved in all detail by Forstneric* * and Prezelj, states that if X has a spray, then X has the Oka-Grauert property [Gro* *, FP1]. We refer to this result as Gromov's Oka principle. Our main result is that the* * Oka- Grauert property is equivalent to excision. 2.1. Theorem. A complex manifold has the Oka-Grauert property if and only if it satisfies excision. Proof. Let X be a complex manifold with the Oka-Grauert property. Consider a fi* *nite cover of a Stein manifold S by Stein open subsets U1; : :;:Um . We have two m-d* *imensional cube diagrams of spaces, one with objects O(Ui1\. .\.Uik; X), and the other wit* *h objects 4 C(Ui1\ . .\.Uik; X) for 1 i1 < . .<.ik m. We have a morphism of inclusions fr* *om the first diagram to the second one, consisting of weak equivalences by assumpt* *ion (here we need to know that the intersection of Stein open sets is Stein; see Lemma 4.* *1). Hence, the induced map between the homotopy limits of the two diagrams is a weak equiv* *alence [GJ, VIII.2.2]. By Theorem 3.4 below, the homotopy limit of the second diagram* * is C(S; X), which by assumption is weakly equivalent to O(S; X). Conversely, assume X is a complex manifold satisfying excision and let S be * *a Stein manifold. We first reduce our problem to the case when S is a domain in Euclide* *an space. As noted above, we may take S to be connected, so S embeds into some Euclidean * *space. Then, by a theorem of Siu [Siu, Cor. 1], there is a connected Stein neighbourho* *od V of S and a holomorphic retraction ae : V ! S. Let : S ,! V be the inclusion, so ae * *O = idS, and we have a diagram _ae*_// O(S; X) oo___O(V; X) * | OE|| | fflffl|ae*//_ fflffl| C(S; X)oo____ C(V; X) * where OE and are the inclusions. Now suppose is a weak equivalence. Since ** *Oae* = id, ae* induces monomorphisms and * induces epimorphisms on all homotopy groups. He* *nce, ae* O OE = O ae* induces monomorphisms on all homotopy groups so OE does too,* * and OE O * = * O induces epimorphisms on all homotopy groups so OE does too, and * *OE is a weak equivalence. To complete the proof we need to show that O(V; X) ,! C(V; X) is a weak equi* *valence when V is a Stein domain in Euclidean space. We want to express V as a finite* * union of open subsets all of whose connected components are convex. This can surely b* *e done in many ways. We shall refer to Whitney's classical lemma on decomposing an ope* *n set in Euclidean space into a union of cubes with special properties [Ste, VI.1]. W* *e get that V = U1 [ . .[.Um , where each Ui is a disjoint union of open cubes with sides p* *arallel to the coordinate axes. (This is not stated explicitly in [Ste], but may easily* * be obtained from there.) Then every intersection U = Ui1\ . .\.Uik is a disjoint union of * *open boxes and hence Stein. Each box is holomorphically contractible in the sense t* *hat the identity map can be joined to a constant map by a continuous family of holomorp* *hic maps, so the inclusion O(U; X) ,! C(U; X) is clearly a weak equivalence. Now we* * look at two m-dimensional cube diagrams of spaces, one with objects O(Ui1\ . .\.Uik;* * X), and the other with objects C(Ui1\ . .\.Uik; X) for 1 i1 < . .<.ik m. As above, we see that the induced map between the homotopy limits of the two diagrams is * *a weak equivalence. Since X satisfies excision, and by Theorem 3.4, this map is the i* *nclusion O(V; X) ,! C(V; X) (at least up to weak equivalence), and the proof is complete. The proof shows that the excision property for a complex manifold X is equiv* *alent to 5 O(S; X) ! O(B; X) being a homotopy limit for every Stein basis B for a binoethe* *rian subtopology on a Stein manifold S, viewed as a subdiagram of the site of S. The proof also shows that the Oka-Grauert property for a complex manifold X * *is equivalent to the inclusion O(V; X) ,! C(V; X) being a weak equivalence for all* * domains of holomorphy V in Cn for all n 1. To some extent it is therefore a matter o* *f taste whether one chooses to work with Stein manifolds or domains of holomorphy in Eu* *clidean space in the present context. If X has the Oka-Grauert property and S is a Stein manifold, it is natural t* *o ask whether the weak equivalence O(S; X) ,! C(S; X) is actually a homotopy equivale* *nce. When S is algebraic, a little topology shows that the inclusion has a right hom* *otopy inverse, so there is a continuous way of associating to each continuous map S !* * X a holomorphic map homotopic to it. Note that we do not assert that the homotopy i* *nverse fixes holomorphic maps. 2.2. Theorem. Let S be an affine algebraic manifold, i.e., a Stein manifold bih* *olomor- phic to an algebraic submanifold of Euclidean space. If X is a complex manifold* * and the inclusion O(S; X) ,! C(S; X) is a weak equivalence, then it has a right homotop* *y inverse. Proof. Being a smooth manifold, X has a countable triangulation [Mun, 10.6], so* * X is homeomorphic to a countable CW complex. Also, S is homotopy equivalent to a fin* *ite CW complex K [Mil, Lemma A.3], and C(S; X) is homotopy equivalent to C(K; X), which has the homotopy type of a (countable) CW complex [FrP, 5.2.5], so C(S; X* *) has the homotopy type of a CW complex. Hence, the natural map a : |sO(S; X)| ! |sC(S; X* *)| ! C(S; X), which is a weak equivalence by assumption, has a homotopy inverse b. L* *et c be the natural map |sO(S; X)| ! O(S; X). By adjunction, a = ic, where i is the inc* *lusion O(S; X) ,! C(S; X). Now i(cb) = ab is homotopic to the identity on C(S; X), so * *cb is a right homotopy inverse for i. 3. Excision and Brown-Gersten descent The main purpose of this section is to establish the excision property of sC(.;* * X) used in the previous section (Theorem 3.4). One feels that it should be possible to ve* *rify this directly using the explicit description of the homotopy limit given in [GJ, VII* *I.2.3] and [Hir, 19.1.10], but rather than attempt this, we give a proof based on Brown-Ge* *rsten descent (Theorem 3.1). Brown-Gersten descent is surely well known among experts* *, but in the absence of a good reference, we have provided a detailed proof. Theorems* * 3.3 and 3.4 are new as far as I know. We start with a brief review of the basic notion* *s of the homotopy theory of simplicial presheaves. Let S be a small Grothendieck site. A simplicial presheaf on S is a contrav* *ariant functor from S to the category sSet of simplicial sets. There is a standard mod* *el struc- ture on the category s PreS of simplicial presheaves on S in which the cofibrat* *ions are monomorphisms, i.e., pointwise injections (where pointwise means at every objec* *t of the 6 site), and a weak equivalence is a map that induces isomorphisms of all homotop* *y sheaves [Jar1, p. 59]. If S has enough points, e.g. if it is the site of a topological * *space, then this is equivalent to the map inducing weak equivalences of all stalks. A weak equivale* *nce is still a weak equivalence with respect to any finer topology. Fibrations are defined b* *y a right lifting property. There is an induced model structure on the full subcategory * *s ShvS of simplicial sheaves on S. More is true: both categories are proper, simplic* *ial model categories [Jar1, Jar2]. There is another model structure on s PreS given by th* *e trivial topology on S, in which the only covers are those consisting of a single isomor* *phism. In this structure, the weak equivalences are the pointwise weak equivalences. The * *words fine and finely shall refer to the former model structure, and the words coarse and * *coarsely to the latter. A morphism of simplicial presheaves which is a weak equivalence * *will be referred to as acyclic. (We are trying not to overuse the word trivial.) A co* *arse weak equivalence is a fine weak equivalence; a fine fibration is a coarse fibration. The concept of a fibrant object is fundamental in homotopy theory. We will n* *ow define several important weaker notions and briefly describe their relationships. We say that a simplicial presheaf G on S satisfies descent if any fine weak * *equivalence from G to a finely fibrant simplicial presheaf on S is coarsely acyclic. Equiva* *lently (using the Whitehead Theorem that a weak equivalence between bifibrant objects is a ho* *motopy equivalence), a finely fibrant model for G is also a coarsely fibrant model for* * G. This notion is invariant under coarse weak equivalences. A finely fibrant simplicia* *l presheaf satisfies descent. It may be shown that a coarsely fibrant simplicial presheaf G on S is both p* *ointwise fibrant and flabby (or flasque), which means that the restriction map G(U) ! G(* *V ) is a fibration for every monomorphism V ! U in S. Now let X be a topological space. We say that a pointwise fibrant simplicial* * presheaf G on X, i.e., a presheaf of Kan complexes, satisfies excision (or two-set excis* *ion) if G(?) is contractible (this is true if G is a sheaf) and whenever U and V are open i* *n X, the square G(U [ V ) ----! G(U) ?? ? y ?y G(V ) ----! G(U \ V ) is a homotopy pullback. This notion was introduced by Brown and Gersten [BG, x* *2], who used the term pseudo-flasque. It is clearly invariant under coarse weak equ* *ivalences. If G is a flabby simplicial sheaf (so G is in particular pointwise fibrant), th* *en G satisfies excision: the diagram G(U) ! G(U \ V ) G(V ) is fibrant in its diagram catego* *ry, so the ordinary pullback G(U [V ) is the homotopy pullback [GJ, VI.1.8]. If G is a* * pointwise fibrant simplicial presheaf satisfying descent, consider the fine weak equivale* *nce from G to its sheafification aG, and the fine weak equivalence from aG to a finely fib* *rant model F in s ShvX. Then F is also finely fibrant in s PreX, so F is flabby and satisfie* *s excision, 7 and since F and G are finely and hence coarsely equivalent, G does too. Under a* * strong finiteness condition on X, the converse holds: this is Brown-Gersten descent. * * We say that a topological space is binoetherian if both the open sets and the irreduci* *ble closed sets satisfy the ascending chain condition. An important example is a space wit* *h a finite topology. 3.1. Theorem (Brown-Gersten descent). For pointwise fibrant simplicial presheav* *es on a binoetherian space, excision is equivalent to descent. Proof. Let G be a pointwise fibrant simplicial presheaf on a binoetherian space* *, and suppose that G satisfies excision. Let G ! F be a fine weak equivalence from G* * to a finely fibrant simplicial presheaf F . Then F satisfies descent and hence exci* *sion. The theorem now follows from the next result. 3.2. Proposition. Let F and G be pointwise fibrant simplicial presheaves sati* *sfying excision on a binoetherian space X. Then a fine weak equivalence G ! F is coar* *sely acyclic. Our argument is an adaptation and explication of Morel and Voevodsky's proof* * of unstable Nisnevich descent in [MV, x3.1.2]. Proof. Let F 0, G0 be coarsely fibrant models in s PreX for F , G respectively.* * Factor the induced map G0 ! F 0as a coarse weak equivalence G0 ! G00followed by a coar* *se fibration G00! F 0. Then G00is coarsely fibrant and it suffices to show that t* *he map G00! F 0is coarsely acyclic. By replacing F , G by F 0, G00, we may assume that* * F and G are coarsely fibrant and hence flabby and pointwise fibrant, and that the fin* *e weak equivalence G ! F is a coarse fibration and hence a pointwise fibration. It suffices to show that for any open set U in X and any vertex x in F (U), * *the fibre of the fibration G(U) ! F (U) over x is contractible (in particular nonempty): * * then G(U) ! F (U) is a weak equivalence (if F (U) is empty, then so is G(U) and this* * is still true). Note that U is binoetherian in the subspace topology, so we may assume * *that U = X. Fix a vertex x in F (X) and consider the simplicial presheaf K on X that associates to an open set V in X the fibre of G(V ) ! F (V ) over the image of * *x in F (V ). Note that K is a pullback of a diagram G ! F * of simplicial presheaves on X,* * where * denotes the final simplicial presheaf. Hence, K is coarsely fibrant, and ther* *efore flabby and pointwise fibrant. We need to show that K is pointwise contractible. Since K is stalkwise contr* *actible (pullbacks commute with filtered colimits, so taking fibres commutes with takin* *g stalks), this follows from [BG, Thm. 1] along with the remark at the end of [BG, x2], on* *ce we know that K satisfies excision. To complete the proof, let us verify this. First of all, since F (?) and G(?) are contractible, so is the fibre K(?). * *Now let U and V be open in X. Since K is flabby and pointwise fibrant, the homotopy pul* *lback of K(U) ! K(U \ V ) K(V ) is the ordinary pullback, so we need to show that t* *he 8 natural map K(U [ V ) ! K(U) xK(U\V )K(V ) is a weak equivalence. By the coglui* *ng lemma [GJ, II.8.13] applied to the natural map from the pullback square K(U [ V ) ----! G(U [ V ) ?? ? y ?y * ----! F (U [ V ) to the pullback square K(U) xK(U\V )K(V ) ----! G(U) xG(U\V )G(V ) ?? ? y p?y * ----! F (U) xF(U\V )F (V ) we would be done if p was a fibration. Since this is not to be expected, we nee* *d to replace p by a fibration in a reasonable way. We shall work in the category of squares of simplicial sets of the type 4 ----! 3 ?? ? y ?y 2 ----! 1 This category carries the pointwise cofibration and the pointwise fibration sim* *plicial model structures [GJ, p. 403]. In both of them, the weak equivalences are the p* *ointwise weak equivalences. We shall write QF for the square F (U [ V )----! F (V ) ?? ? y ?y F (U) ----! F (U \ V ) and similarly for G and K. In the pointwise cofibration structure, factor QG ! * *QF as a weak equivalence QG ! Q followed by a fibration Q ! QF . Using the fibration Q * *! QF , one can show that Q is pointwise fibrant and the maps Q2; Q3 ! Q1 are fibration* *s. Since QG is a homotopy pullback, so is Q, and the natural map Q4 ! Q2 xQ1 Q3 is a weak equivalence. The cogluing lemma applied to the natural map from the pullback square QK ----! QG ?? ? y ?y * ----! QF 9 to the pullback square P - ---! Q ?? ? y ?y * - ---! QF with respect to the pointwise fibration structure (which is right proper) shows* * that QK ! P is a weak equivalence (we invoke the fact that a fibration in the pointwise c* *ofibration structure is a pointwise fibration). Also, P is fibrant in the pointwise cofibr* *ation structure. Hence, it suffices to show that the square P is a homotopy pullback, which is t* *he case if and only if the natural map P4 ! P2 xP1 P3 is a weak equivalence. By the cogluing lemma applied to the natural map from the pullback square P4 ----! Q4 ?? ? y ?y * ----! F (U [ V ) to the pullback square P2 xP1 P3 ----! Q2 xQ1 Q3 ?? ? y p?y * ----! F (U) xF(U\V )F (V ) it suffices to show that the map p is a fibration. Therefore, to complete the * *proof, we need to show that if R ! S is a fibration of squares in the pointwise cofibrati* *on structure, then R2xR1 R3 ! S2xS1S3 is a fibration of simplicial sets. (Having a pointwise * *fibration is not enough.) A square A ----! R2 xR1 R3 ?? ? y ?y B ----! S2 xS1 S3 is the same thing as a square "A----! R ?? ? y ?y B" ----! S of squares, where "Ais the square ? _____//A | || | || fflffl| || A _______A 10 and B" is defined similarly. A map A ! B is an acyclic cofibration if and only* * if the induced map A"! B" is a pointwise acyclic cofibration, and then a lifting in th* *e latter square gives a lifting in the former. Using Brown-Gersten descent, we can now strengthen the excision condition. 3.3. Theorem. Let G be a pointwise fibrant simplicial presheaf satisfying excis* *ion on a topological space X. Let B be a basis for a binoetherian subtopology on an open* * set U in X, viewed as a subdiagram of the site of X. Then G(U) is the homotopy limit of * *G|B. RecallSthat a collection B of open subsets in a topological space is a basis* * for a subtopol- ogy on B if whenever U; V 2 B and p 2 U \ V , there is W 2 B with p 2 W U \ * *V . We do not assume that a basis is closed under intersections. The definition of * *excision refers to the case when B consists of two open sets in X and their intersection. Proof. We may assume that X = U is binoetherian and B is a basis for the topolo* *gy on X. By Brown-Gersten descent, G satisfies descent, so there is a coarse weak equ* *ivalence from G to a finely fibrant simplicialosheafpF on X. Let us show that F |B is f* *ibrant in the diagram category sSet B withotheppointwise cofibration structure. Let A ! * *B be an acyclic cofibration in sSet B and A ! F |B be a morphism. Now A and B yield* *etale spaces over X with sheaves of sections "Aand "Brespectively, such that the indu* *ced map A" ! "Bis a finely acyclic cofibration in s PreX. Of course the diagram F |B yi* *elds the sheaf F itself in this way, so the map A ! F |B factors through A"|B. Since F i* *s finely fibrant, A" ! F factors through B", so A !oFp|B factors through B. This shows * *that F |B is a fibrant model for G|B in sSet B . Hence, the homotopy limit of G|B is* * weakly equivalent to the limit of F |B, which is F (X) since F is a sheaf, and F (X) i* *s weakly equivalent to G(X). The following theorem gives the excision property of C(.; X) used in the pre* *vious section. We shall make brief use the category Space of compactly generated weak* * Haus- dorff spaces [FrP, A.1; May, Ch. 5], "the category of spaces in which algebraic* * topologists customarily work" [May, p. 37]. We denote the internal function complex in sSe* *t by Hom sSet(.; .) as in [GJ, I.5]. 3.4. Theorem. Let X be a smooth manifold and Y be a compactly generated weak Hausdorff space. If B is a basis for a binoetherian subtopology on an open set * *U in X, viewed as a subdiagram of the site of X, then C(U; Y ) is the homotopy limit of* * C(.; Y )|B. Here, smooth means at least once continuously differentiable. For this resul* *t, the most important consequence of X being a smooth manifold is that every open subset of* * X is cofibrant since it has a triangulation [Mun, 10.6]. Surely, the class of space* *s with this property is much larger than the class of smooth manifolds, but I am not aware * *of any description of it. We also need every open subset of X to be normal. 11 Proof. By Theorem 3.3, we need to verify that the simplicial sheaf sC(.; Y ) on* * X satisfies excision. We will show that the simplicial presheaf F = Hom sSet(s.; sY ) on X* * is point- wise fibrant and satisfies excision and that F is coarsely weakly equivalent to* * sC(.; Y ). Note first that F (?) is the singular set of a point, so F (?) is contractible. Consider the functor H = Hom sSet(.; sY ) from sSet opto sSet . By Quillen'* *s Axiom SM7 for a simplicial model category [GJ, II.3.1], if A ! B is a cofibration in * *sSet , then the induced map H(B) ! H(A) is a fibration. Hence, F is flabby and pointwise fi* *brant. If we knew that F was a sheaf, the proof that F satisfies excision would end he* *re, but we do not. The internal function space in Space is kC(.; .), where k is the k-ific* *ation functor from the category of topological spaces to Space and C(.; .) carries the compa* *ct-open topology. By adjunction (at the level of simplicial categories), H = skC(| . |* *; Y ) [GJ, II.3.14], but sk = s, so H = sC(| . |; Y ). If A ! B is a weak equivalence of s* *implicial sets, then the weak equivalence |A| ! |B| has a homotopy inverse, which induces a hom* *otopy inverse for kC(|B|; Y ) ! kC(|A|; Y ), so H(B) ! H(A) is a weak equivalence. F* *inally, since H has a left adjoint [GJ, II.2] (this is one of the defining properties o* *f a simplicial category), H preserves limits, i.e., takes colimits in sSet to limits in sSet . Since H preserves limits and weak equivalences and turns cofibrations into f* *ibrations, H turns homotopy pushouts in sSet , i.e., homotopy pullbacks in sSet op, into h* *omotopy pullbacks. Also, the singular functor s turns homotopy pushouts of cofibrant sp* *aces into homotopy pushouts in sSet . To prove that F satisfies excision, it therefore s* *uffices to show that if U and V are open subsets of X, then the square U \ V ----! U ?? ? y ?y V ----! U [ V is a homotopy pushout. To calculate the homotopy pushout of U U \ V ! V , we factor U \ V ,! U through its mapping cylinder M into a cofibration followed by* * a weak equivalence [May, 6.3], and take the ordinary pushout of the diagram M U \ V * *! V . The homotopy pushout P turns out to be the product (U \ V ) x [0; 1] with U glu* *ed to (U \V )x{0}, and V to (U \V )x{1}. We need to verify that the projection P ! U * *[V is a weak equivalence. It is easy to see that a section of the projection is a hom* *otopy inverse for it, and finding a section is tantamount to finding a continuous function U * *[ V ! [0; 1] equal to 0 on U \ V and 1 on V \ U. Since U [ V is normal, such a function is p* *rovided by the Urysohn lemma. Finally, since every open subset U of X is cofibrant, the weak equivalence |* *sU| ! U has a homotopy inverse, so the induced map sC(U; Y ) ! sC(|sU|; Y ) = F (U) is * *a weak equivalence. Since F satisfies excision, so does sC(.; Y ). 12 4. Complex manifolds as simplicial sheaves on the Stein site Let M be the category of complex manifolds (second countable but not necessaril* *y con- nected) and holomorphic maps. In this section, we shall embed M into a model ca* *tegory, suitable for a homotopy-theoretic interpretation of the Oka-Grauert property. Let S be the category of Stein manifolds and holomorphic maps. This is a sm* *all category (or at least equivalent to one), since a connected Stein manifold can * *be embedded into Euclidean space. We view S as a site with the "usual" topology, in which a* * cover of a Stein manifold S consists of a family of isomorphisms onto Stein open subs* *ets of S which cover S. We only need to verify that covers can be pulled back to covers* * [MM, III.2]. This is implied by the following lemma. 4.1. Lemma. Let f : X ! Y be a holomorphic map between complex manifolds. If X is Stein and V is a Stein open subset of Y , then the preimage f-1 (V ) is Stei* *n. Note that the lemma shows that the intersection of finitely many Stein open * *subsets of a complex manifold is Stein. (The ambient manifold need not be Stein.) Namely, * *for two subsets U and V , the intersection is the preimage of V under the inclusion U ,* *! U [ V . For more sets, iterate this. Proof. Holomorphic functions separate points on U = f-1 (V ) since they do on X* *. To show that U is holomorphically convex, we follow [H"or, 2.5.14]. Let K U be co* *mpact. We need to show that the holomorphic hull K^U of K in U is compact. Since K^X * *is compact and contains K^U, it suffices to show that K^U is closed in U. Now f(K)* * V is compact, so [f(K)V is compact. If h 2 O(V ) and x 2 ^KU, then |h(f(x))| kh O f* *kK = khkf(K), so f(x) 2 [f(K)V. Hence, f maps the closure of K^U into [f(K)V V , so* * the closure is in U. We shall refer to this topology on S as the fine topology and to the trivial* * topology, in which a cover consists of a single isomorphism, as the coarse topology. Not* *e that a point, denoted p, is the final object in S, and the empty manifold ? is the ini* *tial object. Let p be a point in a Stein manifold S. If F is a presheaf on S, we define * *the stalk Fp of F at p to be the filtered colimit of the sets of sections F (U), where U * *is a Stein neighbourhood of p in S. By restricting to the small site of each Stein manifol* *d, we see that the family of stalk functors .p : Shv S ! Set , F 7! Fp, p 2 S 2 S, is fai* *thful, meaning that maps of sheaves are equal if they induce the same maps on all stal* *ks. Since homotopy groups respect filtered colimits of simplicial sets, we see that a map* * f : F ! G of simplicial presheaves on S is a fine weak equivalence (in the sense of Jardi* *ne) if and only if the induced map of stalks fp : Fp ! Gp is a weak equivalence for all p * *2 S 2 S. Let us remark that the stalk functors just defined really are stalks (or poi* *nts of S) in the sense of topos theory: they have right adjoints and preserve finite limits * *as functors Shv S ! Set (so in particular, S has enough points). First of all, since limits* * of sheaves are taken pointwise and finite limits commute with filtered colimits in Set [Ma* *c, IX.2], 13 the stalk functor .p preserves finite limits. It also preserves colimits. A col* *imit of sheaves is the sheafification of the pointwise colimit. In S, sheafification commutes * *with the restriction to the small site of any Stein manifold, because all covers can be * *realized in the manifold. Hence, preservation of colimits can be reduced to the case of a singl* *e manifold, where it holds by the standard result on topological spaces (the Stein open sub* *sets form a basis for the usual topology). It is easily verified that the functor S ! Set* * obtained by restricting .p to representable sheaves is both filtering and continuous, so* * its Kan extension Shv S ! Set is a point in S [MM, VII.5,6], but since .p preserves col* *imits, it is its own Kan extension. The right adjoint of .p can in fact be described explici* *tly [MM, VII.5]: it takes a set A to the "skyscraper sheaf" X 7! hom Set(O(.; X)p; A) on* * S (which does not really look like a skyscraper at all). We will embed M into s ShvS. First note that by the Yoneda lemma [Mac, III.* *2], there is a full embedding of M into the category PreM of presheaves of sets on * *M, given by X 7! O(.; X). It is easy to prove directly that this is still true for the s* *maller site S. 4.2. Proposition. The Yoneda functor M ! PreS is a full embedding. Proof. We need to show that the functor is both faithful and full, i.e., that i* *t induces bijections on all sets of morphisms. Let f; g : X ! Y be maps in M such that f** * = g* : O(.; X) ! O(.; Y ). Let be the map p ! X with image {p} for p 2 X. Since f O =* * g O, we have f(p) = g(p) and f = g. As for fullness, let ff : O(.; X) ! O(.; Y ) be a map in Pre S. Now ff : O(* *p; X) ! O(p; Y ) gives a map f : X ! Y . For a Stein manifold S and p 2 S we have a dia* *gram O(S; X) ---ff-! O(S; Y ) ?? ? y ?y O(p; X) ---ff-! O(p; Y ) where the vertical arrows are induced by the map p ! S with image {p}. Hence, * *for h 2 O(S; X), we have ff(h)(p) = f(h(p)), so ff = f*. Finally, f is holomorphic * *because it preserves analytic discs, mapping O(D; X) into O(D; Y ), where D denotes the op* *en unit disc in C. The Yoneda embedding restricts to a full embedding of M into the category of presheaves of topological spaces on S, if we equip each set O(S; X) with the co* *mpact-open topology. Finally, let us postcompose this functor with the singular functor. T* *his yields an embedding M ! s ShvS, taking a complex manifold X to the simplicial presheaf sO(.; X), which is clearly a sheaf with respect to the fine topology on S (reca* *ll that the singular functor preserves limits). This embedding is no longer full because th* *e singular functor is not full; however, every morphism sO(.; X) ! sO(.; Y ) is given by a* * holomor- phic map X ! Y at the level of vertices. We will often view a complex manifold * *X as an object of s ShvS and write X for sO(.; X). 14 If p is a point in a Stein manifold S and dimp S = m 0, then the stalk of a* * complex manifold X at p is simply the colimit as n ! 1 of sO(_1_nBm ; X), where Bm is * *the open unit ball in Cm and the maps between the scaled balls 1_nBm are the inclusion* *s. This is in fact a homotopy colimit, since (by the identity theorem!) all the maps are c* *ofibrations (just dualize the theory of towers in [GJ, VI.1]). Now for any m 0 and r > 0, * *O(rBm ; X) is weakly equivalent to X itself, so all the stalks of X are weakly equivalent * *to sX. A holomorphic map X ! Y is a cofibration (with respect to either topology) i* *f and only if it is injective. It is a coarse weak equivalence if and only if it indu* *ces a topological weak equivalence O(S; X) ! O(S; Y ) for every Stein manifold S. It is a fine, i* *.e., stalk- wise, weak equivalence if and only if it is a topological weak equivalence. Fib* *rations are somewhat mysterious; we only remind the reader that they are defined by a right* * lifting property with respect to acyclic cofibrations. Examples. Since D and C are holomorphically contractible, they are both coarse* *ly weakly equivalent to a point. The same holds for any star-shaped domain in Eucl* *idean space. The inclusion Dx ,! Cx is a fine but not a coarse weak equivalence. N* *amely, by Liouville's theorem, O(Cx ; Dx ) = Dx , but O(Cx ; Cx ) has infinitely many * *connected components, one for each winding number about the origin. Now let X be a complex manifold. The inclusion sO(.; X) ,! sC(.; X) is a fin* *e weak equivalence of simplicial sheaves on S because every cover has a refinement con* *sisting of sets at which the inclusion is a weak equivalence: take a refinement by bal* *ls, for instance. The Oka-Grauert property is satisfied by X if and only if this inclu* *sion is a coarse weak equivalence. Theorem 2.1 states that this is equivalent to the s* *implicial sheaf sO(.; X) satisfying finite excisionSin the sense of Section 2, meaning th* *at for every finite cover {U1; : :;:Um } in S, sO( Ui; X) is not only the limit but also th* *e homotopy limit of the diagram whose objects are the simplicial sets sO(Ui1\ . .\.Uik; X)* * for 1 i1 < . .<.ik m and whose arrows are induced by restriction maps. This prope* *rty may also be expressed by saying that sO(.; X) is a finite homotopy sheaf. We ca* *n now state Gromov's Oka principle and our interpretation of its conclusion, the Oka-* *Grauert property, as follows. 4.3. Theorem. Let X be a complex manifold. (1) The fine weak equivalence sO(.; X) ,! sC(.; X) of simplicial sheaves on * *the Stein site is coarsely acyclic if and only if sO(.; X) is a finite homotopy sh* *eaf. (2) If X has a spray, then X represents a finite homotopy sheaf on the Stein* * site. Finite excision for pointwise fibrant simplicial presheaves on S is clearly * *invariant under coarse weak equivalences. Let us show that descent implies finite excision. As * *remarked before Theorem 3.1, descent implies two-set excision, but since we are not work* *ing on a topological space (the union of Stein open subsets is usually not Stein), we ca* *nnot simply refer to Theorem 3.3 to get finite excision. A simplicial presheaf satisfying * *descent is coarsely weakly equivalent to a finely fibrant simplicial sheaf F on S, so it s* *uffices to 15 show that F satisfies finite excision. Let S be a Stein manifold. Via itsetale * *space, F |S extends to a simplicial sheaf "Fon S with its usual topology. It suffices to sh* *ow that "F is flabby; we then invoke Theorem 3.3. By the Yoneda lemma, if X is a Stein man* *ifold, then F (X) = Hom s ShvS(O^(.; X); F ), where the sheaf O(.; X) = hom S(.; X) o* *f sets on S has been turned into a simplicial sheaf O^(.; X) in the trivial way (the same* * set in all degrees; all face and degeneracy maps are the identity). If V is an open subset* * of S, let B be a Stein basis for the topology of V , viewed as a subdiagram of the site o* *f S. Then O(.; V ) is the sheaf colimit of the diagram O(.; B), and Hom s ShvS(O^(.; V ); F ) = limHom s ShvS(O^(.; B); F ) = limF (B) = "* *F(V ): If W V are open subsets of S, then the induced map O^(.; W ) ! O^(.; V ) is cl* *early a cofibration (pointwise injection), so by Quillen's Axiom SM7, the restriction m* *ap "F(V ) ! F"(W ) is a fibration, and the proof is complete. Note, finally, that a nondiscrete complex manifold X is never flabby, let al* *one coarsely or finely fibrant. Namely, let Dr be the open disc of radius r centred at the o* *rigin in the complex plane. The inclusion D1 ,! D2 is a monomorphism in S, but since there * *are holomorphic maps D1 ! X that do not extend holomorphically to D2, it is easily * *seen that the restriction map sO(D2; X) ! sO(D1; X) is not a fibration. 5. Partial descent on the quasi-projective site It is natural to ask whether a finite homotopy sheaf on S satisfies descent. Th* *is would turn Gromov's Oka principle into a descent theorem, somewhat analogous to such * *results as Brown-Gersten descent and (unstable) Nisnevich descent in algebraic geometry* *. We do not know the answer: the finiteness properties that make descent possible in al* *gebra _ the Zariski topology being binoetherian, essentially _ do not hold in analysis.* * We hope to address the notion of "analytic descent" in a future paper. In the meantime, le* *t us show how Brown-Gersten descent easily implies partial descent of sorts for quasi-pro* *jective manifolds with the Oka-Grauert property. Let A be the category of quasi-projective complex manifolds, i.e., smooth Za* *riski open sets in projective varieties, and algebraic maps. We put the usual Zariski topo* *logy on A by defining a cover of a quasi-projective manifold X to be a family of isomorph* *isms onto Zariski open subsets of X which cover X (covers can be pulled back to covers be* *cause algebraic maps are Zariski continuous). Taking X to the sheaf of sets O(.; X) * *on the small site A defines an embedding of A into Shv A (it is faithful because the Y* *oneda embedding X 7! hom A (.; X) is). As before, we equip each set of holomorphic m* *aps with the compact-open topology, apply the singular functor, and obtain an embed* *ding of A into the model category s ShvA of simplicial sheaves on A, taking a quasi-pro* *jective manifold X to the simplicial sheaf sO(.; X). This embedding is not full, but e* *very morphism sO(.; X) ! sO(.; Y ) is given by a holomorphic map X ! Y at the level* * of vertices. 16 Now let X be a quasi-projective manifold. We claim that the simplicial shea* *f G = sC(.; X) on A satisfies descent. Let G ! F be a fibrant model for G, i.e., an * *acyclic cofibration to a fibrant simplicial sheaf F on A. Let A be a quasi-projective * *manifold with the Zariski topology. By Theorem 3.4, G|A satisfies excision. So does F |A* *, since it is flabby. Hence, the weak equivalence G ! F is pointwise acyclic by Propositi* *on 3.2, and G satisfies descent. Suppose now that X has the Oka-Grauert property, so sO(S; X) ,! G(S) is a we* *ak equivalence for every Stein manifold S. Every cover in A has a refinement cons* *isting of Stein Zariski open sets, so sO(.; X) ! G is an acyclic cofibration between s* *implicial sheaves on A. Hence, the composition sO(.; X) ! F is a fibrant model for sO(.; * *X), and sO(S; X) ! F (S) is a weak equivalence for every Stein manifold S in A. 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