Extended Hyperbolicity

EXTENDED HYPERBOLICITY SIMONE BORGHESI AND GIUSEPPE TOMASSINI Abstract. Giv en a complex space X , we cosidered the problem of ﬁnd- ing a hyp erb o lic mo del of X . This is an ob ject I p ( X ) with a morphism i : X → I p ( X ) in such a wa y that I p ( X ) is “hyperb ol ic” i n a suitable s ense and i is as close as possible to b e an isomorphism. Using the theory of mo del categories, we found a deﬁnition of hy p erb olic simpl icial sheaf (for the strong topology) that extends the classical one of Bro dy for complex spaces. W e prov e the existence of hyperboli c models for an y simplicial sheaf. F ur thermore, the morphism i can b e take n to b e a coﬁbration and an aﬃne weak equiv alence (in an algebraic setting, Morel and V o evodsky called it an A 1 we ak equiv a- lence). Imitating one p ossibl e deﬁnition of homotop y groups f or a topological space, we deﬁned the holotopy groups f or a simplicial sheaf and show ed that their v anishing in “positive” degrees i s a necessary condition for a sheaf to b e h yp erb olic. W e deduce that if X is a complex space w i th a non zero holotopy group in p ositive degree, then its hyperb olic mo del (that in general wi ll only b e a si mplicial s heaf ) cannot be w eakly equiv alen t to a hyperb ol ic complex space (in particular is not itself hyperb olic). W e ﬁnish the manuscript by applying these results and a top olo gi c al r e alization functor , constructed in the previous section, to pro ve that the hy p erb olic models of the complex pro jectiv e spaces cannot b e weakly equiv alent to hyperb olic complex spaces. Contents 1. Int ro duction 2 2. Basic constructions 7 2.1. Sheav es and s implicial ob jects: the categor ie s F T ( S ) and ∆ op F T ( S ) 8 2.2. Simplicial lo c alization 14 2.3. Notations 15 2.4. Aﬃne lo calization 17 2.5. Hyperb o lic simplicial sheaves 19 3. Hyper b olicity and B ro dy hyperb olic it y 22 4. Holotopy g roups 33 2000 Mathematics Subje ct Classiﬁc ation. 32Q45, 18G30, 18G55. Key wor ds and phr ases. Kobay ashi Hyperboli c spaces, Simplicial shea v es, Homotopical algebra. Supported b y the MU RST pro j ect “Geomet ric Prop erties of Real and Complex M anifolds”. 1 2 BOR GHESI AND TOMASSINI 5. The top olog ical r ealization functor 38 5.1. Remarks o n homoto py colimits 43 6. Some applica tions 44 References 47 1. Introduction The no tion o f hyper b olic spac e was given by Kobay ashi in [9]. It is based on the existence o f cer tain intrinsic distances, orig inally introduced to gene r alize Sch warz Lemma to higher dimensional complex spaces. Let D ⊂ C b e the unit disc en- dow ed with the Poincar´ e distance ρ . In v ie w of the Sch w arz-P ick Lemma every holomorphic map f : D → D is a cont ractio n for ρ . Let X b e a complex space. A chain of holomorphic discs b etw een t wo p oints p , q in X is a set α = { f 1 , · · · , f k } of holo morphic maps D → X such that there are po ints p = p 0 , p 1 , · · · , p k = q in X a nd a 1 , b 1 , · · · , a k , b k in D with the prop er ty f i ( a i ) = p i − 1 and f i ( b i ) = p i , i = 1 , · · · , k . The length of α is deﬁned a s (1) l ( α ) = k X i =1 ρ ( a i , b i ) The Kob ayashi pseudo distanc e d K o b on X is deﬁned a s (2) d X K o b ( p, q ) = inf α l ( α ) where α v aries through the family o f all chains o f holomor phic discs joining p and q . F or quasi-pr o jective v arieties the ps eudo distance of Ko bayashi can be deﬁned by mea ns of chains of algebra ic curves (s e e [6]). The contraction prop erty holds with res p ect to the Kobayashi pseudo distances: if f : X → Y is a holomorphic ma p b etw een co mplex spac e s , we hav e (3) d Y K o b ( f ( p ) , f ( q )) ≤ d X K o b ( p, q ) , for every p, q ∈ X . In particular, d Y K o b is in v arian t by biholomor phisms. It follows that d D K o b = ρ . EXTENDED HYPERBOLICITY 3 W e hav e d C K o b ≡ 0, d C ∗ K o b ≡ 0. More genera lly , o ne ha s d G K o b ≡ 0 for every connected co mplex Lie gr oup G (see [10]). A co mplex space X is sa id to b e hyp erb olic (in the sense of Kobayashi) if d X K o b is a distance. The unit disc D is hyperb olic, wher eas C is not. C \ A with card A ≥ 2 is hyperb olic. A compact complex curve of genus g ≥ 2 is a hyper b o lic space [10]. X is s a id to b e hyp erb olic mo dulo C , where C is a clo sed subset (usually a closed complex subspa ce), if for every pair o f distinct p oints x, y ∈ X \ C we hav e d X K o b ( x, y ) > 0 . If a c o mplex spa ce Y is C -co nnected (i.e. for any p 6 = q po int s in Y there exists a holomor phic function f : C → Y such tha t p, q , ∈ f ( C )) then, by v irtue of the contraction pro p e rty the only holo morphic maps with v alues in a hyperb olic spa ce X a re the cons tant ones. In particular , every holo mo rphic map C → X is constant. The crucia l fact is that for co mpact co mplex spaces the conv erse is also true. This is the conten t of the fundamental theorem of Br o dy (cfr. [10], [12]). This result motiv ates the following deﬁnition: a co mplex space X is sa id to b e Br o dy hyp erb olic if every holomorphic ma p f : C → X is c onstant. As well as for hyperb o lic it y we hav e the notion of Br o dy hyp erb olicity mo dulo a close d subset C : X is said to b e Br o dy hyp erb olic mo dulo C if every non constant holomorphic map f : C → X satisﬁes f ( C ) ⊂ C . A Kobay ashi hyperb olic space is Bro dy hyperb olic but the converse is in general not true. Indeed Mark Green constructed a Z ariski op en set W in P 2 (the t wo dimensional complex pro jective space) , deleting four lines in general po sition and three po ints outside the four lines, which is Bro dy hyperb olic but no t K obay ashi hyperb olic [12]. Related to h yp erb olicity ar e s ome basic conjectures whic h motiv ated several impo rtant paper s in Algebr aic and Analytic Ge o metry . (1) A gener ic hyper surface of degr ee ≥ 2 n + 1 in P n is hyperb olic; (2) The co mplement of a hypersurface of degree ≤ 2 n in P n is not hyperb o lic; (3) A gener ic hypersurfac e of degr ee ≥ n + 2 in P n is hyperb o lic mo dulo a prop er closed subv ariety; 4 BOR GHESI AND TOMASSINI (4) a smo oth pro jectiv e h yp erb olic v ariety has a n ample cano nical bundle (Kobay ashi’s conjecture); (5) a smo oth algebraic v ariet y is o f genera l type if and only if it is hyper b olic mo dulo a pr op er a lgebraic subset (Lang’s conjecture [13], [14]). F or the basic material as well as a discussion of the geometric mea ning o f these conjectures we r efer to [10], [12], [13], [14] and the bibliogr aphy there. In this pap er we will consider the following problem: given a c omplex spa ce X , co ns truct a ”hyper b o lic mo de l” of X i.e. a “hyperb olic” ob ject Ip ( X ), in a sense the “closest” hyperb olic ob ject to X endow ed with a cano nical natur al map c X : X → Ip ( X ) having the following universal prop er ty: if Y is hyperb olic a holomorphic map f : X → Y factoriz es through Ip ( X ) i.e. we hav e a commutativit y diagram X c X / / f   Ip ( X ) ˜ f | | y y y y y y y y y Y . One p ossible w ay to do this would be to co nsider the quo tien t top olo g ical space X/ R where R is the equiv alence relation: x ∼ y iﬀ d X K o b ( x, y ) = 0 or, be a ring in mind Bro dy’s Theorem, if and only if they b elong to the image of a holomor phic map C → X . This approa ch has t wo o ddnes s es. O ne is that X/ R is in genera l v ery diﬀerent from X , indeed X/ R is just a p oint for C -connected space s X . On the other hand, X / R will have in gener al no co mplex s tructure (even in a weak sens e), th us it will b e impo ssible to de ﬁne a Kobay ashi psedo distance on this quotient in order to hav e an useful concept o f hype rb olicity on it. Regarding this, it is worth mentioning the nice pa per of Campana [4] where a c o ncept of Kobay ashi pseudo distance is deﬁned for orbifolds. Then, for any v ariet y which is smo o th and bimer omorphically equiv alen t to a K¨ ahler manifold he constr ucted an orbifold C ( X ) called the c or e of X and a meromor phic function c X : X → C ( X ). F urther more, he co njectured tha t the ge ne r ic ﬁb er of c X has a v anishing Kobayashi metric and C ( X ) is Bro dy hyperb olic mo dulo a prop er subv ariety . EXTENDED HYPERBOLICITY 5 In this pap er w e developed a diﬀerent appr oach. W e used techniques pioneered by Quillen in [17] and lar gely employed in [16], which we dr e w inspira tion fro m in wr iting the technical sectio ns of this paper . W e construct an (unstable ) homo- topy categor y of c omplex spaces H , who s e ob jects include (homotopy) cla sses of complex spa ces. Unlike the classical ho mo topy catego r y of top olo gical space s, the category H r eﬂects the complex structure of the o b jects. The pro cedure in volv es an enlargement of the categor y of complex spaces to a new ca tegory containing as full sub c ategory the o ne o f complex spac es with holomor phic functions. In this bigg er category we deﬁne a notion of hyperb olicity which we prove that it restric ts to the Bro dy hyperb olicity for complex spaces. Using this notion, we show that in each class of c o mplex spaces lies a h yp erb olic represe n tative Ip ( X ), which in gener al will not be a complex space. It follows that Ip will b e (weakly equiv alen t to) a po int if and o nly if X is. Such corr esp ondence is functorial a nd there exists a canonical morphism c X : X → Ip ( X ) satisfying the universalit y prop erty describ ed ab ove. c X and e f will b e morphisms o f the homotopy ca teg ory in gener a l, but the c o mp o si- tion e f ◦ c X is a class represented b y a holomorphic function and the commut ativity is as ho lomorphic functions as opp osed to ”‘up to homotopy”’. Co ncerning the ob ject Ip ( X ), we will prov e that the class of P n cannot hav e a h yp erb olic co mplex space a s r epresentativ e, wher eas in the cla s s of C , the p o int can b e taken as hype r- bo lic complex space representativ e. Ip ( X ) is given by a co mplicated cons tr uction even if X is a complex space , although its top olo gic al r e ali zation (see Section 6) is a top olo gical spac e homoto pic equiv alent to the top ologica l space underlying X . The pr o cedure to construct the ca tegory H follows clo sely the one describ ed in [16] which works in a quite general context. It follows tha t almost all the results prov ed here are v a lid for alg ebraic schemes o f ﬁnite type over a no etheria n ba se, as well. The main idea is to co ns truct a categ o ry o bta ined fro m another b y “adding” the inverses of cer tain morphis ms . In the ca se of the categor y Compl of complex 6 BOR GHESI AND TOMASSINI spaces with the strong ly topolo gy and holomorphic maps, w e wish to add the in- verse to the cano nical map p : C → pt (the canonica l pro jection A 1 B → B in the algebraic case) alo ng with all its base changed ma ps. Such a catego r y , which we denote as p − 1 Compl , exists, how ever, to ma ke it us able, it s hould be obtained as the homotopy category asso cia ted to a mo del stru ctur e (see [17]) on Comp l . In gener al, deciding whether a lo calized ca tegory S − 1 C is equiv a lent to the homotopy category asso ciated to a mo del structure on C is a very co mplica ted tas k . This has b een prov ed in the case of derived ca tegories and the homotopy ca tegory of to p o logical spaces. There a re only partial r e s ults on this issue, if we assume that the categ o ry C is a homotopy ca tegory itself and p osseses a “simplicial str ucture”. The easiest wa y to replace a categor y C with one endowed of such simplicial structure is to consider ∆ op C , the categor y of simplicial ob jects in C . Then, we may try to give to ∆ op C a simplicial mo del structure. If a ll this is s uccessful, the ho motopy categor y asso ciated to such simplicial mo del structure is a go o d candidate to start with for establishing whether w e can lo calize with res pe ct o f so me morphism by using a n appropria te mo del s tructure. In our situatio n, the catego ry Co mpl is replac e d with F T ( S ), the categor y of sheav es over the site of complex s pa ces endowed with the strong top ology T (in the a lgebraic ca se, this will deno te the ca tegory of s heav es ov er the site of smo oth schemes of ﬁnite type over a no e ther ian base endo wed with a top olo gy not ﬁner than quasi c o mpact ﬂat topo lo gy). The reaso n for doing this lies mainly in the fact tha t not a ll diagr ams admit colimits and the existing ones in Compl o ften are unsuitable to do ho motopy theo ry with (see Section 2 .1 for more details on this). On the other hand, F T ( S ) is complete and co complete and the col- imits hav e a “suitable” shap e. W e than pro ceed with the pro gram descr ibe d a b ove in o rder to inv ert p : C → pt . W e end up with the categor y H s which is deﬁned as the homotopy catego ry a s so ciated to the simplicial mo del structure o n ∆ op F T ( S ). The mo rphism p in the categor y H s ﬁts in the Bousﬁeld framework [1], and lies in- side the c la ss of weak equiv alences in an appropriate mo del str ucture on ∆ op F T ( S ). The ass o ciated homotopy categ o ry will b e denoted by H and so metimes by H olo EXTENDED HYPERBOLICITY 7 when we w is h to stress that we ar e in the holomor phic setting. Any ob ject of the site repres ent s a cla ss in H and in the case it is a complex space, its hyperb o lic mo del e X will be only a simplicia l s heaf on Com pl . The no tio n o f hyperb olicity fo r a simplicia l sheaf X is g iven in the Deﬁnition 2.4. In the par ticular case X = X is a compa ct co mplex space , in view of Theor em 3.1 and B ro dy’s Theorem, w e con- clude that X is h yp erb olic accor ding to o ur deﬁnition if and only if it is K o bay ashi hyperb olic (see Corollar y 3.1). Th us, the Deﬁnition 2.4 is a genera liz ation of the classical co ncept of hyperb o lic it y fo r co mplex spaces . In the section 4 we a sso ciate certain sets to each o b ject o f H olo which hav e a natural group struc tur e in p ositive simplicial degree. They a re called holotopy sets or gr oups when a pplicable (see Deﬁnition 4.1). W e prov e that the v a nishing of some of the holoto py gro ups of a complex s pa ce X is a necessa ry condition for the hyper b olic mo del Ip ( X ) to b e iso morphic in H olo to some hyperb olic complex space. In the following section w e construct an useful functor for explicit computations: the top olo gic al r e alizatio n functor . T o a simplicial sheaf it asso ciates a top ologica l space in s uch a way few r easonable prop er ties are sa tisﬁed (cfr. Deﬁnition 5.1). In the last section, as an application o f some of our r esults, we s how that Ip ( P n ) is not weakly equiv a lent to a Bro dy h yp erb olic spa ce for any n > 0 an that the same holds for any complex space whose universal covering is C N for so me N > 0. The ﬁrs t author wishes to thank Cales Casacub erta for having given him the chance o f visiting the Universitat de Ba rcelona and discussing with him to pics ab out lo ca lization o f catego ries. 2. Basic constructions In this pap e r with P n we will denote the n - th dimensio nal pro j ective complex space. The g eneral problem we are dealing with is to mo dify the catego r y o f complex spaces to a catego ry wher e the consta nt mor phism p : C → p t is inv ertibile. The task o f inv erting mor phisms in a catego ry , ca n b e accomplished by starting from an arbitra r y category C with r esp ect to a g iven family S of morphis ms satisfying 8 BOR GHESI AND TOMASSINI suitable compatibility conditions (cfr . [7]). The categor y S − 1 C that we obtain is called the lo c aliza tion of C with r esp ect to S . In this kind o f generality , S − 1 C is not pra c tical to work with. In this sense, r easonable catego ries ar e the ”homotopy categorie s” asso ciated to a mo del stru ctur e in the sense of Q uillen (i.e. endo wed with a ”go od deﬁnition” of we ak e quivalenc e [17]). In this sec tio n we reca ll the main r esults of [16 ]. The constructions made there hold in the g eneral context of a site with enough p oints in the sense of [8]. W e restrict o urselves to the site S T of complex spa ces with the stro ng top ology or tha t of schemes of ﬁnite t yp e ov er a no etherian scheme B of ﬁnite dimension, endow ed with a Gr othendieck top ology which is weaker o r a s ﬁne as the quasi compac t ﬂat top ology . 2.1. Shea v es and sim plicial ob jects: the categories F T ( S ) and ∆ op F T ( S ) . Let S be the categ o ry of complex spaces or schemes of ﬁnite t yp e ov er a no ether ian scheme B . If we wish to do so me k ind o f homo topy theory on it, we should chec k the s ha p e of colimits of certain diagra ms. Reca ll that given a diagram D A i / / f   X B (4) in a small categor y C , an ob ject co lim D in C is the colimit o f D if and only if colim D ﬁts in the commutativ e diagr am A i / / f   X p   B g / / colim D (5) and Hom C (colim D , X ) is the limit o f the diagra m Hom C ( A, Z ) Hom C ( X, Z ) i ∗ o o Hom C ( B , Z ) f ∗ O O (6) in the category o f se ts fo r any Z ∈ C . In other words, this last condition means that Ho m C (colim D , X ) are pairs of morphisms ( α, β ), α ∈ Hom C ( X, Z ) and β ∈ Hom C ( B , Z ) with the pro per ty that i ∗ α = f ∗ β . The deﬁnition of colimit of an EXTENDED HYPERBOLICITY 9 arbitrar y diagram is similarly reduced to the one of limit in the c a tegory of s ets by applying H om C ( , Z ). W e are pa r ticularly interested in colimits of diagr ams o f the kind A i / / f   X pt (7) where i is an injection. In this pap er , such colimits will sometimes b e called quotient of X by A along i . In general, it may happ en that the quotient do es not exist in the catego ry S or if it exists, it is diﬀerent from the one taken in the underlying category of top olo gical space s . Examples 2.1 . 1) Let D b e the diagr ams (8) C − 0 i / / f   C pt P 1 i / / f   P 2 pt where i a re the canonica l embeddings. Then the colimits of D in Co mpl are just a po int in b oth ca s es, unlike their resp ective co limits in the category of top ologica l s paces. 2) Let D b e the diagr am Z i / / f   C pt (9) where i is the canonical injection. D has no co limit in S . Indeed, by contradiction, let Z = colim D in S , p : C → Z the corr esp onding cano nica l holomorphic function a nd x = p ( Z ). Since there exists a non constant holomorphic function h : C → C s uch that h ( n ) = 0 for every n ∈ Z , Z cannot b e just the point x , moreover p − 1 ( x ) = i ( Z ). Let U b e a rela tively compact neigh b ourho o d of x a nd { z ( n ) } ⊂ p − 1 ( U ) a seque nc e with no accumulation p oints. If h : C → C is a holomor phic function sa tisfying h ( n ) = 0 a nd h ( z ( n ) ) = n for every n ∈ Z , no holo morphic function g : Z → C exists such that g ◦ p = h . 10 BOR GHESI AND TOMASSINI A simila r a r gument can b e used to pr ove that the diagr am C i / / f   C × C pt (10) where i is the injection C → { 0 } × C , ha s no colimit in S . 3) Let D b e the diagr am { 0 } ∪ { 1 } i / / f   A 1 k pt (11) where A 1 k is the aﬃne line ov er a ﬁeld k and i is the embedding of the corres p o nding rational po ints. Then, since the k - algebra of the p oly no mials P ( x ) of the form a + x ( x − 1) Q ( x ), a ∈ k , is no t ﬁnitely g enerated, D has no colimit in the catego ry of the a lgebraic schemes of ﬁnite type ov er k . W e ther efore enla rge S to a categor y which contains the colimits o f a ll dia g rams and, at the same time, hav e a “rea sonably g o o d” sha pe from o ur p o int o f view. Such a categor y is F T ( S ): the ob jects a re sheaves of sets on a site S endow ed with the Grothendieck to po logy T a nd morphisms ar e maps of sheav es of se ts. Reca ll that a she af of sets on S T (or a n a rbitrar y site) is a controv ariant functor F : S T → S ets satisfying the following conditions: (1) F ( ∅ ) = { pt } , wher e pt is the ﬁnal o b ject of S T ; (2) let q : E → X b e a cov ering for the to p o logy T , q 1 and q 2 resp ectively the canonical pro jections E × X E → E ; then (12) F ( X ) q ∗ → F ( E ) q ∗ 1 ⇒ q ∗ 2 F ( E × X E ) is an exa ct sequence of sets i.e. q ∗ F ( X ) = { a ∈ F ( E ) : q ∗ 1 ( a ) = q ∗ 2 ( a ) } Let Y ( X ) := Hom S ( · , X ). The functoria l equa lity Hom S ( A, B ) = Hom F unt ( S op ,S ets ) ( Y ( A ) , Y ( B )) EXTENDED HYPERBOLICITY 11 is known a s Y oneda Lemma. The Y o neda embedding is a faithfully full functor Y : S ֒ → F un ( S op , S ets ). If the topo logy T is not ﬁner than the quasi co mpact ﬂat top olo g y , then the image of Y is contained in the full sub categ ory F T ( S ). Theorem 2.1. L et X ∈ S T and T a top olo gy not ﬁ n er than t he quasi c omp act ﬂat top ol o gy or t he str ong top olo gy in the holomorphic c ase. Then t he functor Hom S ( · , X ) is a she af for the top olo gy T . Pro of. In the algebra ic case, we res trict the pro of to the case in which S T is a site of shemes of ﬁnite t yp e over a base as we ha ve b een assuming fro m the very beg inning. Then the conclusion follows fr o m the theo rem of Amitsur [15]. Assume now that S T is the s ite o f co mplex s paces and let q : E → Z b e an op en cov ering of the co mplex spac e Z . Then, the seque nc e (13) E × Z E q 1 ⇒ q 2 E q → Z is exa c t as sequence of sets. W e hav e to prov e that the s equence o f sets (14) Hom S ( E × Z E , X ) q ∗ 1 ⇔ q ∗ 2 Hom S ( E , X ) q ∗ ← Hom S ( Z, X ) is exact, as well. Supp ose that q ∗ 1 f = q ∗ 2 f with f ∈ Ho m S ( E , X ). Since q is contin uous, surjective and Z has the quotient topolo gy induced by q , a pplying the functor Hom T op ( · , X ) to the exact sequence (13) we obtain an ex a ct sequence, he nce a contin uous map f ′ : Z → X s uch tha t f = f ′ ◦ q . It follows that f ′ is holomorphic, f b eing holomor phic and q a lo cal biholomorphism.  The category F T ( S ) is c omplete and co co mplete. Indeed, the limit of a diagr am D in F T ( S ) is the functor U lim D ( U ) which is a shea f for the top olo gy T . As for the colimit, it is deﬁned as a T ( U co lim D ( U )) wher e a T is the asso ciated s heaf. In particula r it p osse s ses t wo canonical ob jects: an initial shea f ∅ , the sheaf tha t asso ciates the empty set to any elemen t of the s ite, except fo r the initial ob ject o f the site S to whic h it asso cia tes the one p oint set and the ﬁnal s heaf, which we will denote as pt or S pe c B if the ob jects of the site are complex spaces o r schemes o ver B , resp ectively . 12 BOR GHESI AND TOMASSINI W e now would like to consider the lo c alized catego ry p − 1 F T ( S ), where p : C → pt (or p : A 1 → S pec k ). Mor eov er, we wish the lo calized ca tegory to have supple- men tary structures such as the ones we would g et if p − 1 F T ( S ) were e q uiv alen t to the homo topy catego ry of an appro priate mo del structure on F T ( S ). Bas ically , a mo del structure o n a ca tegory C is the data of three classes of mor phis ms: w eak equiv alences, co ﬁbrations and ﬁbrations satisfying ﬁve a xioms C M1 , · · · , CM5 (see [17]) with the r equest that, in addition, the factoriza tions of CM5 are functorial. W e do not know ab out the exis tence of such mo del structur e on F T ( S ). This is a particular cas e of the mor e general and co mplicated question on whether a lo - calized category S − 1 C is equiv alen t to the homoto py categor y ass o ciated to s ome mo del structure on C . Some re sults of this kind are known in the case C itself is a ho motopy ca tegory (see [1]). T o us e them, we are forc ed to embed F T ( S ) in the “simplest” ca tegory we know that is endow ed of a mo del str ucture, namely ∆ op F T ( S ), the catego ry of s implicial ob jects in F T ( S ). A simplicia l obje ct X in C is a sequence {X i } i ≥ 0 of o b jects o f C with a s e q uence ∂ n i : X n → X n − 1 of morphisms for n ≥ 1 , i = 0 , 1 , · · · , n called fac es and a seq uence σ n i : X n → X n +1 of morphisms for n ≥ 0, i = 0 , 1 , · · · , n called de gener ations , satisfying the following conditions 1) ∂ i ∂ j = ∂ j − 1 ∂ i if i < j 2) σ i σ j = σ j +1 σ i if i ≤ j 3) ∂ i σ j =      σ j − 1 ∂ i if i < j identity if i = j or i = j + 1 σ j ∂ i − 1 if i > j + 1 A morphism f : X → Y of t wo simplicial ob jects X = {X i } i ≥ 0 , Y = { Y i } i ≥ 0 of C is a seq ue nc e { f i } i ≥ 0 of morphisms f i : X i → Y i which make the diagr ams X i f i / / σ n i   Y i σ n i   X i +1 f i +1 / / Y i +1 X i f i / / ∂ n i   Y i ∂ n i   X i − 1 f i − 1 / / Y i − 1 commutativ e. EXTENDED HYPERBOLICITY 13 With this notion of morphism, the family of simplicia l ob jects of C forms a categ ory denoted b y ∆ op C . Given X ∈ C w e denote by the sa me symbol the c onstant simplicia l obje ct deﬁned by X i = X , ∂ n i = σ n i = Id X , fo r every i, n . Suppo se that C ha s a ﬁna l ob ject ∗ , direct pr o ducts a nd direc t copro ducts. Let [ n ] be the set { 0 , 1 , · · · n } Then, for every integer n ≥ 0, denote by ∆[ n ] the simplicial ob ject that at the level m has as many copies o f ∗ as nondecrea sing monotone functions [ m ] → [ n ]. The m + 1 injective functions [ m − 1] → [ m ] induce the faces and the m surjective functions [ m ] → [ m − 1] induce the degenera cies of ∆[ n ]. On each copy of ∗ they ac t as the identit y mo rphism. Notice that in ∆[ n ] n there is only one nondegener ate elemen t, na mely the o ne corresp o nding to the iden tity . F or example, ∆[1] is describ ed as ∆[1] i = ∐ i +2 j =1 ∗ for each i ≥ 0 and of the three ∗ in degree 1, tw o o f them are the degener ations o f of the ∗ in degree 0 . The t wo ∗ in degree zero are the image s through the fa c e morphisms of the nondegener ate ∗ in degree 1. Remark 2.1. F or every simplicial o b ject X , the pair of ∗ in degree 0 deﬁnes tw o morphisms ǫ 0 and ǫ 1 : X → X × ∆[1]. Let X , Y t wo ob jects of ∆ op C . Deﬁnition 2.1. A homotopy b etwe en two morphi sms f , g : X → Y is a morphism H : X × ∆[1] → Y such that H ◦ ǫ 0 = f , H ◦ ǫ 1 = g . In particular, this deﬁnition g ives a notion of homo topy for ob jects and mor- phisms of C . Examples 2.2 . 1) Let ∆ n top = { ( t 0 , t 1 , · · · , t n ) ∈ R n +1 : 0 ≤ t i ≤ 1 , Σ i t i = 1 } . The co llection { ∆ n top } n forms a co simplicial top o lo gical space ∆ • top with the standard cofac e morphisms ∂ i (inclusion of the face missing the vertex v i ) and co degenera tions σ i (proiection from v i on the cor risp onding face). 14 BOR GHESI AND TOMASSINI 2) Let C b e the ca tegory o f sets. An ob ject A • = { A i } i ≥ 0 of ∆ op C is called a simplicia l set . The ge ometric al r e ali zation of A • is the top olog ical space | A • | = ∐ n A n × ∆ n top ( ∂ i ( a ) , t ) ∼ ( a, ∂ i ( t )) . A morphism φ : A • → B • of simplicial ob jects is s aid to b e a we ak e quiva- lenc e if its top ologica l r e a lization | φ | : | A • | → | B • | is a weak equiv alence, i.e. the homor phis ms | φ | ∗ : π k ( | A | , a ) → π k ( | B | , | φ | ( a )), be t ween the homotopy groups are iso morphisms, for all k > 0 a nd a bijection for k = 0. 3) Let T op b e the ca tegory of topo logical spaces with contin uous maps. Then the functor Sing : T op → ∆ op Ins , which asso cia tes to a top ologica l space K the simplicial set Ho m T op (∆ • top , K ) is a functor that is left adjoint to A • | A • | . The pair of adjoint functors ( Sing , | | ) (15) ∆ op Ins | | / / T op Sing o o sends simplicia l homotopie s in the sense of Deﬁnition 2.1 to homoto pie s of top ologica l s paces a nd viceversa. A simplicial ob ject in F T ( S ) is s aid to be a simplicial she af . F or the time b eing, we will consider F T ( S ) as the full sub categ o ry of ∆ op F T ( S ), identiﬁ ed with constant simplicial sheaves. 2.2. Simplicial lo cali zation. The following will endow ∆ op F T ( S ) with a model stucture in the sense of Quillen: Deﬁnition 2.2. A morphism f : G → F of simplicial she aves is a we ak e quivalenc e if for every p oint x of a c omplex sp ac e or a scheme over B , f x : G x → F x is a we ak e quivale nc e of simplicia l sets ( G x and F x b eing the r esp e ctive stalks over x of F and G ). An injective mor phism f : X → Y is s aid to b e a simplicial c oﬁbr ation . A lifting in a commutativ e squa re of morphisms EXTENDED HYPERBOLICITY 15 A q / / j   X f   B r / / Y (16) is a morphism h : B → X which mak es the diagra m co mm utative. In such situatio n we say that j has the left lifting pr op erty with re s pe ct to f and f has the right lifting pr op erty with res pe ct to j . A morphism f : X → Y is called a ﬁ br ation if a ll diagrams (16) admit a lifting, for a ll acyclic c oﬁbr ations j (coﬁbration a nd weak equiv alence simultaneously). The c lasses of weak equiv alences, coﬁbra tions and ﬁbrations give ∆ op F T ( S ) a structure of simplicial model category as shown in [1 1]. Under these assumptions, there exists a loc a lization o f ∆ op F T ( S ) with resp ect of the w eak equiv a lences. In other words, there exists a categor y whic h we will denote by H s and a functor l : ∆ op F T ( S ) → H s which ha s the prop er ties 1) if f is a weak equiv alence, l ( f ) is an isomor phism; 2) the prop erty is universal, namely , if a nother ca tegory C exists and it is endow ed with a functor l ′ : ∆ op F T ( S ) → C with the sa me prop erty as l , then ther e e x ists a unique functor u : H olo s → C such that l ′ = u ◦ l . An o b ject X of ∆ op F T ( S ) 1) is called c oﬁbr ant if ∅ → X is a coﬁbra tion; 2) is called ﬁbr ant if X → pt is a ﬁbr a tion. 2.3. Notations. 1) W e denote pt the simplicial constant shea f deﬁned as the a sso ciated sheaf to the the presheaf which asso ciates to an ob ject of S the set consisting of one element. The p ointe d c ate gory asso ciated to ∆ op F T ( S ) is the categor y ∆ op • F T ( S ) whose ob jects ar e the pairs ( X , x ) wher e X ∈ ∆ op F T ( S ) and x : pt → X is a mor phism; a mor phism of pair s ( X , x ) → ( Y , y ) is a 16 BOR GHESI AND TOMASSINI morphism f : X → Y such that f ◦ x = y . As p o int ed shea f, pt will stand for ( pt , p t ). There is a pair of adjoint functor s (17) ∆ op F T ( S ) + ⇄ t ∆ op • F T ( S ) where t is the forg etful functor and + is deﬁned by : X X + with X + := X ∐ pt , p o in ted by pt . 2) Let f : Y → X b e a morphism of (p ointed) simplicial sheav es. The symbol cof ( f ) denotes the colimit of the diagr am Y f / /   X pt (po int ed by the image of Y ) where pt is a p oint. co f ( f ) is called the c oﬁbr e of f . If f is a coﬁbr ation the co ﬁbr e of f is sometimes denoted by X / Y . 3) Let X and Y b e p ointed simplicia l sheaves. The s hea f X ∨Y is, by deﬁnition, the co limit o f pt / /   X Y po inted b y the ima g e of pt . 4) The p o int ed simplicial sheaf X ∧ Y is deﬁned by X × Y / X ∨ Y . 5) The simplicia l po int ed cons tant sheaf S 1 s is deﬁned b y ∆[1] /∂ ∆[1 ] where ∂ ∆[1] is the s implicia l subsheaf of ∆[1] costisting in the union o f the image s of the fac e morphisms of ∆[1]. F or n ∈ N we set S n s = S 1 s ∧ n · · · ∧ S 1 s . Remark 2. 2. Performing the sa me constructions as for ∆ op F T ( S ) we obtain a homotopy ca tegory H s • . F o r a mo re co mplete de s cription o f the main pr op erties of H s and H s • we r efer to [17] e [16]. Her e we only r ecall a pro p o sition that will b e used later. EXTENDED HYPERBOLICITY 17 Prop ositi o n 2. 1. L et i : Y → X b e a simplicial c oﬁbr ation of p ointe d simplicial she av es. Then, for every p ointe d simplici al she af Z , the morphism i induc es a long exact se quenc e of p ointe d sets and gr oups (s e e the pr o of of L emma 4.1 ) (18) Hom H s • ( Y , Z ) i ∗ ← Hom H s • ( X , Z ) π ∗ ← Hom H s • ( X / Y , Z ) ← Hom H s • ( Y ∧ S 1 s , Z ) i ∗ ← Hom H s • ( X ∧ S 1 s ) π ∗ ← Hom H s • ( X / Y ∧ S 1 s , Z ) · · · This prop osition is a par ticula r case of Pro p o sition 4 ′ of [17]. The Y oneda embedding, induces a functor Y s : S T → H s which is a full em- bedding (see the Prop o s ition 1.1 3 and Remark 1.14 of [16]). Ho wev er, in gener al, it is more diﬃcult to desc r ib e the morphisms b etw eeen ob jects in H s . Indeed, Hom H s ( Y , X ) is obtained as a quotient of the set of diagra ms Y s ∼ ← Y ′ → X of ∆ op F T ( S ) where s is a weak equiv alence. H s (or its p ointed v ersio n) is the a ppropria te ca tegory in which we are go ing to inv ert p : C → p t . In the next sectio n we will give a mo del structur e to ∆ op F T ( S ) whose weak equiv alences co nt ain p , a nd ar e in a sense the “ smallest” class co n taining all the base changemen ts o f p as well. Such w eak equiv alences are written in terms of morphisms in H s and the homoto py c a tegory asso ciated to this mo del structure is the lo caliza tion of H s with r esp ect to the weak equiv alences. 2.4. Aﬃne locali zation. Unless otherwis e men tioned, the res ults pre s ented in this subsection a re taken from section 3.2 of [16]. Deﬁnition 2.3. A simplicial she af X ∈ ∆ op F T ( S ) is said to b e A 1 -lo c al (or C lo c al in the c omple x c ase) if the pr oje ction Y × A 1 → Y induc es a bije ction of sets Hom H s ( Y , X ) → Hom H s ( Y × A 1 , X ) for every Y ∈ ∆ op F T ( S ) . In what follows we describ e a ne w s tructure o f mo dels on ∆ op F T ( S ), whic h we will call aﬃne . A mor phism f : X → Y is ca lled: 18 BOR GHESI AND TOMASSINI 1) an aﬃne (or A 1 in the algebr ai c c ase or C in the c omple x c ase) we a k e quiv- alenc e if, for every A 1 -lo cal simplicial sheaf Z ∈ ∆ op F T ( S ) f ∗ : Hom H s ( X , Z ) → Ho m H s ( Y , Z ) is a bijection; 2) an aﬃne c oﬁbr ation if it is injective; 3) an aﬃne ﬁbr ation if all diag rams (16) a dmit a lifting, where j is any aﬃne coﬁbration and a ﬃne weak equiv alence. An o b ject X o f ∆ op F T ( S ) is called 1) A 1 - ﬁbr ant if the cano nical mor phism X → ∗ is an aﬃne ﬁbration; 2) A 1 - c oﬁbr ant if ∅ → X an aﬃne co ﬁbration. Theorem 2.2. (cfr. The or em 3.2, [16] ) The st ructur es liste d ab ove endow ∆ op F T ( S ) of a mo del str u ctur e, which wil l b e c al le d aﬃne mo del structure or A 1 mo del s truc- ture . The lo calized catego ry with res pec t of the aﬃne w eak equiv alences is denoted a s H a nd its p ointed version as H • . Remark 2.3. (1) An y ob ject of ∆ op F T ( S ) is b oth (simplicially) coﬁbrant a nd A 1 -coﬁbrant. (2) If f : Y → X is a simplicial weak eq uiv a lence (resp ectively a simplicial coﬁbration) then it is an aﬃne weak equiv alence (res p ectively an aﬃne coﬁbration). Therefor e, the aﬃne lo ca liz ation functor ∆ op F T ( S ) → H factors as ∆ op F T ( S ) → H s → H , where the ﬁrst functor is the simplicial lo calization a nd the second is the ident ity on ob jects and identit y on the fractions representing morphisms. How ever, the functor H s → H is not a n equiv alence of categor ies. (3) The sa me classes of p ointed morphisms, give ∆ op • F T ( S ) a mo del structure. Prop ositio n 2.1 ho lds for H a s well EXTENDED HYPERBOLICITY 19 Prop ositi o n 2.2. L et j : Y → X b e an aﬃne c oﬁbr a tion (i.e an inje ct ion of sim- plicial p oi nte d she aves). Then, for every simplicial p ointe d she af Z , the morphism j induc es long exact se quenc e of p ointe d s et s and gr oups (19) Hom H • ( Y , Z ) j ∗ ← Hom H • ( X , Z ) π ∗ ← Hom H • ( X / Y , Z ) ← Hom H • ( Y ∧ S 1 s , Z ) j ∗ ← Hom H • ( X ∧ S 1 s ) π ∗ ← Hom H • ( X / Y ∧ S 1 s , Z ) · · · The pro of o f such a statement is the sa me as for the P rop ositio n 2 .1. 2.5. Hyp erb oli c s impli ci al sheav es . Let us go back to the concept of hyperb o l- icity . Deﬁnition 2.4. A simplicial she af X is said to b e hyperb olic if it is A 1 -lo c al. L et C b e a simplicial su bshe af of X . The simplicial she af X is said to b e hyperb olic mo d C if X / C is hyp erb olic. Deﬁnition 2.5. A hyp erb olic r esolution of X is a morphism of simplici al she aves r : X → e X wher e e X is a hyp erb o lic simplicia l she af and r is an aﬃne we ak e quivalenc e. A hyp erb o lic r esolution fun ctor is a pair ( I , r ) wher e I is a functor ∆ op F T ( S ) → ∆ op F T ( S ) and r is a natural tra nsformation Id → I s uch that every morphism X → I ( X ) is a hyperb olic resolution. F r om P r op osition 2.1 9 o f [16] we derive the following, fundamental result: Theorem 2. 3. Ther e exist s a hyp erb olic r esolution functor ( Ip , r ) with the fol lowing pr op erties: 1) for every X ∈ ∆ op F T ( S ) the simplicial she af Ip ( X ) is hyp erb olic and (sim- plicially) ﬁbr ant; 2) r is an aﬃne e quivalenc e and a c oﬁbr ation; 3) let H s, A 1 b e the ful l sub c ate gory in H s of A 1 -lo c al (hyp erb olic) obje cts. Ip sends an aﬃne we ak e quivalenc e to a simplicia l we ak e quivalenc e, henc e it 20 BOR GHESI AND TOMASSINI induc es a functor L : H s → H s, A 1 , that factors as H s → H → H s, A 1 , wher e the ﬁrs t functor is the identity on obje ct s (se e also R emark 2.3 (2)); 4) the c anonic al immersion I : H s, A 1 ֒ → H s is a right adjo int of L . F urthermor e, H s, A 1 is a c ate gory e quivalent to H . Given X = X ∈ F T , Ip ( X ) is the hyperb olic simplicial sheaf asso c iated to the simplicially constant shea f X . Howev er, due to its rather inv olved construction, the use o f Ip ( X ) is pr oblematic even in the ca se when X is a complex space or a scheme ov er k . Therefore, in g eneral, the previous result shall be considered as an existence theorem. Nev ertheless, it may o c c ur tha t, in some particula r cases, the class of Ip ( X ) in H could b e repr esented b y an under standable ob ject, o r even by a hy- per b olic space (e.g. 3 .1). In order to give a more precise idea of the diﬃculties inv olves, let Ho m ( X , Z ) b e the right adjoint functor to Y Y × X , wher e the ob jects ar e simplicial sheav es. Let us deﬁne Sing A 1 • ( X ) to b e the simplicial sheaf { Hom (∆ n A 1 , X n ) } n ≥ 0 , where ∆ • A 1 is the cosimplicia l s heaf such that ∆ n A 1 = A n for every n and the s tr ucture morphisms ar e as describ ed in pag e 88 of [1 6]. T he n, the class Ip ( X ) is deﬁned to b e the simplicial shea f ( Ex ◦ Si ng A 1 • ) ω ◦ Ex ( X ) where X → Ex ( X ) is a ﬁbra nt simplicial reso lution a nd ω is a suﬃciently larg e ordinal. W e conclude this section by a short discussion on morphisms in lo calized cat- egories. Mor phisms in a lo ca lized catego ry S − 1 C can b e ex pr essed in ter ms of morphisms of C using the so called c alculus of fr actio ns . More precisely , Hom S − 1 C ( X , Y ) = {X s ← X ′ f → Y : s ∈ S, f ∈ Ho m C ( X ′ , Y ) } ∼ where the elements o f the n umerator ar e called fr actio ns and ∼ is an equiv alence betw een fractions. EXTENDED HYPERBOLICITY 21 If the lo calizatio n is as so ciated to a model structure C (as it happ ens for H s and H ),we know that there ar e ob jects X , Y such that, Hom S − 1 C ( X , Y ) is a quo- tien t of Hom C ( X , Y ) ; for instance, if X is c o ﬁbrant a nd Y is ﬁbrant. Under these assumptions it can b e proved that (20) Hom S − 1 C ( X , Y ) = Hom C ( X , Y ) ∼ l where f ∼ l g if and only if the morphism f ∐ g facto r s through a cylinder obje ct Cyl ( X ): Cyl ( X ) can " " E E E E E E E E E X ∐ X O O f ∐ g / / Y . (21) W e recall that a cylinder asso ciated to a n ob ject X of a ca tegory C endowed with a mo del str uc tur e is an o b ject Cyl ( X ) ∈ C with morphisms X ∐ X i → Cyl ( X ) can → X such that c an ◦ i = can ◦ id X ∐ id X and can is a weak e q uiv alence. Cylinder ob jects alwa ys exis ts in a catego ry C endow ed with a mo del structure. F ur thermore the morphism i can be c hosen to b e a coﬁbration, can a ﬁbration and the corrisp ondence X Cyl ( X ) functorial. If C = ∆ op F T ( S ), a cylinder ob ject for the a ﬃne mo del s tructure asso ciated to X may b e taken to be X × A 1 where i is the morphis m X ∐ X → X × A 1 determinated by the inclusions at 0 a nd 1 (i.e. by the morphisms X → X × { 0 } , X → X × { 1 } ) and can the pro jection onto X . W e a lready observed that every ob ject of ∆ op F T ( S ) is coﬁbrant for b oth the mo del structures o n ∆ op F T ( S ). Conse q uent ly , if Y is ﬁbr a nt (resp ectively s implicially ﬁbra nt ), Hom H ( X , Y ) (resp ectively Hom H s ( X , Y ) ) is a quotient set o f Ho m ∆ op F T ( S ) ( X , Y ) . In the sequel, this fact will be extensively used. Lemma 2.1. F or any simplici al she af X , the morphism r : X → Ip ( X ) is universal in the c ate gory H (r esp e ctively in the c ate gory H s ) in t he fol lowing sense: for any 22 BOR GHESI AND TOMASSINI hyp erb olic obje ct Y and morphism f : X → Y in H , t her e exists a un ique morphism ˜ f : Ip ( Y ) → Y in H s . Pro of. Consider the commutativ e squar e X r X / / f   Ip ( X ) Ip ( f )   Y r Y ∼ = / / Ip ( Y ) . (22) By deﬁnition of A 1 weak equiv alence, r Y is a simplicial weak eq uiv a lence, since b oth Y and Ip ( Y ) a re hyper b olic (i.e. A 1 lo cal). The map ˜ f is deﬁned as r − 1 Y ◦ Ip ( f ). Note that Ip ( f ) is a mor phism in H s .  Corollary 2 .1. L et X and Y b e shaves with Y hyp erb olic and f : X → Y b e a morphism of she aves. Then the c omp osition ˜ f ◦ r is a morphism of she aves and the c ommutativity of t he diagr am X r / / f   Ip ( X ) ˜ f | | y y y y y y y y y Y (23) is in the c ate gory of she aves, i.e. it is strictly c ommutative and not only “up to homotopy” in H s . Pro of. By the previous lemma, w e have commutativit y in the categor y H s . Remark 1.14 of [16] implies tha t Ho m ( X, Y ) = Ho m H s ( X, Y ) since b oth X and Y ha ve simplicial dimension ze r o. There fore, eq uality of the morphisms f and ˜ f ◦ r in H s is an equality of morphisms of sheaves.  3. Hyperbolicity and Brod y hyperbolicity In this section we will compar e the diﬀerent notions of hyperb olicity that w e hav e introduced above. In particular, we prove that a s implicia l sheaf re pr esented by a complex space X is hyperb olic if and only if X is Bro dy hyper bo lic. This is a corolla r y of the following EXTENDED HYPERBOLICITY 23 Theorem 3. 1. A she af X ∈ F T ( S ) is a hyp erb olic she af if and only if the pr oje ction U × A 1 → U induc es a bije ction Hom F T ( S ) ( U, X ) → Hom F T ( S ) ( U × A 1 , X ) for every obje ct U ∈ S T . Mor e over, u nder this hyp othesis, for every Y ∈ F T ( S ) ther e exist s a bije ctio n (24) Hom H ( Y , X ) ∼ = Hom H s ( Y , X ) ∼ = Hom F T ( S ) ( Y , X ) . Remark 3.1. I f in (24) the sets ha ve a group structure induced (up to homotopy) by a group str ucture on Y or by a co group structure (up to homotopy) on X , the bijection is a group isomor phism. Before be g inning the pro o f, we ﬁx, by the fo llowing co mm utative diagra m H s L   S T Y s 2 2 e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e e Y A 1 , , Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y / / F T ( S ) cost / / ∆ op F T ( S ) L s 6 6 n n n n n n n n n n n n n n L A 1 ( ( P P P P P P P P P P P P H s, A 1 ∼ = H I O O (25) the names of the functors inv olved in the pr o of. Notice tha t, the ﬁr st functor o n the left is the Y o neda embedding a nd L ar e the lo calization functor s. Pro of of Theorem 3.1. First of all we hav e the following bijections of sets (26) Hom H ( Y , X ) ∼ = Hom H ( L ( Y ) , X ) ∼ = Hom H s, A 1 ( L ( Y ) , X ) ∼ = Hom H s ( Y , X ) . The left end side bijection is a c o nsequence of the fact that the ca nonical morphism Y → L ( Y ) is a n a ﬃne equiv alence, the second one follows from the equiv alence betw een H a nd H s, A 1 (Theorem 2.3) by deﬁnition of hyperb olicity o f a simplicial sheaf. Finally , the third one follows from the fact that ( L , I ) is a pa ir of adjoint functors (Theo r em 2.3.(4)). Assume now that X = X and Y = Y are sheaves. Using the r esults quoted in [16, Remark 1.14 , p. 5 2 ] one shows that 24 BOR GHESI AND TOMASSINI Lemma 3.1. L et X , Y b e she aves. Then Hom H s ( Y , X ) ∼ = Hom F T ( S ) ( Y , X ) . This res ult implies the seco nd assertion o f Theorem 3 .1. Indeed, if X is a h yp erb olic sheaf, fro m Lemma 3.1 c ombined with the a b ove cons iderations we get Hom H ( Y , X ) = Hom H s ( Y , X ) = Hom F T ( S ) ( Y , X ) for every sheaf Y . Moreover, Lemma 3.1 also implies the ﬁr st assertion in the following weak er form: given a sheaf X , the pro j ection U × A 1 → U induces a bijection Hom H s ( U, X ) → Hom H s ( U × A 1 , X ) for every U ∈ S T if a nd o nly if it induces a bijection Hom F T ( S ) ( U, X ) → Hom F T ( S ) ( U × A 1 , X ) . Thu s, in order to ﬁnish the pro of of Theorem 3.1 in the g eneral case, it is suﬃcient to prove the following: for every U ∈ S T , the pro jection U × A 1 → U induces a bijection Hom H s ( U, X ) → Hom H s ( U × A 1 , X ) if a nd only if X is hyperb olic. F or this we use the following Lemma 1.16 of [16]: Lemma 3.2. L et Σ a s et of obje cts of F T ( S ) such that, for every U ∈ S T , ther e exists an epimorphism F → U wher e F is a dir e ct sum of elements of Σ . Then ther e exist a functor Φ Σ : ∆ op F T ( S ) → ∆ op F T ( S ) and a natur al tr ansformation Φ Σ → Id with the fol lowing pr op erties: given Y 1) for every n ≥ 0 the she af of sets Φ Σ ( Y ) n is a dir e ct su m of she av es b elonging to Σ ; 2) the morphism Φ Σ ( Y ) → Y is b oth a (simplicial) we ak e quivalenc e and a lo c al ﬁbr ation (i.e. the morphism induc es on the stalks a Kan ﬁbr ation of simplicia l sets). EXTENDED HYPERBOLICITY 25 Since w e will refer often to this lemma, we a re going to rec all here ho w to construct the functor Φ Σ . Let f : X → Y b e a morphis m of simplicia l sheav es; deﬁne Ψ Σ ,f as the colimit of ∐ D n ,n ≥ 0 F × ∂ ∆[ n ] / /   X ∐ D n ,n ≥ 0 F × ∆[ n ] (27) where D n is the set of the co mm utative squares of the kind F × ∂ ∆[ n ] / /   X f   F × ∆[ n ] / / Y (28) and F ∈ Σ. Let α 1 : Ψ Σ ,f → Y be the ca nonical mo rphism and Φ m +1 Σ ,f be Ψ Σ ,α m . The Φ i Σ ,f form a direct sys tem of coﬁbra tions { Φ 1 Σ ,f · · · ֒ → Φ i Σ ,f ֒ → Φ i +1 Σ ,f ֒ → · · · } whose colimit we w ill deno te as Φ Σ ,f . Such a simplicial sheaf factors functorially f : X → Φ Σ ,f → Y . The functor that asso ciates to a simplicial sheaf Y the simplicial shea f Φ Σ , ∅→Y satisﬁes the pr op erties of the Lemma. In view of the Y oneda L emma, we see that we ca n ta ke as Σ the sheav es repre- sented b y o b jects in S T . Indeed, for every sheaf W , we hav e a sur jective morphism (even a s pre s heav es) (29) ∐ U ∈ S T ∐ s ∈ W ( U ) Hom F T ( S ) ( · , U ) → W. Thu s, by L e mma 3.2, given an ar bitrary simplicial sheaf Y there exists Y ′ such that Y = Y ′ in H s and Y ′ n = ∐ n i U n i with U n i ∈ S T . By deﬁnition, if X is hyperb olic (i.e. A 1 -lo cal), the pr o jection p : Y × A 1 → Y induces a bijection p ∗ : Hom H s ( Y , X ) → Hom H s ( Y × A 1 , X ) for every Y ∈ F T ( S ). In particular, this holds if Y = U ∈ S T , so using Lemma 3.1 we co nclude that for every U ∈ F T ( S ) (30) p ∗ : Hom F T ( S ) ( U, X ) → Hom F T ( S ) ( U × A 1 , X ) is a bijection. 26 BOR GHESI AND TOMASSINI Conv ersely , a ssume that (30) is a bijection for every U ∈ S T . Then, for every U ∈ S T , p ∗ : Hom H s ( U, X + ) → Hom H s ( U × A 1 , X + ) is a bijection. Let sk n Y b e the simplicial s heaf deﬁned by sk n Y = ( ( sk n Y ) i = Y i , se i ≤ n, ( sk n Y ) i = ∐ σ u j degene rations σ u 1 ◦ · · · ◦ σ u i − n ( Y n ) , se i > n. Such a n o b ject is c a lled the n -skeleton of Y . T he immer sion i n : sk n − 1 Y ֒ → sk n Y is a coﬁbration and for Y = Y ′ , the coﬁbre sk n Y ′ /sk n − 1 Y ′ is iso morphic to the sheaf ∐ n i U n i ∧ S n s . W e use the following coﬁbra tion sequence s : (31) sk n − 1 Y ′ + → sk n Y ′ + → ∨ n i U n i + ∧ S n s → sk n − 1 Y ′ + ∧ S 1 s (32) ∨ n sk n Y ′ + → dirlim n sk n Y ′ + = Y ′ + → ∨ n ( sk n Y ′ + ∧ S 1 s ) id −∨ i n ∧ id S 1 − → ∨ n ( sk n Y ′ + ∧ S 1 s ) . Notice that we are forced to take separa te base p oints, s ince in the algebraic c a se, we cannot ass ume that a simplicia l sheaf Z can b e co nsidered as a p ointed simplicial sheaf. If n = 1 the sequence (31) bec o mes (33) ( ∐ 0 i U 0 i ) + ֒ → sk 1 Y ′ + → ∨ 1 i U 1 i + ∧ S 1 s → ( ∐ 0 i U 0 i ) + ∧ S 1 s . Thu s the following sequence (34) ( ∐ 0 i U 0 i × A 1 ) + ֒ → ( sk 1 Y ′ × A 1 ) + → ( ∨ 1 i U 1 i ∧ S 1 s ) ∧ A 1 + → ∐ 0 i U 0 i + ∧ S 1 s ∧ A 1 + is a coﬁbration sequence as well. The pro jection p : A 1 → pt maps the latter sequence to the former. Applying Hom H s • ( , X + ) we get the long exact sequence of p ointed sets and groups EXTENDED HYPERBOLICITY 27 (35) Hom H s • (( ∐ 0 i U 0 i ) + , X + ) ← Hom H s • ( sk 1 Y ′ + , X + ) ← Hom H s • ( ∨ 1 i U 1 i + ∧ S 1 s , X + ) ← Hom H s • (( ∐ 0 i U 0 i ) + ∧ S 1 s , X + ) · · · as a par ticula r case of the exa ct sequence (1 8). The mo r phism p ∗ induces maps from the sequence (35) to the one corr esp onding to Y × A 1 + . W e a re go ing to prov e that p ∗ is a bijection of p ointed sets, from A = Hom H s • (( ∐ 0 i U 0 i ) + , X + ) , B = Hom H s • ( ∨ 1 i U 1 i + ∧ S 1 s , X + ) , C = H om H s • ( ∐ 0 i U 0 i + ∧ S 1 s , X + ) . p ∗ is bijective from A , b ecause by the a djunction (1 7), we g et A = Hom H s ( ∐ 0 i U 0 i , X + ) . Since dir ect sums of classes in H s are represented by direct sums in ∆ op F T ( S ), we hav e that A = ∐ 0 i Hom H s ( U 0 i , X + ) and we conclude by using the ass umption we hav e on X . Regarding the pointed set B w e arg ue as follows: a ﬁbrant mo del of X + is of the kind ˜ X + , where ˜ X is a ﬁbrant mo del of X , th us it is a nonconnected simplicial shea f. On the other hand, ∨ 1 i U 1 i + ∧ S 1 s is a pointed co nnected simplicial sheaf. Since B is a quotient set of Ho m ∆ op • F T ( S ) ( ∨ 1 i U 1 i + ∧ S 1 s , ˜ X + ), w e conclude that B = ∗ , the constant ma p to the base point, b ecaus e Hom ∆ op • F T ( S ) ( ∨ 1 i U 1 i + ∧ S 1 s , ˜ X + ) is. The same a rgument works for Hom ∆ op • F T ( S ) ( ∨ 1 i U 1 i + ∧ S 1 s ∧ ( A 1 + ) , ˜ X + ) . Thu s p ∗ is an is omorphism on B . The same arg ument shows that p ∗ is also an isomorphism on C . By the Five Lemma, we conclude that p ∗ is an isomor phism from Ho m H s • ( sk 1 Y ′ + , X + ). Similarly , w e prove that the co ﬁbration exac t sequences (31) yield that p ∗ is bijectiv e fr om Hom H s • ( sk n Y ′ + , X + ), for every n ≥ 0. 28 BOR GHESI AND TOMASSINI Since p ∗ is bijectiv e from C = Hom H s • (( ∐ 0 i U 0 i ) + ∧ S 1 s , X + ) the same holds for Hom H s • ( sk 1 Y ′ + ∧ S 1 s , X + ) and consequently for Ho m H s • ( sk n Y ′ + ∧ S 1 s , X + ). Then, using the exa c tness of sequences (32), we conclude that p ∗ is bijective from Hom H s • ( Y ′ + , X + ) = Hom H s ( Y ′ , X + ) = Hom H s ( Y , X + ) , th us fr o m Hom H s ( Y , X ). Theorem 3.1 is completely proved.  Lemma 3.3. L et X ∈ S T and p : U × A 1 → U b e the pr oj e ction. Then the map p ∗ : Hom F T ( S ) ( U, X ) → Hom F T ( S ) ( U × A 1 , X ) is bije ctive for every sm o oth s cheme U if and only if (36) p ∗ k ( u ) : Hom F T ( S ) ( S pec k ( u ) , X ) → Hom F T ( S ) ( A 1 k ( u ) , X ) is, for every ﬁ nite ﬁeld extension S p ec L → S pec k , p L : A 1 L → S p ec L b eing the pr oj e ction. Pro of. W e have just to prov e that the bijectivit y of p ∗ L for every L ﬁnite extension of k implies the bijectivity of p ∗ for every smo oth scheme U . The mo rphism p : U × A 1 → U is a faithfully ﬂat cov ering, thus, by faithfully ﬂat descent w e have the following ex act seq ue nce of sets 0 → Hom ( U, X ) p ∗ → Hom ( U × A 1 , X ) p ∗ 1 ⇒ p ∗ 2 Hom (( U × A 1 ) × U ( U × A 1 ) , X ) . In order to pr ove the sur jectivit y o f p ∗ , we hav e to show that p ∗ 1 = p ∗ 2 . Notice that ( U × A 1 ) × U ( U × A 1 ) = U × A 2 and p ∗ 1 and p ∗ 2 are induced by the pr o jections on the factors o f A 2 to A 1 . Thus, given α ∈ H om ( U × A 1 , X ), we prov e that α ◦ p 1 = α ◦ p 2 : U × A 2 → X. By hypothesis , any map A 1 L → X factors thr ough S pe c L for any ﬁnite extens io n L/k . In particular , α ◦ p 1 and α ◦ p 2 coincide on the closed po int s of U × A 2 . Since the unio n of all c lo sed p oints o f U × A 2 is an everywhere dense subset for the Zaris ki top ology , we conclude that α ◦ p 1 = α ◦ p 2 .  EXTENDED HYPERBOLICITY 29 Corollary 3.1. L et X b e a c omp a ct c ompl ex sp ac e. Then X is Kob ayashi hyp erb olic if and only if it is hyp erb olic ac c or ding to the deﬁnition 2.4 . Pro of. Conseq uence of Theorem 3.1, Lemma 3.3 and Bro dy’s Theorem.  Corollary 3.2. L et X b e a c omplex sp ac e, C a close d c omplex subsp ac e of X . Then X is hyp erb o lic m o dulo C in the sense of Br o dy if and only if X/C is a hyp erb olic she af ac c or ding to the deﬁnition 2.4. Pro of. Let S T be the site o f co mplex spaces. B y de ﬁnitio n, the shea f o f S T given by Y Hom F T ( S ) ( Y , X/C ) is the asso ciated s heaf fo r the stro ng top olo gy to the presheaf which ass o ciates to a complex space Y the co limit o f Hom S ( Y , C ) / /   Hom S ( Y , X ) Hom S ( Y , pt ) . If X/ C is a hyperb olic sheaf, then, by Theorem 3.1, we obtain tha t the mor phism Hom F T ( S ) ( pt , X/C ) → Hom F T ( S ) ( C , X/ C ) is a bijection. Assume, b y co ntradiction, that there exis ts a no n constant holo - morphic map f : C → X suc h that f ( C ) 6⊂ C . Then f re pr esents an element in Hom F T ( S ) ( C , X/ C ) which is not in the imag e o f Hom F T ( S ) ( pt , X/ C ) which is absurd. Conv ersely , if X is Bro dy-h yp erb olic mo dulo C , o ne has Hom S ( C , X ) = ( X − C ) ∐ Ho m S ( C , C ) . On the o ther hand, we o bserve that Hom F T ( S ) ( C , X/ C ) is precisely e q ual to the colimit of Hom S ( C , C ) / /   Hom S ( C , X ) Hom S ( C , pt ) . This follows from the fact that the new sections that w e would get by taking the asso ciated sheaf are of the form ( f , g ) wher e f : U → X , g : V → X are holomor phic 30 BOR GHESI AND TOMASSINI maps, { U, V } is an o p e n covering o f C (w e may ass ume b oth U and V to b e connected) and f ( U ∩ V ), g ( U ∩ V ) a re co ntained in C . In this situation, we have that b o th f ( U ) and f ( V ) are contained in C , a s well. Therefore, ( f , g ) = ( U → pt , V → pt ) = C → pt and we alr eady have this section in Hom F T ( S ) ( C , X/ C ). Consequently , Hom F T ( S ) ( C , X/ C ) = ( X − C ) ∐ Hom S ( C , pt ) but the latter se t is Hom F T ( S ) ( pt , X/ C ), thus X /C is a hype r b olic sheaf by Theor em 3.1.  Let us discus s some exa mples of hyperb o lic r esolutions of complex spaces. Roughly sp eaking, Ip ( X ) ”enla rges” X by a dding a simplicia l structure whic h trivializes passing from H s to H . If X is a B ro dy hyperb olic complex spac e , Ip ( X ) is iso- morphic to X in the ca tegory H s . If X is no t Br o dy h yp erb olic the simplicial structures added to Ip ( X ) have the task to ” make constant” (up to simplicia l ho- motopy , hence in H s ) all morphisms C → X . Passing from H s to H , X a nd Ip ( X ) bec ome iso morphic o b jects . Examples 3.1 . 1) Ip ( C ) is a simplicial sheaf isomor phic to a p oint in H . Indeed, C ∼ = pt in H and the h yp erb olic resolutio ns prese r ve aﬃne equiv a- lences. T his fact is not surpr ising b eca use if we wan t to mak e a ll morphisms C → C homotopically c o nstant, in particular this must b e true for the iden- tit y C → C . 2) F o r the same reas on, Ip ( C n ) ∼ = pt in H for every n ∈ N . 3) More gener ally , if p : V → X is a vector bundle, Ip ( p ) : Ip ( V ) ∼ = → Ip ( X ) in H s bec ause p is a C weak eq uiv alence. Therefore, if X is a hyperb o lic complex space, then Ip ( V ) ∼ = X in H s and hence in H . 4) If X is a complex spa ce and Ip ( X ) is repre s ented b y a hyper b o lic complex space Y , then Y is unique up to isomor phis ms (cfr. the lemma b elow). In general, this is not the ca s e; e.g. in the next section we will show tha t Ip ( P n ) ca nnot b e C -equiv alent to a hyperb olic complex space. EXTENDED HYPERBOLICITY 31 In the ca se Ip ( X ) admits a hyperb o lic complex space a s representativ e, then such a spac e is unique up to biho lomorphism: Lemma 3.4. L et X b e a simplicial she af, Y , Y ′ hyp erb olic c omplex sp ac es such that Ip ( X ) = [ Y ] H = [ Y ′ ] H . Then Y ′ and Y ar e isomorphic c omplex sp ac es. Pro of. L et S b e the category o f c o mplex spa c es. By hypothesis , ther e exists an isomorphism φ : Y ∼ = Y ′ in H , namely a mor phis m ψ : Y ′ → Y in H such that ψ ◦ φ = id Y and φ ◦ ψ = id Y ′ in H . Since Y e Y ′ are complex h yp erb olic spaces, and in particular C -ﬁbrant ob jects by Cor ollary 3.1 (see also the end of Section 2), φ and ψ can b e r epresented b y morphisms φ ′ : Y → Y ′ and ψ ′ : Y ′ → Y in ∆ op F T ( S ). More precisely , we may supp ose that φ ′ and ψ ′ are holomorphic maps, Y , Y ′ being complex spaces and S ֒ → ∆ op F T ( S ) b eing a full immersion. Mor e ov er, the fact that φ , ψ ar e inverse to each o ther means tha t ψ ′ ◦ φ ′ ∼ id Y , φ ′ ◦ ψ ′ ∼ id Y ′ as holomorphic maps, wher e f ∼ g if and only if there exists a holo morphic map H : W × C → V s uch that H | W × 0 = f e H | W × 1 = g (cfr. equation (20)). Since b oth Y , Y ′ are h yp erb olic, H must b e constant along the ﬁbres which are is omorphic to C , thu s f ∼ g if and only if f = g as maps. In particula r, ψ ′ ◦ φ ′ = id Y and φ ′ ◦ ψ ′ = id Y ′ .  In some cases, we ca n e x tend some results known for hyper b o lic complex spaces to hyperb o lic sheaves: Lemma 3.5. Le t F T ( S ) b e the c ate gory of she aves of sets on t he site of c omplex sp ac es with the str ong top olo gy and F b e a hyp erb olic she af. Then Hom F T ( S ) ( P n , F ) = F ( pt ) for any n ≥ 1 . In other wor ds, any she af map fr om P n to a hyp erb olic she af F must b e c onstant. 32 BOR GHESI AND TOMASSINI Pro of. Consider the case n = 1 ﬁrst. Let P 1 = U 0 ∪ U 1 be an op en cov ering with U 0 = P 1 \ { 0 } and U 1 = P 1 \ {∞} . Then the squa r e U 0 ∩ U 1   i 0 / /  _ i 1   U 0   U 1 / / P 1 (37) is co ca rtesian in the catego ry of s heav es. Thus Hom F T ( S ) ( P 1 , F ) = lim      Hom F T ( S ) ( U 0 ∩ U 1 , F ) Hom F T ( S ) ( U 0 , F ) i ∗ 0 o o Hom F T ( S ) ( U 1 , F ) i ∗ 1 O O      . (38) Since U 0 ∼ = U 1 ∼ = C , we hav e that Hom F T ( S ) ( U j , F ) = Hom F T ( S ) ( pt , F ) = F ( pt ) for j = 0 , 1 b eca use of the theo rem 3.1. Moreov er, i ∗ j are injective be cause they hav e a r etraction given by f ∗ where f : pt → U 0 ∩ U 1 is any p oint. W e conclude the sta tement of the lemma in the case of P 1 by noticing that the image of i ∗ 0 coincides with the one of i ∗ 1 . Cons ider no w the op en cov ering of P n given by U 0 = P n \ P n − 1 ∼ = C n and U 1 = P n \ {∞} , where ∞ co incides with the p oint (0 , 0 , · · · , 0) ∈ U 0 = C n . W e get a co cartes ian square like (37) with P n replacing P 1 . The pr evious arg umen t car ries through in the general ca se. The only thing to c heck is that H om F T ( S ) ( U 1 , F ) = F ( pt ). Notice that the canonical pro jectio n p : U 1 → P n − 1 is a rank o ne vector bundle. Locally on P n − 1 (for the strong top ology) it is V × C , where V is a n op en aﬃne of P n − 1 . Hence (39) p ∗ V : Hom F T ( S ) ( V , F ) → Hom F T ( S ) ( V × C , F ) are bijections for all V , since F is hyperb olic. Glueing these data for V ra nging on an op en aﬃne cov ering of P n − 1 , we get that (40) p ∗ : Hom F T ( S ) ( P n − 1 , F ) → Hom F T ( S ) ( U 1 , F ) is a bijection. By inductive a ssumption, we conclude that Hom F T ( S ) ( U 1 , F ) = F ( pt ) . EXTENDED HYPERBOLICITY 33  4. Holotopy groups Throughout this section, S T will denote the site of co mplex spaces endow ed with the strong topo logy . A simplicial ob ject of S T is, by deﬁnition, a simplicia l c omplex sp ac e . If we forg et the complex structure, we could study the ob jects of S T by mea ns of the classic a l homo to py gro ups. Isomorphis m classe s of homotopy groups a re in- v arian t under homeo morphisms hence, a for tio ri, under biholomorphisms, how ev er, they do no t r eﬂect the existence and the prop erties of the complex structure. A rather natura l mo diﬁcation o f the deﬁnition of homotopy ena bles us to attach to every simplicial shea f on S T t wo families { π par i,j ( X ) } i,j , { π ip er n,m ( z 1 , z 2 )( X ) } m,n of s ets (cfr. Deﬁnition 4.1) which, for p ositive simplicia l deg rees, hav e a canonical gro up structure and ar e inv ar iant under biholomo rphisms. W e will use these gro ups in Section 6 to show that there exist complex spaces (e.g P n ) whose hyper b o lic reso- lutions (cfr. Deﬁnition 2.5) are not is omorphic to the class of hyperb olic complex spaces, not even in the categor y H . Deﬁne the p ar ab olic cir cle by S 1 par = C / (0 ∐ 1) , and we denote by S n par the s he a f S 1 par ∧ n · · · ∧ S 1 par . Let D ⊂ C be the unit disc and z 1 6 = z 2 t wo p oints of D . W e deﬁne the hyp erb oli c cir cle S 1 ip er ( z 1 , z 2 ) by S 1 ip er ( z 1 , z 2 ) = D/ ( z 1 ∐ z 2 ) and we denote by S n ip er ( z 1 , z 2 ) the shea f S 1 ip er ( z 1 , z 2 ) ∧ n · · · ∧ S 1 ip er ( z 1 , z 2 ). The quotients deﬁning parab olic and h yp erb olic cir cles are taken in the category F T ( S ), even though, in view of a theorem of Cartan (cfr . [5]) the se t theor e tic quotients hav e a complex struc tur e. Deﬁnition 4.1. L et X b e a simplicial she af on S T . Deﬁne 34 BOR GHESI AND TOMASSINI (41) π par i,j ( X , x ) = Hom H • (( C − 0 ) ∧ j ∧ S i − j par , ( X , x )) for i ≥ j ≥ 0 , (42) π ip er n,m ( z 1 , z 2 )( X , x ) = Hom H • ( S n ip er ( z 1 , z 2 ) ∧ S m par , ( X , x )) for n, m ≥ 0 . These sets ar e c al le d r esp e ctively par ab olic holotopy p ointed sets of X (or gr oups in t he c ase they ar e ) and hyper bo lic holo topy pointed se ts of X (or gr ou ps in the c ase they ar e). Remark 4.1 . The deﬁnitions ab ove are compatible with the classical ones of algebraic topo logy . Mo re precisely , let H top be the (unstable) homoto py ca te- gory of topo logical spaces (i.e. the lo c a lization o f the category of topo logical spaces with resp ect to the usual weak equiv alences); then we hav e π n ( X, x ) = Hom H top (( S n , p ) , ( X, x )) for every top olog ical space X . Mor eov er, the topo logi- cal r e a lization functor (cfr. Section 5) pr ovides functor ial g r oup ho mo morphisms π par i,j ( X, x ) → π i − j ( X, x ) a nd π ip er n,m ( z 1 , z 2 )( X , x ) → π m ( X, x ) for any complex spa ce X . Lemma 4 . 1. The s et s π par i,j , π ip er n,m have a c anonic al gr oup structure for i > j > 0 and m > 0 . Pro of. The ﬁrst step consists in proving that S 1 par ∼ = S 1 s in H . Consider the c o ﬁbration seque nc e (43) 0 ∐ 1 ֒ → C → S 1 par → S 1 s → C ∧ S 1 s → · · · where 0 ∐ 1 and C are p ointed b y 0. Since C ∼ = pt in H , we have C ∧ S 1 s ∼ = pt in H . Applying the functor Hom H • ( , Z ), in view of P rop ositio n 2.2, w e o btain lo ng exact se quences o f sets and, from these, the iso morphism Hom H • ( S 1 par , Z ) ∼ = Hom H • ( S 1 s , Z ) for every Z ∈ ∆ op • F T ( S ). It follows that S 1 par ∼ = S 1 s in H . The simplicia l ob ject S 1 s is a cog roup (ob ject) in H s (and conseq uent ly in H ). It is suﬃcient to o bs erve EXTENDED HYPERBOLICITY 35 that, if a str is the asso ciated sheaf for the s tr ong top ology , S 1 s ∼ = a str ( Sing ( S 1 )) in H s and that S 1 is a cogr oup in H top with pr o jection p : S 1 → S 1 / ( { i } ∐ {− i } ) homeo ∼ = S 1 ∨ S 1 as str uc tur al map. Then, applying to p the functor a str ( Sing ( )) we get a morphis m [ S 1 s ] → [ S 1 s ∨ S 1 s ] = [ S 1 s ] ∨ [ S 1 s ] in H s which satisﬁes the prop erties making it a comultiplication. These pr op erties are formulated in such a wa y to induce o n the se ts H om H s ( S 1 s , Z ) a natura l group structure. The same holds for Ho m H ( S 1 s , Z ).  Theorem 4.1. L et X b e a hyp erb ol ic s he af . Then the gr oups π par i,j ( X, x ) , π ip er n,m ( X, x ) vanish for i − j > 0 and any m > 0 . Pro of. W e be g in with proving that Hom H • ( Y ∧ S 1 par , X ) = 0 for every po inted complex spa ce ( Y , { y } ). By deﬁnition (cfr. Sectio n 2.3), Y ∧ S 1 par = Y × C / R where R is the complex s pace Y × (0 ∐ 1 ) ∪ y × C . Since Y × C / R is a shea f and X is a ﬁbra nt space, by Theorem 3 .1 we conclude that (44) Hom H ( Y × C /R , X ) = Hom F T ( S ) ( Y × C /R, X ) = = { f ∈ Hom F T ( S ) ( Y × C , X ) : f | R = constant } . Moreov er, since Y × C and X a re complex spaces, we hav e Hom F T ( S ) ( Y × C , X ) = Ho m olom ( Y × C , X ) . X is Bro dy h yp erb olic hence, for every y ∈ Y , the r e striction of a holomorphic map f : Y × C → X to y × C is c onstant. F urther more, if f ∈ Hom F T ( S ) ( Y × C /R, X ) then f is co nstant on Y × 0 ⊂ R a nd conseq uent y on the whole Y × C . It follows that, if f is p ointed, then f must b e constant with ima g e x , the base point of X . 36 BOR GHESI AND TOMASSINI This shows tha t Hom H • ( Y ∧ S 1 s , X ) = x for any p o int ed co mplex spac e Y a nd any hyperb olic po int ed sheaf X . W e would like no w to prov e the same r esult with Y b eing replaced by a quotient sheaf W = Y / Z . Consider the following co mm utative dia gram R / /   pt   Z × C / /   Y × C / /   Y × C R C / / Y Z × C (45) where the t wo s quares ar e co car tesian. Consider now the t wo new co c artesian squares Z × C / /   Y × C R   R / /   pt   C / / P Y Z × C / / W × C R = W ∧ S 1 par . (46) By chasing the diag ram (45) and using that Y Z × C a nd Y × C R are colimits of the relev an t diagrams , we ﬁnd tw o sheaf maps P → W ∧ S 1 par and W ∧ S 1 par → P that are mutually inv erses. By deﬁnition of P we ha ve (47) Hom H • ( W ∧ S 1 par , X ) = Ho m H • ( P, X ) = Hom F T ( S ) • ( P, X ) = = Hom F T ( S ) • (( Y × C ) /R , X ) × Hom F T ( S ) • ( Z × C , X ) Hom F T ( S ) • ( C , X ) . Recall that ( Y × C ) /R = Y ∧ S 1 par , thus, by the ﬁrst part o f the pro of of the prop osition, Ho m F T ( S ) • (( Y × C ) /R, X ) = x , the base p o int of X . The same holds for Ho m F T ( S ) • ( C , X ) b e cause, by a ssumption, X is Bro dy hyperb olic. Ther efore, Hom H • ( W ∧ S 1 par , X ) = x EXTENDED HYPERBOLICITY 37 for a n y quotient shea f W , a nd in particular for W = ( C \ 0) ∧ j ∧ S i − j − 1 par , W = S n ip er ( z 1 , z 2 ) ∧ S m − 1 par (see Deﬁnition 4.1).  Corollary 4.1. L et X b e a simplicia l she af. A ssume t hat π par i,j ( X , x ) 6 = 0 for i − j > 0 or π ip er n,m ( z 1 , z 2 )( X , x ) 6 = 0 for m > 0 . Then Ip ( X ) is n ot C we akly e quivalent to a hyp erb olic she af . In p articular, if X is a c omplex sp ac e such t hat π par i,j ( X , x ) 6 = 0 for i − j > 0 or π ip er n,m ( z 1 , z 2 )( X , x ) 6 = 0 for m > 0 , then X is not a Br o dy hyp erb olic c omple x sp ac e. Pro of. The pro ofs for the tw o ca ses are similar so we consider only the case of the parab olic ho lotopy g r oups. By deﬁnition, π par i,j ( X , x ) = Hom H • (( C \ 0 ) ∧ j ∧ S i − j s , ( X , x )) and this set is a quo tient of Hom ∆ op • F T ( S ) (( C \ 0 ) ∧ j ∧ S i − j s , ( e X , e x )) where ( e X , e x ) is an C -ﬁbrant p ointed simplicial sheaf C -weakly equiv alent to ( X , x ). In particular, we may ass ume that e X is the hyperb olic reso lution Ip ( X ) of X . If Ip ( X ) were C -weakly equiv alent to a B ro dy hyperb olic co mplex space X ′ , then π par i,j ( X , x ) would b e a quotient of Hom ∆ op • F T ( S ) (( C \ 0 ) ∧ j ∧ S i − j s , ( X ′ , x ′ )) which for i − j > 0 cons is ts only in the constant ma p with v alue x (cfr. Theorem 4.1).  Remark 4. 2. As mentioned in sectio n 2, to relate holotopy gro ups of a complex space X with morphisms in ∆ op F T ( S ) it is necessar y to replace X with its hy- per b olic mo del Ip ( X ). Then we know that π i,j ( X, x ) will b e a quotient o f the set Hom ∆ op • F T ( S ) ( S i,j , Ip ( X )), where S i,j is a p ointed mo del of the relev an t spher e. 38 BOR GHESI AND TOMASSINI 5. The topological realiza tion functor F r om now on, CP n will de no te the complex pro j ective s pace s e e n as top ologic a l space. W e would like to compare o b jects in H a nd H ( k ) with the top olog ical spaces, ob jects of the top ologica l (unsta ble) homotopy categor y H top . W e will show that there e x ists a functor t olo : H → H top which e x tends the functor which asso ciates the under lying top olo gical space to a complex spa c e. In the algebra ic case extends the co rresp onding functor which a sso ciates to an a lgebraic v ariety over C , the top olo gical space of its (Zar iski) closed po ints. The genera l c ase only applies to the site of smo oth v arieties ov er a ﬁeld k whic h admits a n embedding i in C . I t inv olves pas sing from a simplicial sheaf ov er k to a simplicial sheaf over C by means of i ∗ (or, more pr ecisely , b y means o f its total left der ived functor). Recall tha t for a sheaf F and a morphism o f sites φ : S 1 → S 2 , the sheaf φ ∗ F on S 1 is deﬁned as the asso ciated sheaf to the pr esheaf whose sections are ( φ ∗ F )( U ) = colim V F ( V ), where the colimit is taken over all the mo rphisms U → φ − 1 V for U ∈ S 1 and any V ∈ S 2 . Deﬁnition 5.1. L et ( S , I ) b e a site with interval (cfr. Se ction 2.3 [16] ) e quipp e d with a r e ali zation funct or r : S → T op t o the c ate gory of top olo gic al sp ac es. D enote by H ( S ) the I homotopy c ate gory whose obje cts ar e simplicial s he av es over S . Then a fun ctor t r : H ( S ) → H top with values in the uns t able homotopy c ate gory of top olo gic al sp ac es is c al le d a top o logical rea liz a tion functor if the following pr op erties ar e satisﬁe d: (1) if X ∈ ∆ op F ( S ) is a simplicia l set, then the class t r ( X ) c an b e r epr esente d by the ge ometric r e aliza tion | X | ; (2) if F is the she af Ho m S ( , X ) , wher e X ∈ S , then t r ( F ) c an b e r epr esent e d by r ( X ) ; (3) t r c ommutes with dir e ct pr o duct s and homotopy c ol imits. EXTENDED HYPERBOLICITY 39 Theorem 5.1. The sites with interval ( Comp l , C ) and (( S m/k ) T , A 1 k ) admit a top o- lo gi c al r e ali zation functor, pr ovi de d that k c an b e emb e dde d in C and T is not ﬁ n er then the ﬂat t op o lo gy. Pro of. Let φ : ( S 1 , I 1 ) → ( S 2 , I 2 ) be a r easona ble co un tinuous map of sites with in- terv al (cfr. Deﬁnition 1 .49 [1 6]). Consider the functor φ ∗ : ∆ op F ( S 2 ) → ∆ op F ( S 1 ) obtained b y applying the inv erse imag e functor on ea ch comp onent of the simpli- cial sheaf on S 2 . A cla ssical result in mo del catego ries assures the ex istence of the total left der iv ativ e betw een ho motopy catego ries of a functor, provided tha t such a functor se nds weak equiv alences b etw een coﬁbrant ob jects to weak equiv alences. In the ca se of φ ∗ , we will not b e able to prove this for every s implicial shea f on S 2 and the r elev an t I model categ ories. How ever, we can get the sa me result in the fol- lowing wa y . W e cons ider the full category of I 2 lo cal ob jects H s,I 2 ⊂ H s int ro duced in the Theorem 2.3, whic h is equiv a lent to the I homotopy ca tegory H ( S 2 , I 2 ) by the same theorem. Such a categor y has the prop erty that a morphism is a n I 2 weak eq uiv alence if and only if it is a simplicial weak eq uiv a lence. Thus, to show that φ ∗ admits a total left derived functor b etw een the I ho motopy catego ries, it is suﬃcient to show that φ ∗ sends s implicial weak equiv alences betw een I 2 lo cal ob jects (since every ob ject is coﬁbra nt ) to simplicial w eak e quiv alences. Actually , since the pro p er ty fo r a simplicia l sheaf X to be I 2 lo cal is inv a riant under simpli- cial weak equiv alences on X (cfr. Deﬁnition 2 .3) to v alidate the same conclusion it suﬃces to show a w eaker condition: there exists a (simplicia l) reso lutio n functor Φ and a natur a l tra ns formation Φ → id with the pro pe r ty that φ ∗ sends simplicial weak equiv alences b etw een simplicial shaves o f the kind Φ( X ) → Φ( Y ) to sim- plicial weak equiv alences for all I 2 lo cal simplicial sheaves X and Y . But this is precisely the statement of Pro p osition 1.57 .2. of [16] where Φ is ta ken to be Φ Σ int ro duced in Lemma 3.2. This shows the e xistence of the total left derived functor L φ ∗ : H ( S 2 , I 2 ) → H ( S 1 , I 1 ) of φ ∗ . E xplicitely , it is deﬁned as follows: let X be a s implicial shea f ov er S 2 , then L φ ∗ ( X ) is r epresented b y the simplicial sheaf φ ∗ (Φ Σ ( Ip ( X ))), where Ip ( X ) is the I 2 lo cal simplicial sheaf ment ioned in the The- orem 2 .3. This deﬁnition is well po sed o n H ( S 2 , I 2 ) b eca use of the ab ov e remar ks 40 BOR GHESI AND TOMASSINI and the fact that, if X a nd X ′ represent the same class in H ( S 2 , I 2 ), then Ip ( X ) and Ip ( X ′ ) a re simplicially weak equiv alen t. W e will now conside r the case of the site with interv a l ( Co mpl , C ) since the algebraic case when k = C is entirely similar. W e set the realization functor r : S → T op to b e the one which as so ciates the underlying top ologica l spa ce X top to a complex space X . Let pt b e the site with interv al w ho se only nonempty ob ject is the ﬁnal ob ject p t and ψ b e the trivia l mor phism of sites with in terv al pt → Comp l . Notice that a simplicial sheaf on pt is just a simplicial set. W e take the interv al I in pt to b e the consta nt simplicial set Hom ( pt , C ). The functor ψ ∗ sends a simplicia l sheaf X on Compl to the simplicial set X ( pt ). Thus, ψ ∗ ( C ) = I so that, in par ticular, it is I contractible. Because o f this, the functor ψ is said to be a r e asonable contin uous map of sites with in terv al (cfr. Deﬁnition 3.1 6, [16]) and L ψ ∗ has a particularly nice descr iption: L ψ ∗ ( X ) is represented by the simplicial sheaf ψ ∗ Φ Σ ( X ) where Σ is the class of repr esentable sheav es o n Co mpl (see Lemma 3.15. of [16]). Let T l c open be the ca tegory o f lo cally co ntractible top o logical spaces. W e now endow the images o f L ψ ∗ by a struc tur e of top ologica l spaces in order to obtain a functor H (∆ op F ( Compl )) → H (∆ op T l c open ). If Y is a simplicial shea f tha t in e a ch degree is a disjo int union of repre s entable sheav es ∐ j ∈ J Y j , then w e set θ Y := Y top , where Y top is the s implicial top olo gical space having the top olog ical space ∐ j ∈ J Y top j in the corres p o nding degree . Since ψ ∗ is reaso nable, L ψ ∗ ( X ) = [ ψ ∗ Y ] wher e Y is any representable simplicial shea f equipp ed with a s implicia l weak eq uiv alence Y → X . An y tw o such mo dels will give r ise to simplicially weak equiv alent inv erse images by Prop ositio n 1.57 .2. [16], thus, in particular, I weak equiv alent. This shows that the deﬁnition o f θ induces a functor H = H (∆ op F ( Compl )) → H (∆ op T l c open ) which we will call θ , as well. Remark 5.1. H (∆ op T l c open ) is a full s ub ca tegory of H (∆ op F open ( T lc open )). T he latter categ o ry is the I homotopy categ ory tak ing as interv al the shea f I = H om cont ( , C ). Such a n interv al is an ob ject of ∆ op T l c open , thus we ca n see H (∆ op T l c open ) a s the EXTENDED HYPERBOLICITY 41 lo calized categ o ry with resp ect to the I = C weak equiv alences, co nsidering C as constant simplicia l to po logical s pace a nd no longer only a s co nstant s implicial set. Prop ositi o n 5.1. Ther e is an e quivalenc e of c ate gories γ : H (∆ op T l c open ) ∼ = H top . Pro of. (sketch) It is a par ticular case of Pr op osition 3.3 of [16 ]. Here we wr ite the deﬁnition of the functor γ : H (∆ op T l c open ) → H top which g ives the equiv alence of catego r ies. Let X be a simplicial lo ca lly contractible top olo gical space. Since for any top olo g ical space Z in T l c open there is an op en cov ering ∐ i U i → Z , with U i contractible for all i , b y Lemma 3.2, X admits a (simplicial) weak equiv alence e X → X with e X j = ∐ i j U i j . In turn, ˜ X is I w eakly equiv a lent to X ′ , wher e X ′ is the simplicia l set with X ′ j = ∐ i j pt , b eca use e X and X ′ are ter mwise weakly equiv alent and o f Pro po sition 2.14, [16]. The eq uiv a lence of categor ies is deﬁned as [ X ] [ |X ′ | ] where |X ′ | is the geo metr ic realiza tio n of the s implicia l set X ′ .  Remark 5.2. If X is a topo lo gical space in T l c open , then |X ′ | is weakly equiv alen t to | S ing • ( X ) | . But this top ologica l s pa ce is w eakly equiv alen t to X itself, th us the constant s implicia l top olog ical spa ce X is sent by γ to a top ologica l space weakly equiv alen t to X in the classical s ense of homo topy theor y . Let D b e a small catego ry and ∆ op F T ( S ) D be the category of functors from D to ∆ op F T ( S ). W e will deno te by ho colim( D ) a homotopy c olimit of D o n the category ∆ op F T ( S ). That is a pair ( k , a ) c onsisting in a functor k : ∆ op F T ( S ) D → ∆ op F T ( S ) which takes ob jectwise weak equiv alences in ∆ op F T ( S ) D to weak equiv- alences in ∆ op F T ( S ) and a natural transformatio n a : k → co lim D . Suc h functor can b e obtained by ﬁr st tak ing a s uitable coﬁbra nt diagram replacement of a n element in ∆ op F T ( S ) D and comp os ed with the ordinar y colimit functor (cfr. [2]). W e set t olo : H → H top to be the functor γ ◦ θ . Prope rty (1) o f Deﬁnition 5.1 follows b y deﬁnition of γ . Pro p er ty (2) is a conseq ue nce o f Remark 5.2. As for the pr op erty (3), we hav e that ψ ∗ commutes with limits by deﬁnition. Since direct pro ducts in the homotopy c ategorie s are r epresented by dir ect pro ducts of 42 BOR GHESI AND TOMASSINI ob jects, we hav e that t olo commutes with dir ect pro ducts. ψ ∗ has a right a djo int, namely ψ ∗ , th us it is right ex act. Mor eov er, ψ ∗ sends coﬁbrations (se ctionwise injections) to co ﬁbrations. On the other hand, the sa me holds for the re s olution functor of Lemma 3.2 Φ: if i is a sectionwise injection, then Φ( i ) is a s ectionwise injection by deﬁnition of Φ; furthermor e, Φ commutes with colimits, since its v alue on ob jects ha s b een deﬁned a s a colimit. In particular , if D is a coﬁbrant diagram in ∆ op F ( Compl ), Φ( D ) is coﬁbra nt a nd we conclude that ψ ∗ Φ( D ) is co ﬁbrant as well and also that colim( ψ ∗ Φ( D )) ∼ = ψ ∗ Φ ∗ (colim( D )). This shows tha t, for a ny diagram D , L ψ ∗ (ho colim( D )) = ho colim( L ψ ∗ ( D )), since the former class can b e represented by colim( ψ ∗ Φ( D ′ )) for any co ﬁbrant replacement D ′ ∼ → D b ecause ψ ∗ Φ( D ′ ) is a coﬁbrant diagra m. Therefore, θ commutes with homotopy colimits. Recall that the equiv a lence γ is deﬁned to b e the functor that, to a class represe nted by a simplicial top olog ical space X , asso ciates the class in H top represented by | (Φ S ( X )) ∼ | wher e S is the class of contractible to po logical spac e s and the o pe r ation ∼ re places each c ontractible top ologica l space with a p oint. Beca us e of the deﬁnition of Φ S we see that ∼ sends injections to injectio ns and commutes with co limits. Before pro ceeding to inv estigate the prop erties of the functor | | , we need to r ecall the mo del structures inv olved in the catego r ies. The functor | | is deﬁned on the ca tegory of simplicial sets and takes v alues in the ca teg ory of top olo g ical spaces. The mo del str ucture for the category of simplicial sets is: let f : X → Y b e a map of simplicial sets , then f is: (1) a we ak e quivalenc e if | f | is a we ak homotopy e quivalenc e (se e b elow); (2) a c oﬁbr atio n if it is an inje ction; (3) a ﬁ br ation if f has the right lifting pr op erty with r esp e ct to acyclic c oﬁbr a- tions. Let X 0 ֒ → X 1 ֒ → X 2 → · · · b e a sequential dir ect s ystem of topo lo gical spac e s such tha t for each n , ( X n , X n +1 ) is a relative CW complex. Then we will s ay that EXTENDED HYPERBOLICITY 43 the cano nical function X 0 ֒ → co lim X i is a gener alize d r elative CW inclus ion . A contin uous function b etw een top ologic a l spaces f : X → Y is (1) a we ak e quivalenc e if f ∗ : π ∗ ( X, x ) → π ∗ ( Y , f ( x )) is a gr oup isomorphism for ∗ ≥ 1 and a bije ction of p o inte d sets if ∗ = 0 ; (2) a c oﬁbr atio n if it is a r etr act of a gener alize d r elative CW inclusion; (3) a ﬁbr ation if it is a Serr e ﬁbr ation. The functor | | preser ves coﬁbratio ns and also it commutes with colimits, b eca use it has a right adjoint, namely the functor Sing ( ). In conclusion, the functor γ commutes with homotopy colimits, and so do es the topo logical realiza tion functor t olo .  5.1. Remarks on homotopy colimits . The pr actical use of the top ologica l real- ization functor requires few remarks on the diﬀerences b etw een (homotopy) colimits of diagr ams in the categor y H top and the category H . Let us consider the colimit of the diagr am C \ 0   / /   C pt . (48) In the category of complex spaces, this is just a p oint. How ever, we have previous ly inferred in this manuscript that the colimit of such a diagram in the categ o ry o f sheav es o n Compl is no t (weakly equiv alent to) the consta nt shea f to a p o int. Indeed, its class in the resp ective homotopy categories pla ys the role of the t wo dimensiona l sphere S 2 = CP 1 , or, more precisely , of the shea f repres ent ed by CP 1 , who se class is by no means isomorphic to the one o f the p oint. As a diagra m of to po logical spaces, its colimit is no t a p oint, but it is not an appro priate mo del for S 2 . W e should p oint out that the diag r am (48) is a coﬁbrant diagr a m for the aﬃne mo del structure in the ca tegory ∆ op F ( Compl ), but it is not coﬁbrant in the categor y of top ologica l s paces fo r the mo del structure deﬁned ab ov e. This a pparent o ddness 44 BOR GHESI AND TOMASSINI disapp ears if we consider homotopy co limits instead. F or instance, t olo ( S n par ) ∼ = t olo ( S n ip er ( z 1 , z 2 )) ∼ = S n for any n ≥ 0 and z i ∈ D , t olo ( C \ 0) ∼ = S 1 and we hav e na tural maps of pointed sets (res pe c tively of groups ) π par i,j ( X , x ) → π i ( t olo ( X ) , t olo ( x )) and π ip er n,m ( X , x ) → π n + m ( t ( X ) , t olo ( x )) . 6. Some applica tions In this last section we are going to consider few applica tio ns of the theory devel- op ed so far. W e will b e gin with examples of complex spaces that are not C weakly equiv alen t to any complex hyper b o lic space. Deﬁnition 6 .1. We wil l say that a c omplex sp ac e is weakly hyper b o lic if is C we akl y e quivalent to a Br o dy hyp erb ol ic c omplex sp a c e. W e r ecall a preliminar y r e s ult (cfr. Le mma 2.15 [16]): Lemma 6.1. The p ointe d simplicial she af ( C \ 0) ∧ S 1 par is c anonic al ly we akly e quiv- alent to P 1 . Pro of. Consider the diagram D ( C \ { 0 } , { 1 } ) / /   ( C , { 1 } ) ( C \ { 0 } , { 1 } ) ∧ ∆[1] . (49) If D ′ is ano ther diag ram X f / / i   Y Z (50) in H then colim D ∼ = colim D ′ in H if ther e exists a morphism of dia g rams D → D ′ such that the morphisms are weak aﬃne equiv alences. Consider the diagr ams D ′ e EXTENDED HYPERBOLICITY 45 D ′′ ( C \ { 0 } , { 1 } ) / /   pt ( C \ { 0 } , { 1 } ) / /   ( C , { 1 } ) ( C \ { 0 } , { 1 } ) ∧ ∆[1] pt (51) and the morphisms f = (( C , { 1 } ) → pt , id ) : D → D ′ g = ( id , ( C \ { 0 } , { 1 } ) ∧ ∆[1] → pt ) : D → D ′′ . The mo rphisms f and g induce a ﬃne weak equiv alences c o lim D → colim D ′ and colim D → colim D ′′ . Identifying colim D ′ with ( C \ { 0 } , { 1 } ) ∧ S 1 s and colim D ′′ with C / ( C \ { 0 } ), we conclude that ( C \ { 0 } , { 1 } ) ∧ S 1 s ∼ = C / ( C \ { 0 } ) . The squar e C \ { 0 } / /   C   P 1 \ {∞} / / P 1 (52) is coca rtesian in F T ( S ), he nc e the coﬁbres of ho r izontal morphisms are is omorphic. W e der ive C / ( C \ { 0 } ) ∼ = P 1 / ( P 1 \ {∞} ) in F T ( S ). But P 1 / ( P 1 \ {∞} ) ∼ = P 1 in H , since P 1 \ {∞} ∼ = pt in H . Remark 6 .1. In the pro of o f Lemma 4.1 we hav e already seen that S 1 par is weakly equiv alen t to S 1 s . W e a re now go ing to apply the theor y develop ed so far to prove that Theorem 6.1. F or any n > 0 , P n is not we akly hyp erb olic. In other wor ds, Ip ( P n ) c annot b e r epr esente d in H by a Br o dy hyp erb olic c ompl ex sp ac e. Pro of. In view of Corolla ry 4.1, it is suﬃcient to show that 46 BOR GHESI AND TOMASSINI π par 2 , 1 ( P n , ∞ ) = Hom H • (( C \ { 0 } ) ∧ S 1 par , ( P n , {∞} )) 6 = 0 or equiv alent ly , by Lemma 6.1 and Remar k 6.1, that Hom H • ( P 1 , ( P n , {∞} )) 6 = 0 . Our candidate to represent a no nzero class is the ca no nical em b edding i : P 1 ֒ → P n . The top olog ical rea liz ation yields a group ho mo morphism t : π par 2 , 1 ( P n , ∞ ) → π 2 ( CP n , ∞ ) . t olo ( i ) : CP 1 ֒ → CP n is the canonica l inclusion and not nu ll homotopic, since CP n is obtained by CP 1 by a tta ching cells of dimensio n 4 a nd a b ov e, hence it is an equiv alence up to dimension 2 and in pa rticular t olo ( i ) ∗ : Z = π 2 ( CP 1 , ∞ ) → π 2 ( CP n , ∞ ) is an isomor phis m. In co nclusion t [ i ] 6 = 0, thus [ i ] 6 = 0 ∈ π par 2 , 1 ( P n , ∞ ).  Prop ositi o n 6 .1. L et X b e a c omplex sp ac e and p : e X → X a c onne cte d c overing c omple x sp ac e. Assume t hat X is we akly hyp erb olic and let f : C → X b e a nonc onstant holomorphi c function. Then for any lifting ˜ f of f t o e X , ˜ f ( C ) c ontains just one p oint in e a ch ﬁb er of p or e quivalently p | ˜ f ( C ) is a biholomorphism for any such f and ˜ f . Pro of. Let X be w eakly hyper b olic. Assume, by a contradiction, that there exist a nonconstant holomor phic function f : C → X and a lifting ˜ f : C → ˜ X such that a 6 = b ∈ p − 1 ( x ), x ∈ X , a, b ∈ ˜ f ( C ). F or the pur p o ses o f this pro of, we ca n ass ume that ˜ f (0) = a and ˜ f (1) = b . Then we hav e the following commutativ e dia gram: C ˜ f / / q   ˜ X p   C / { 0 } ∐ { 1 } α / / X (53) EXTENDED HYPERBOLICITY 47 where α sends the cla ss of { 0 } ∐ { 1 } to x ∈ X . W e hav e tha t [ α ] 6 = 0 ∈ π par 1 , 0 ( X, x ). Indeed, [ α top ] 6 = 0 ∈ π 1 ( X top , x ). Consider the comp osition [0 , 1 ] g → C / { 0 } ∐ { 1 } α top → X top , where g is a path from 0 to 1 in C . If α top ◦ g is no t homotopic to a constant relatively to { 0 , 1 } , then α top is not homotopic to a consta nt . But, by construction, α top ◦ g lifts uniquely to a pa th in e X top starting from a and ending in b , hence α top ◦ g cannot be homotopic to a cons tant relatively to { 0 , 1 } . This shows that π 1 ( X top , x ) 6 = 0 which is absurd since X is weak hyperb olic .  The Prop ositio n 6.1 in particular implies the following Corollary 6.1. A ny c omplex sp ac e X whose universal c overing s p ac e is C n for some n ≥ 1 , is not we akly hyp erb olic. Pro of. Let p : C n → X b e the universal covering of X . Let a 6 = b ∈ p − 1 ( x ), x ∈ X . A complex line l ⊂ C n passing throug h a, b provides a ho morphic map f : C → X which do e s not satisfy the conclusion of P rop osition 6 .1.  References [1] Bousﬁeld A. K. , Constructions of factorization systems i n categories. J. Pur e Appl. Alg. 9 (1977), 207-220. [2] Bousﬁeld A. K., Kahn D. M., Homotop y Limits, Completions and Lo calizations. L e ctur e Notes i n M athematics . Berlin, Springer-V erl ag. 304 . [3] Br o dy R., C ompact manifolds and hyperboli cit y . T r ans. Amer. Math. So c . 235 (1978) , 213- 219. [4] Campana F., Orbifolds, sp ecial v arieties and classiﬁcation theory , Ann. Inst. F ourier (Gr eno- ble) 54 (2004) 499–630. [5] Car tan H., Quotien ts of complex analytic spaces. 1960 Cont ributions to f unction theory (In ternat. Collo q. F unction Theory , Bomba y , 1960) pp. 1–15 T ata Institute of F undamen tal Researc h, Bomba y (Review er: R. C. Gunning) 32.49 (32.32). [6] D email ly J. P ., Lempert L. , Shiﬀman B., Al gebraic approximation of holomorphic maps from Stein domains to pro jectiv e manifolds, Duke Math. J. 76 (1994), 333-363. [7] Gabri el P ., Zisman M. Calculus of F r actions and Homotopy Theory . Berlin, Heidelberg, New Y ork: Springer V erlag, 1967. [8] Gr othendiec k A., Artin M., V erdier J.-L. Th ` eorie des topos et cohomologie ` etale des sch ` emas. (SGA4), L e ctur e Note s in Math . Heidelb erg-Springer. 26 9 , 270 , 305 (1972-1973) . [9] K oba ya shi S., Hyp erb olic Manifold and Holomorphyc Mappings. Mar c el Dekker , N ew Y ork, 1970. [10] Kobay ashi S. Hyp erb olic complex spaces. Springer- V erlag . 318 . (1998). [11] Jardine J. F. Simplicial preshea ves. J. Pur e Appl. Algebr a . 47 (1987), 35-87. [12] Lang S., Introduction to Complex Hyp erb ol ic Spaces. Springer-V erlag 1987. [13] Lang S., H yperb olic and Diophan tine Analysis, Bul l. AMS. 14 (1986), 95-118. 48 BOR GHESI AND TOMASSINI [14] Lang S. , Survey of Diophantine Geometry , Numb er The ory III , EMS 60 , Spri nger V erlag 1991. [15] M ilne J. ´ Etale Cohomology . Princ eton Universit y Pr ess . 33 . (1980). [16] M orel F., V o ev o dsky V. A 1 -Homotop y theory of sche mes. Publ. Math. IHES 90 , (1999) 45- 143. [17] Qui l len D. Homotopical Algebra. Le ctur e Notes in Math. Berlin, Springer-V erl ag. 43 (1973). Universit ` a degli Studi di Milano - Bicocca - Piazz a d ell ’A teneo Nuov o, 1 - 20126, Mil a n o, It al y e-mail: simon e.borg hesi@ unimib.it Scuola Normal e S u periore di Pisa, Piazz a d ei Ca v alieri 7, 56126 Pisa, It al y e-mail: g .tomas sini@ sns.it

Extended Hyperbolicity

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