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. Since a*
*ny two
fibrant models for the same object in s ShvA are pointwise weakly equivalent, t*
*his holds
for every fibrant model for sO(.; X), and we have proved the following "Stein d*
*escent
theorem".
5.1. Theorem. Let X be a quasi-projective manifold and F be a fibrant model f*
*or
sO(.; X) in s ShvA. If X has the Oka-Grauert property, then sO(S; X) ! F (S) i*
*s a
weak equivalence for every quasi-projective Stein manifold S.
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cians, vol. I (Berlin, 1998), Doc. Math., extra vol. I (1998), 579-604.
[Vo2] ______, Voevodsky's Seattle lectures: K-theory and motivic cohomology, A*
*lgebraic K-theory
(Seattle, WA, 1997), Proc. Symp. Pure Math. 67, Amer. Math. Soc., 1999.
Department of Mathematics, University of Western Ontario, London, Ontario N6*
*A 5B7,
Canada
E-mail address: larusson@uwo.ca
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