Vector bundles on contractible smooth schemes

V ector bundles on contractible smo oth sc hemes Ara vind Asok Department of Mathematics Universit y of W ashington Seattle, W A 98195 asok@math. washington.e du Bren t Doran ∗ School of Mathematics Institute for Adv anced Study Princeton, N J 08540 doranb@mat h.ias.edu Abstract W e discuss algebr aic vector bundles on smo o th k - s chemes X con tractible from the stand- po int of A 1 -homotopy theo ry; when k = C , t he smo oth manifolds X ( C ) are contractible as top ologica l spaces. The integral algebraic K-theory and integral motivic cohomolog y of such schemes are that o f Sp ec k . O ne might hop e that furthermor e, and in analogy w ith the classification of top olo gical vector bundles on manifolds, alge br aic vector bundles on such schemes are all isomor phic to tr ivial bundles; this is almo s t certa inly true when the scheme is affine. How ev er, in the non-affine ca se this is false: we show that (ess e ntially) every smo o th A 1 -contractible strictly quasi- affine scheme that a dmits a U -torsor whose total spa ce is a ffine, for U a unipo ten t g roup, pos s esses a non-tr ivial vector bundle. Indeed we pro duce explicit arbitrar y dimensio nal families o f non-iso morphic such schemes, with each scheme in the fam- ily equipped with “as ma ny” (i.e., ar bitrary dimensional mo duli of ) non-isomor phic vector bundles, o f every sufficiently lar ge rank n , a s one desires ; neither the schemes nor the vector bundles o n them are dis tinguishable by algebr aic K- theory . W e also discus s the trivia lity of vector bundles for cer tain smo oth c omplex affine v a rieties whose underlying complex mani- folds are contractible, but that are no t neces sarily A 1 -contractible. 1 In tro duction In th is note, we stud y the set of isomorphism classes of ve ctor bun dles on smo oth k -schemes that are con tractible in the sense of A 1 -homotop y theory (as introduced in [MV99]); s u c h sc hemes will b e called A 1 -con tractible. W e wish to stress three coun ter-in tuitiv e p oin ts. First, as our main results show, there are lots of these, b oth of the sc hemes and of the b u ndles on a t ypical fi x ed suc h sc heme (see T heorem 1.2, and Corollaries 3.1 and 3.4). Second, they arise quite naturally and explicitly , s o should n ot b e considered p athologica l. Third, the standard cohomology theories (at least those theories repr esen table on the A 1 -homotop y category) are completely in s ensitiv e to th ese structures, and so are missin g a sur prising amount of algebro-geometric data. Regarding the third p oin t let us b e more sp ecific right from the start. Since m otivic cohomol- ogy is representa ble in th e A 1 -homotop y categ ory (see [V o e01] Theorem 2.3.1 1 ), A 1 -con tractible ∗ This material is based up on work su p p orted by the National Science F oun dation, agreement No. DMS-0111298. 1 The p roof of this fact requires, at th e moment, that k b e a p erfect field. 1 2 1 Introduction sc hemes ha v e the motivic cohomology of Sp ec k and so, for in stance, hav e n o non-trivial alge- braic cycles. Similarly , and more im p ortant ly for our present pu rp oses, since algebraic K-theory is r epresent able in the A 1 -homotop y catego ry (see [MV99] § 4 Th eorem 3.13), one kn o ws that the algebraic K -theory of an y A 1 -con tractible smo oth k -scheme is isomorphic to that of Sp ec k ; already from K 0 (Sp ec k ) ∼ = Z this imp lies that all vect or b undles are stably trivial. Giv en an A 1 -con tractible smo oth scheme X , it is th erefore n atural to ask wh ether all the v ector bund les on X are in fact trivial, esp ecially giv en that top ological v ector bund les on op en con tractible m anifolds are trivial. Indeed, recalling the Quillen-Su slin theorem for affine space (itself th e protot ypical smo oth A 1 -con tractible s cheme, and the on ly one kn o wn b efore [AD07]), one may view this as a generalized Serre problem. W e sho w there is a stark d ic hotom y b et w een the affine and str ictly quasi-affine cases: in the affine case, the answ er seems to b e y es, whereas in the qu asi-affine case w e p ro v e that the answe r is a resound ing no and construct explicit counte r- examples in abu ndance. What is esp ecially in teresting is that n one of the standard means f or distinguishing v ector bund les on a sc heme (e.g., Chern classes, algebraic K-theory , algebraic cycles) can pla y an y role at all; fr om th eir stand p oint, all the bun d les are ind istinguishable from a trivial b undle. On th e other hand , it follo ws from Corollary 4.2 th at there are mo duli of strictly quasi-affine surfaces wh ic h are not A 1 -con tractible (nor are the complex sur faces ev en contrac tible in th e sense of manifolds) and ye t they admit only trivial v ector bun dles. Th us h a ving non-trivial v ector bund les is b y n o means a n ecessary feature of b eing strictly q u asi-affine. Represen tabilit y prop erties of the functor “isomorphism classes of vector bundles” W e pu t the ab ov e discussion in a broader con text. Let S m k denote the category of separated, finite type, s mo oth schemes defined o v er k . The A 1 -homotop y category is constructed by embed- ding the catego ry S m k in a larger category of sp ac es , equipp ing that category with th e structure of a mo d el category and then formin g the asso ciated homotop y category . T he catego ry of sp aces is take n to b e the categ ory of simp licial Nisnevich shea v es on S m k . The homotop y category can b e formed by lo calizing along t wo classes of morphism s: fi rst, along the simplicial we ak e qu i va- lenc es and second along the A 1 -we ak e quivalenc es . W e refer the reader to ([MV99 ] § 2 T heorem 3.2) for precise details regarding this constru ction. Here and through th e remaind er of the p ap er [ · , · ] s and [ · , · ] A 1 will d enote the set of simplicial homotopy classes of maps and A 1 -homotop y classes of maps b et w een spaces. Let V ( · ) (resp . V n ( · )) d enote the functor that assigns to an ob ject X ∈ S m k the set of isomorphism classes of (ran k n ) lo cally f ree shea v es on X . Let B GL n denote the usu al sim p licial classifying sp ace defin ed in [MV99] § 4.1, then by ibid. § 4 Pr op osition 1.16, one kno ws that the s et of simp licial homotop y classes of maps [ X , B GL n ] s can b e iden tified w ith H 1 N is ( X, GL n ). Using a version of “Hilb er t’s Th eorem 90” (i.e., that GL n is a “sp ecial group ” in the sense of S erre), we kno w th at the last group is isomorphic to H 1 Z ar ( X, GL n ) whic h is, essentiall y by construction, isomorphic to V n ( X ). Ideally , one hop es that V n ( X ) d escends to a fun ctor on the A 1 -homotop y catego ry and is repr esen table by the sp ace B GL n , i.e., th at [ X , B GL n ] A 1 = V n ( X ). 3 1 Introduction P ositiv e results In the case n = 1 this ideal scenario is the realit y , w ithout restriction on X . Recall th at a s mo oth sc heme X is called A 1 -rigid (see [MV99] § 3 Example 2.4), if for any smo oth sc heme U , the map H om S m k ( U, X ) − → H om S m k ( U × A 1 , X ) indu ced by pullbac k along the p ro jection U × A 1 − → U is a bijection. Morel and V o ev od sky sho w that, s in ce the sheaf G m is A 1 -rigid, B G m = B GL 1 is in fact A 1 -lo cal (see ibid. § 3 Definition 2.1). Thus, [ X , B G m ] A 1 = [ X , B G m ] s and one concludes V 1 ( X ) = H 1 Z ar ( X, G m ) = [ X, B G m ] A 1 (see ibid. § 4 Prop osition 3.8). F urthermore, Morel argues (see [Mor] Theorem 3) th at if one restricts V n ( · ) to a functor on the catego ry of sm o oth affine sc hemes then again this ideal is r ealized, at least if n 6 = 2 (it is exp ected that n = 2 works as well, but the details remain to b e written out). Consequen tly , if X is an affine A 1 -con tractible smo oth k -sc heme, then ev ery v ector bund le on X (of rank n 6 = 2) is isomorp h ic to a trivial b u ndle. Negativ e results Unfortunately , for n ≥ 2, the f unctors V n ( X ) cannot descend to f unctors on the h omotop y catego ry withou t a r estriction on X : it has long b een kno wn (and wa s p ointed out to us by Morel) that ev en w ith X = P 1 , the canonical m ap V ( P 1 ) − → V ( P 1 × A 1 ) indu ced by pull-bac k via the p ro jection m orphism is not a bijection, as w e discuss in § 2. Observe that this means the space B GL n is not A 1 -lo cal for n > 1 ( cf. , [MV99] p . 138). Indeed, one can show that A 1 -lo calit y of B GL n is equiv alen t to the assertion that, for any smo oth sc heme X , and i = 0 , 1, the canonical map H i N is ( X, GL n ) − → H i N is ( X × A 1 , GL n ) is a bijection (com bine Pr op osition 1.16 of [MV99] § 4 and [Mor04] Lemma 3.2.1); Morel has called this latter cohomologica l condition on a group “strong A 1 -in v ariance.” Nev ertheless, Morel’s r esults might mak e one hop e that some form of homotop y inv ariance holds for the functor V n ( X ) for general n b ey ond the affine case. F or instance, p erh ap s any A 1 -w eak equiv alence of smo oth sc hemes f : X → Y where X is affine would in duce a b ijection f ∗ : V ( Y ) → V ( X ); when, in addition, Y is affine this is tru e by the discuss ion ab ov e. R emark 1.1 . In deed, by the Jouanolou-Thomason homotopy lemma (see e.g., [W ei89 ] Prop osition 4.4), give n an y sm o oth scheme Y admitting an ample family of line bun dles (e.g., a quasi- pro jectiv e v ariety) , there exists a smo oth affine sc heme X and a Zariski lo cally trivial smo oth morphism with fib er s isomorphic to affine spaces f : X − → Y . In particular, this morphism is an A 1 -w eak equiv alence, so th e ab o v e na ¨ ıv e hop e w ould redu ce the stud y of v ector b undles on suc h schemes to the case of affine v arieties! Un fortunately , Theorem 1.2 sho ws that this is false. Nev ertheless, Morel’s results com bined with the Jouanolou-Thomason homotop y lemma giv e the f ollo wing general picture. Sup p ose Y is a sm o oth sc heme o v er a p erfect field k (according to our con v en tions, this means Y is separated, r egular and Noetherian and th us admits an ample family of line bun dles). As long as n is a strictly p ositiv e in teger 6 = 2, then for any smo oth affine sc heme X that is A 1 -w eakly equiv alent to Y , we hav e a b ijection [ Y , B GL n ] A 1 ∼ = V n ( X ). Alternativ ely , h omotop y inv ariance might hold for a sligh tly b roader class of v arieties than affine ones, sa y f or quasi-affine sc hemes with “nice enough” affine closures. I n [AD07], u sing 4 1 Introduction tec h niques for studying unip oten t group actions d ev elop ed in [DK07 ], w e constructed many ex- amples in charact eristic 0 of non-isomorp h ic strictly qu asi-affine A 1 -con tractible smo oth sc hemes. Using this construction for arbitrary k , we will see that neither of the ab o v e generalizations are p ossible; it seems Morel’s r esu lts are in fact the strongest one can exp ect. F urtherm ore, w e will see that any attempt to qu an tify the lac k of homotop y inv ariance must accoun t for arbitrarily man y non-isomorphic ve ctor bu n dles. Sp ecifically , our main goal in this pap er is to pro v e the follo w ing result. Theorem 1.2. L e t k b e a field. Supp ose X is a finite typ e , smo oth, affine A 1 -c ontr actible k -scheme e quipp e d with a fr e e ev erywher e stable action of a split c onne cte d unip otent gr oup U . i) The quotient X/U exists as a smo oth A 1 -c ontr actible q u asi-affine scheme. ii) If X/U is affine, then for every p ositive inte g er n , the pul l- b ack map V n ( X/U ) − → V n ( X ) is a bije ction. iii) If X is isomorphic to affine sp ac e and X/U is affine, then every ve ctor bund le on X/U is isomorph ic to a trivial bund le. iv) If X/U is not affine, but admits a smo oth quasi-affine closur e with at le ast one c o dimension ≥ 2 b oundary c omp onent, then X/U admits non-trivial ve ctor bund les of r ank m f or al l sufficiently lar g e m . R emark 1.3 . One might susp ect that an y qu asi-affine sc heme th at is not affine has non-trivial v ector bu ndles, but this is false in general. I ndeed, one can show that all v ector bund les on th e complemen t of finitely man y p oin ts in A 2 are trivial. (W e will generalize th is fact in Corollary 4.2.) R emark 1.4 . The b oun dary comp onent condition in ( iv ) ab o v e is imp osed for ease of p ro of; it almost certainly can b e remo v ed, and is satisfied for instance wh en Sp ec k [ X ] U is s m o oth, w hic h is true in the generic case of our construction. W e ha v e asked (see [AD07 , Aso]), in analogy with the structur e theory of contrac tible manifolds, wh ether any smo oth A 1 -con tractible v ariet y can b e realized as suc h a quotien t of affine space by the free action of a unip oten t group . A p ositiv e solution to th is qu estion wo uld h a v e the follo wing consequence: r emoving the b ound ary comp onent condition in Theorem 1.2 implies the in teresting d ic hotom y that al l n on-affine smo oth A 1 -con tractible v arietie s ha v e a n on-trivial v ector bu ndle, wh ereas al l affine ones would only ha v e trivial vecto r bundles. In § 3, we will exp and on this theorem b y placing a lo w er b ound on “h ow many” n on-trivial v ector bund les su c h a qu asi-affine A 1 -con tractible v ariet y can ha v e, and thereby constru ct large dimensional families of examples with as many n on-trivial v ector b u ndles as one lik es, all in - distinguishable from the trivial bund le f rom the p oint of view of algebraic K-theory (or of any in v ariant representable in the A 1 -homotop y category). 5 1 Introduction Con tractible complex affine algebraic v arieties In addition, w e will visit the generalized Serre pr oblem as discussed in [Za ˘ ı99] § 8. W e will sa y that a scheme X ov er C is top ologically cont ractible if X ( C ) equipp ed with its usual stru cture of a complex manifold is contrac tible as a top ologica l space. Th e generalized Serre p roblem asks: if X is a smo oth complex affine algebraic v ariet y which is top ologicall y con tractible, then are all algebraic vecto r b undles on X isomorphic to trivial bundles? Note that if X is an A 1 - con tractible smo oth sc heme o ver C , then X is necessarily top ologically con tractible (see [AD07] Lemma 2.5). How eve r, not all top ologically con tractible complex v arieties are A 1 -con tractible (see [Aso ]); for example any top ologically cont ractible smo oth complex surface of log-general t yp e is not A 1 -con tractible (in fact suc h surfaces can b e sh o wn to b e A 1 -rigid ). W e will observe in § 4, putting together r esults of sev eral authors, that the generalize d Serr e problem is true for all top ologic ally con tractible smo oth complex v arieties of dimension ≤ 2; consequently there are p ositiv e dimens ional mo duli of smo oth sur faces that eac h admit only trivial v ector bun dles. Finally , we will present some examples of top ologically con tractible sm o oth complex 3-folds all of wh ose v ector bund les are isomorph ic to trivial bu ndles. Con v en tions and Definitions The word “sc heme” will mean separated sc heme, lo cally of fi n ite t yp e o v er a fi eld k . T he w ord “v ariet y” w ill mean redu ced, fi nite t yp e s c heme. A scheme X is called A 1 -con tractible if th e canonical morph ism X − → Sp ec k is an A 1 -w eak equiv alence in the sense of [MV99] § 3 Definition 2.1. A sc heme X is called q u asi-affine if there exists an affin e scheme ¯ X and an op en immersion X ֒ → ¯ X ; we will refer to qu asi-affine sc hemes that are not affine as strictly quasi-affine schemes. If X is an y sc heme, w e let V ec ( X ) denote the category of fin ite r ank lo cally fr ee O X -mo dules and, as ab o v e, V ( X ) will denote the set of isomorp h ism classes of ve ctor bu ndles on X . Throughout, U will denote a sp lit c onne cte d unip oten t k -group. Sp litness of U implies that U admits an increasing filtration by normal sub groups w ith sub-quotient s isomorphic to G a , and in particular that U is isomorphic to affine sp ace as a k -scheme. Observe that if c har( k ) > 0, then split un ip oten t groups c an hav e non-trivial fi n ite subgroup s (e.g., th e kernel of th e Ar tin -S c hreier morphism G a − → G a ). Actions of groups on sc hemes are alwa ys assumed to b e left actions; actions will b e called free if th ey are sc heme-theoreticall y fr ee, i.e., the action m orp hism is a closed immersion. If X is a scheme equ ipp ed with an action of U , then X/U will denote the geometric quotien t of U by X , if it exists as a sc heme. A U -torso r o v er a scheme X will b e a triple ( P , π , U ) consisting of a faithfully fl at, fin ite present ation morph ism π : P − → X , from a left U -sc heme P , suc h that the canonical morphism U × P − → P × P is an isomorphism on to P × X P . O bserve that in this situation, U acts freely on P (see [MFK94] Lemma 0.6) and X is a geometric quotien t of P by U . Our n otatio n and termin ology will follo w [AD07] un less otherwise men tioned. How eve r, the reader need not b e familiar with the r esults of ibid. , as long as she tak es on faith Theorems 3.10 and 4.11 therein: in essence, (a) there is a (computable) notion of an everywhere stable U -actio n on an affine s c heme X , (b) it is equ iv alen t to X b eing endo w ed w ith the structure of a U -torsor 6 2 V ector bundles and U -torsors o v er a qu asi-affine scheme X/U , and (c) in certain circumstances we can explicitly iden tify the complemen t of the op en imm ersion of X/U in Sp ec k [ X/U ] u sing geometric in v arian t theory . Ac kno w ledgemen ts W e wo uld lik e to thank F abien Morel for in teresting discussions around the topic of v ector bun dles in A 1 -homotop y theory , an d in particular for p ointing out the qu asi-pro jectiv e counter-e xample for rank 2 v ector bu n dles. W e wo uld also like to thank Jacob L u rie for a u seful conv ersation that help ed simp lify our p ro ofs. Both of these interact ions we re facilitated by a wo rkshop in T op ology at the Banff I nternational Researc h Station. Finally , we w ould lik e to thank th e referee for useful commen ts. 2 V ector bundles and U -torsors If q : X → X/U is a U -torsor, we observe the induced m ap q ∗ : V ( X/U ) → V ( X ) can ha v e ve ry differen t c haracter d ep endin g on wh ether X/U is affine or non-affine. The affin e and strictly quasi-affine cases will b e us ed to p r o v e T heorem 1.2. The affine case: q ∗ is a bijection Lemma 2.1. Supp ose q : X → X/U i s a U -torsor with X/U a smo oth, affine scheme. Then q induc es a bije ction q ∗ : V ( X/U ) ∼ − → V ( X ) . If in addition X is isomorphic to affine sp ac e, then every ve ctor b u nd le on X/U is isomorphic to a trivial bund le. Pr o of. Lindel prov ed (see [Lin82]) that if Y is a smo oth affine k -sc heme then p ullbac k via the pro jection map Y × A n − → Y induces a bijection V ( Y ) ∼ − → V ( Y × A n ). According to th e hypotheses, q : X − → X/U equip s the triple ( X, q , U ) with the stru cture of a U -torsor o v er X/U and X/U is affine. Observe th at for any affine sc h eme Y , H 1 ( Y , G a ) = H 1 ( Y , O Y ) = 0 b y [Gro61] Th´ eor ` eme 1.3.1. As U is split, an in ductiv e argument shows that H 1 ( Y , U ) = 0 for any su ch Y . Th us ( X , q , U ) m ust b e a trivial U -torsor ov er X/U , whence X ∼ = U × X/U . Thus, th e first r esult follo w s from the discussion of the p revious paragraph. 2 The fin al statemen t, wh er e X is assumed to b e affine space, no w follo w s from the Qu illen- Suslin theorem (see e.g., [Qui76]) that all ve ctor bun dles on affine space are isomorphic to trivial bund les. A quasi-pro jectiv e coun terexample: failure of surjectivit y Consider the p r o jection morphism p 1 : P 1 × A 1 − → P 1 . W e will sh o w that th e pu ll-bac k map p ∗ 1 : V ( P 1 ) − → V ( P 1 × A 1 ) is not a b ijection. By Grothendiec k’s description of the category 2 See the p ro of of Corollary 3.2 in the App end ix to [AD07] for details (note that b ecause of th e splitness assumption, th ere is no restriction on the b ase field) . 7 2 V ector bundles and U -torsors of v ector bundles on P 1 , we kno w that ev ery lo cally free sh eaf on P 1 is isomorph ic to a direct sum of rank 1 locally free sh ea v es. A vec tor bund le on P 1 × A 1 isomorphic to a p ull-bac k of a v ector bund le F on P 1 is n ecessarily isomorphic to the (external) tensor pro duct of F and O A 1 . There is a ran k 2 v ector bund le E on P 1 × A 1 whose restriction to P 1 × { 0 } is trivial and whose restriction to P 1 × { 1 } is isomorphic to O (1) ⊕ O ( − 1). This means E is n ot isomorphic to the pull-bac k of any b undle on P 1 . The strictly quasi-affine case: failure of injectivity No w assume q : X − → X/U is a U -torsor w ith X/U not affine. The existence of A 1 -con tractible strictly quasi-affine X/U will b e pr ov ed in § 3. More generally , for the rest of this subsection, and in p articular for the statemen ts of Lemmas 2.2, 2.3 and 2.4, we assu me we are in the follo w ing situation: i) X/U is an op en dense su bsc heme of a finite type smo oth sc heme X/U , w ith the inclusion denoted j : X/U ֒ → X/U , ii) w e denote by Z the closed complemen t of X/U in X/U equip p ed with the r educed in d uced sc heme structure and assume it is non-empty . In this situation, we hav e a lo calization s equ ence in G -theory (see [Sri96] Prop osition 5.15) : · · · − → G 1 ( X/U ) − → G 0 ( Z ) − → G 0 ( X/U ) − → G 0 ( X/U ) − → 0 . As b oth X/U and X/U are fi nite typ e sm o oth schemes, w e know by P oincar ´ e d ualit y (see [Sri96] § 5.6) that G i ( X/U ) ∼ = K i ( X/U ) an d G i ( X/U ) ∼ = K i ( X/U ). Since X/U is smo oth and A 1 -con tractible, it follo w s that K i (Sp ec k ) − → K i ( X/U ) is an isomorphism . Lemma 2.2. If X/U is A 1 -c ontr actible, the lo c alization se quenc e gives a short exact se quenc e 0 − → G 0 ( Z ) − → G 0 ( X/U ) − → Z − → 0 . Pr o of. Since X/U is A 1 -con tractible, it follo w s th at G 1 ( X/U ) ∼ = G 1 (Sp ec k ) ∼ = k ∗ , G 0 ( X/U ) ∼ = G 0 (Sp ec k ) ∼ = Z . W e just need to sho w th at the b oundary map G 1 ( X/U ) − → G 0 ( Z ) is trivial, or equiv alen tly , that the morphism G 1 ( X/U ) − → G 1 ( X/U ) is surj ectiv e. T o see this, observe that eac h pair ( V , α ) consisting of a v ector bun d le on X/U and an automorphism α of V represent s an element of G 1 ( X/U ). No w, the map G 1 ( X/U ) − → G 1 ( X/U ) is induced b y restriction. Since G 1 ( X/U ) ∼ = k ∗ , w e can repr esen t an y class in this group by a pair consisting of a tr ivial bundle and an automorphism corresp on d ing to multiplica tion by an elemen t of k ∗ . S uc h a p air can b e extended to giv e a class in G 1 ( X/U ). (This pro duces a splitting of the map G 1 ( X/U ) − → G 1 ( X/U ) by the canonical morph ism G 1 (Sp ec k ) − → G 1 ( X/U )). Lemma 2.3. If X/U is a smo oth A 1 -c ontr actible op e n dense su bscheme of a finite typ e smo oth scheme X/U , then ther e exists a non-trivial ve ctor bund le on X/U . 8 2 V ector bundles and U -torsors Pr o of. First, observe that G 0 ( Z ) is alwa ys non-trivial. Th us, using Lemma 2.2, the map j ∗ : K 0 ( X/U ) − → K 0 ( X/U ) alwa ys has a k ernel. In particular, K 0 ( X/U ) h as a generator w hic h is not isomorp hic to [ O X/U ]. Cho osing any ve ctor bun d le representing the isomorphism class of this n on-trivial generator giv es the result. No w un der th e hyp othesis that the b oun dary comp onen t is of co d imension at least t wo , non- isomorphic bu ndles on X/U w ill restrict to non-isomorphic bun dles on X/U . In deed, this follo ws b y the follo w in g “we ll-kno wn” resu lt ab out r estrictions of vect or b undles on n ormal v arieties. Lemma 2.4. Assume that the c omplement of X/U in X/U is of c o dimension at le ast two. Then the r estriction functor j ∗ : V ec ( X/U ) − → V ec ( X/U ) is ful ly-faithful. F urthermor e , the Pi c ar d gr oups of X/U and X/U ar e isomorphic. Pr o of. Since X/U is dense in X/U , the restriction fun ctor is faithful (an y morp hism is uniquely determined by r estriction to the generic p oin t). T o c hec k th at the fu nctor is fu ll, it suffices to sh o w that giv en any pair of lo cally free shea v es V 1 and V 2 on X/U , any morp hism ϕ | X/U : V 1 | X/U − → V 2 | X/U extends to a morp hism ϕ : V 1 − → V 2 . By assump tion, X/U h as complemen t of co d imension ≥ 2 in X/U , and X/U is smo oth and hence normal. Observe th at the canonical morp hism O X/U − → j ∗ O X/U is an isomorphism (since regular fun ctions on X/U extend to regular fun ctions on X/U b y normalit y). Giv en V i as ab ov e, w e can c ho ose an op en cov er U i of X/U on which V i trivialize. C onsider the ind uced op en co v er X/U ∩ U i of X/U . Any morp hism ϕ X/U : V 1 | X/U − → V 2 | X/U is sp ecified by a matrix of regular functions on th e X/U ∩ U i . By the extension p rop erty of regular fun ctions m en tioned ab ov e, this matrix extends u niquely to give a morph ism ϕ : V 1 | U i − → V 2 | U i ; furthermore, these m orphisms glue to give the required extension. In terpreting line bundles in terms of ˘ Cec h co cycles, the extension prop ert y of regular functions on sm o oth sc hemes sho ws that lin e b undles on X/U extend to line bun dles on X/U . R emark 2.5 . Note that an y quasi-affine v ariet y admits a canonical op en immersion in to the sp ectrum of its rin g of regular functions, wh ic h (b y defi nition of a geometric quotien t) here is isomorphic to S p ec( k [ X ] U ). Although this last scheme is affine by d efinition, it is we ll-kno wn (Hilb ert’s 14th Pr oblem) that it need n ot b e No etherian, though it is kno wn to b e lo cally of fin ite t yp e (wh ence our con v en tions). W e established in [AD07] Corollary 3.18 (iii) th at in fact the complemen t of X/U in S p ec( k [ X ] U ) consists of co dimension ≥ 2 affine s ubschemes. In particular, whenev er k [ X ] U is finitely generated, then X/U adm its co d im en sion ≥ 2 affine “closures”. The smo othness hyp othesis on the partial compactification we imp ose is for the tec hnical con v enience of id entifying K-theory with G -theory (via Lemma 2.3), and almost certainly could b e remo v ed. F or the general case, (when Sp ec k [ X ] U is neither smo oth nor finitely generated), one w ould ha v e to replace G -theory by Thomason’s K-theory (see [TT 90]). W e b eliev e all the lemmas, with the exception of Lemma 2.3, go through in this setting: one n eeds a more su b tle argument to extract a v ector bund le from a class in Thomason K -theory . In an y case, the ab o v e results hold for an y A 1 -con tractible s m o oth v ariet y Y that admits a smo oth partial compactificati on Y (w e assume Y is a v ariet y) suc h that Y \ Y has co d imension ≥ 2 in Y . 9 3 “Low er b ounds” on the failure of A 1 -inv ariance Pro of of Theorem 1.2 Pr o of of The or em 1.2 (i). S upp ose X is a smo oth affine A 1 -con tractible sc heme admitting a fr ee ev erywhere stable action of a unip oten t group U . Sin ce u nip otent groups in p ositiv e c haracteristic can hav e non -trivial fin ite sub groups, ev erywhere stabilit y implies prop ern ess of the action of U on X but n ot necessarily th at the action is free. Imp osing this ad d itional condition, the same pro of as that of T heorem 3.10 of [AD07] sh ows th at in this situation a q u otien t X/U exists as a quasi-affine smo oth sc heme (indeed, the pro of of lo c. cit. shows existence of such a quotien t is equiv alen t to the action b eing free and ev erywhere stable). No w, using § 3 Ex amp le 2.3 of [MV99], together with fact that U is a sp ecial group (i.e., all U -to rsors are Zarisk i lo cally trivial) we conclude that fur thermore X/U is an A 1 -con tractible smo oth sc heme (see also [AD07] Key Lemm a 3.3). Pr o of of The or em 1.2 (ii), (iii). Statemen ts (ii) and (iii) f ollo w immediately from Lemma 2.1. Pr o of of The or em 1.2 (iv). C om bining Lemmas 2.2, 2.3, and 2.4, we obtain the required non- trivial v ector bund le on X/U . W e can refin e this s tatemen t h ow ever. Note that P ic ( X/U ) is necessarily trivial by L emma 2.4, th us the non-trivial generator corresp onds to a ve ctor bundle of rank m ≥ 2. F urtherm ore, th e v ector bu ndle representing the non-trivial class on X/U is not stably trivial either so taking direct sums with th e trivial b u ndle pro duces non-trivial vecto r bund les on X/U of any rank ≥ m . Restricting these bu ndles to X/U pro d uces non-trivial v ector bund les of th at same rank. R emark 2.6 . T ak e X = A n . If a unip oten t group U acts freely and everywhere stably on X with a s trictly quasi-affine quotien t A n /U satisfying the h yp otheses T heorem 1.2, we kn o w there is a n on-trivial vecto r b undle on A n /U . Th e p u ll-bac k of this ve ctor bu n dle to A n is necessarily trivial, thus we s ee that pull-bac k by the qu otien t morphism do es not ind uce an injection on isomorphism classes of vec tor bu ndles, at least once we ha v e shown suc h an example exists. 3 “Lo w er b ounds” on the failure of A 1 -in v ariance W e must no w show that Theorem 1.2 d escrib es a large class of s chemes. W e are n ot trying to present a classification, so we simply supply a class of examples that work o v er an arbitrary fi eld. The basic idea is to rewrite the problem of fi nding U -torsors with A n as total space by “linearizing”, i.e., by restricting fr om a linear U -representat ion W to a U -inv arian t subv ariet y isomorphic to A n . Th e sim p lest cases to consider are G a -equiv arian t closed immersions of A n as h yp ersu rfaces in W , c hosen so that A n inherits the stru cture of a G a -torsor f rom a larger G a -torsor – namely an app ropriate op en sub s c heme W ′ ⊂ W . More sp ecifically , the geometric p oin ts of W ′ will b e “stable” p oin ts of W w ith tr ivial isotropy; suc h a set can b e exp licitly id entified with the help of a mo d ified Hilb ert-Mumford numerica l criterion from geometric in v ariant theory (GIT). F urthermore, giv en such a G a -equiv arian t closed immersion, we can ident ify the “b ound ary” lo cus (i.e., complemen t) of A n / G a in Sp ec( k [ A n ] G a ) again using geometric inv ariant theory . (F or 10 3 “Low er b ounds” on the failure of A 1 -inv ariance more details on this p oin t of view, w e refer the reader to [AD07] and [DK07].) With some care one can thereby arr ange that the conditions of Theorem 1.2 are satisfied. Let V denote the standard 2-dimensional repr esentati on of S L 2 . By abuse of n otation, we will write V instead of A ( V ), and furtherm ore, we c ho ose co ord inates u, v on V thr ou gh ou t and write 0 for the origin of V . W e em b ed G a ֒ → S L 2 as the su bgroup of lower triangular matrices. Recall that S L 2 / G a ∼ = V \ 0; th us V is an S L 2 -equiv arian t completion of S L 2 / G a , and w e can iden tify the iden tit y coset [ e ] in S L 2 / G a with { (0 , 1) } ∈ V . Via this em b edding G a ֒ → S L 2 , an arbitrary S L 2 -represen tation W can b e considered as a G a -represen tation b y restriction. An y giv en S L 2 -orbit in V × W is con tained either in 0 × W or the complement; if it is in the complemen t, th en it restricts to a G a -orbit in [ e ] × W . Similarly any S L 2 -in v arian t subs c heme of ( V \ 0) × W r estricts to a G a -in v arian t subsc heme of [ e ] × W . W e argued in [AD07] (Theorem 3.10, Lemma 4.5, and T heorem 4.11) , assu m ing k was of c haracteristic 0, us in g f aithfully flat descen t and the functorialit y for quasi-affine m aps in GIT, that if the geometric p oin ts of an S L 2 -in v arian t subsc heme Y in ( V \ 0) × W are stable for the S L 2 -action on V × W , then the corresp onding G a -in v arian t subscheme X in W is a G a -torsor o v er a quasi-affine v ariet y . F u rthermore we show ed the complemen t of X/ G a in Sp ec( k [ X ] G a ) is the GIT S L 2 -quotien t of the b ound ary of the closure Y of Y in V × W (i.e., the complemen t of the quotien t is the quotient of the complement ). Note that G a acts freely on X if and only if S L 2 acts freely on Y . As we explained in the p ro of of Theorem 1.2 (i), if furthermore Y w as con tained in the op en subsc heme of V × W wh ere S L 2 acts freely , the same result holds in arbitrary c haracteristic. Corollary 3.1. F or any inte g e rs n ≥ 4 , and any m ≥ 1 , ther e e xi sts a strictly quasi-affine A 1 -c ontr actible smo oth scheme of dimension n with at le ast m non-isomorphic, stably trivial, non-trival ve ctor bund les of every r ank l for sufficiently lar ge l . Pr o of. Let W = V ⊕ 3 with co ord inates { w 1 , . . . , w 6 } . Th e Hilb ert-Mumf ord numerical criterion applied to S L 2 -orbits in the S L 2 -represen tation V × W and then restricted to G a -orbits in [ e ] × W , implies, as ju s tified ab o v e, that all geometric p oints in the complement of the subscheme d efi ned b y { w 1 = 0 , w 3 = 0 , w 5 = 0 } are stable for th e G a -action; in particular, V × W \ { w 1 = 0 , w 3 = 0 , w 5 = 0 } / G a is a qu asi-affine geometric qu otien t. F or f + 1 a 1-v ariable p olynomial with no rep eated ro ots and constan t term 1, consider a G a - in v ariant hyp ersur f ace X giv en by the G a -in v arian t equation w 1 = 1 + f ( w 3 w 6 − w 4 w 5 ). Ob serv e first that by the pr eceding p aragraph all geometric p oint s of X are stable, since any non-stable p oint must satisfy w 1 = w 3 = w 5 = 0 which violates th e hyp ersurf ace equation b ecause 0 6 = 1. It follo w s th at the geometric qu asi-affine quotien t X/ G a exists. Second, X is isomorphic to A 5 via the closed immersion w 2 = z 1 , . . . , w 6 = z 5 , where { z 1 , . . . , z 5 } are co ordinates on A 5 . T hird, note that f ( w 3 w 6 − w 4 w 5 ) is S L 2 -in v arian t; it follo ws easily that the asso ciated hyp ersur face equation defining Y in V × W is uw 2 − v w 1 = 1 + f ( w 3 w 6 − w 4 w 5 ), wh ere ( u, v ) are the co ordin ates on the first factor of V . F ourth , again by the Hilb ert-Mumford criterion, all geometric p oint s of Y are stable with resp ect to the S L 2 -action on V × W , so in particular the S L 2 action on Y is p rop er, and the affine geometric qu otient Y /S L 2 exists. The b oundary B = Y \ Y of Y , equiv alentl y the closed su bsc heme of ¯ Y defined by the sim ultaneous v anishing of u and v , is explicitly giv en b y f ( w 3 w 6 − w 4 w 5 )+ 1 = 0 in 0 × W ; it is S L 2 - 11 3 “Low er b ounds” on the failure of A 1 -inv ariance in v ariant , co dimens ion 2 in Y , and its geometric p oints are all S L 2 -stable. As p er the discussion preceding this C orollary , this means the complemen t of X/ G a = A 5 / G a in Sp ec( k [ A 5 ] G a ) is given b y B /S L 2 , whic h is n ecessarily again cod imension 2. Observe that by the Jacobian criterion, the fact that f + 1 has no rep eated r o ots implies that b oth Y and B are smo oth schemes for an y k . Also, B has m disj oin t comp onent s, where m is the degree of f . Recall that an action is called set-theoretically free if its stabilizers at k -p oint s are trivial. F ur thermore, prop er, set-theoreticall y f ree actions are free ( cf. [AD07 ] Lemm a 3.11). A direct computation shows the only geometric p oin ts ha ving non-trivial stabilizers for the G a - action on W lie in the non-stable lo cus, so G a acts freely on X an d h ence S L 2 acts freely on Y . Also note that S L 2 acts freely on B in 0 × W ; indeed B is a finite disjoint u nion of rank 2 v ector bun d les o v er S L 2 . Consequently S L 2 acts not only prop erly but also set-theoretic ally freely , hen ce freely , on all of Y , so Y /S L 2 is smo oth. Denote the S L 2 -quotien t of the b ou n dary B by Z ; th en Z is smo oth and co d imension 2 in Y /S L 2 ∼ = Sp ec( k [ A 5 ] G a ). Indeed, Z in these examples is isomorph ic to a d isjoin t union of m copies of the affine plane. It follo ws that for any m ≥ 1, w e can choose f and h ence X so that K 0 ( Z ) is isomorp hic to Z ⊕ m . T h us Lemma 2.2 shows that K 0 (Sp ec k [ A 5 ] G a ) ∼ = Z ⊕ m +1 . As v ector bu ndles repr esen ting differen t classes in K 0 are not stably equ iv alen t, b y taking dir ect sums with trivial bu ndles w e get m non-trivial, n on-isomorphic b undles in eve ry su fficien tly large rank. Th en restriction, by Lemma 2.4, give s th e d esir ed bu ndles on A 5 / G a . Higher dimensional examples immediately follo w by taking other rep resen tations; for example, W = V ⊕ 3 ⊕ k r , w here k denotes the trivial representat ion and A 5+ r is pr esen ted as a hyp ersurf ace with the same equation as ab ov e. R emark 3.2 . W e exp ect it is p ossible to construct a smo oth quasi-affine 3-dimensional v ariet y with the desired prop erties via unip oten t quotients of an affine sp ace. How eve r, we do not b eliev e there are an y smo oth A 1 -con tractible surfaces other than A 2 (see also Remark 4.4 ); this is known to b e true o v er C (see [Aso ]). R emark 3.3 . The quasi-affine quotien t scheme in the simplest case of the construction fr om Corollary 3.1 (where f is the iden tit y so that A 5 is defined by w 1 = 1 + ( w 3 w 6 − w 4 w 5 )) is v ery p leasan t to visualize. An easy compu tation with inv arian ts pr esen ts S pec ( k [ X ] G a ) as a quadric h yp ersu rface in A 5 . When k = C this ma y b e though t of as the complexification of a sphere, that is, T ∗ ( S 4 ). Here B is a single affine plane: o v er C , B is the cotangen t plane at a p oint, so the complemen t of B is clearly contract ible as a complex manifold. Th is particular example of a quasi-affine cont ractible complex v ariet y , with a different presenta tion, was kno wn to Winkelmann [Win90]. In the other direction, giv en a d esired b ound ary we can often pic k the d efining hyp ersurface equation for X so as to yield a quotien t with that sp ecified b oundary . V aryin g the b oundary in a family ma y b e realized by v arying the defining hypers urface equation in the fixed G a - represent ation W . By arranging for a b ound ary Z w ith a large K 0 ( Z ), w e can then by the ab o v e pro cess get sm o oth A 1 -con tractible sc hemes with arbitrarily m any n on-isomorphic v ector bund les, and indeed find arb itrary dimensional families of su c h schemes. 12 3 “Low er b ounds” on the failure of A 1 -inv ariance Corollary 3.4. F or any inte gers n ≥ 6 , m ≥ 1 , and l ≥ 1 ther e exists an m -dimensional smo oth scheme S and a smo oth morphism f : X − → S of r elative dimension n whose fib ers ar e strictly quasi-affine A 1 -c ontr actible smo oth schemes, p air-wise non-isomorph ic, e ach of which p ossesses at le ast l -dimensional mo duli of stably trivial, non-trivial ve ctor bund les in every suitably lar ge r ank. Pr o of. W e use n otatio n and terminology as in the pro of of Corollary 3.1. Consid er W = V ⊕ 4 , with co ord inates { w 1 , . . . , w 8 } , as an S L 2 -represen tation and hence a G a -represen tation (wh ere G a ֒ → S L 2 as lo w er triangular matrices, as b efore). T hen the hyp ersurface w 1 = 1 + f ( w 3 w 6 − w 4 w 5 , w 3 w 8 − w 4 w 7 , w 5 w 8 − w 6 w 7 ) in W is isomorphic to A 7 ; the closed immersion is determined b y fun ction w 2 = z 1 , . . . , w 8 = z 7 , wh ere { z 1 , . . . , z 7 } are the co ordinates on A n . It is easily c hec k ed that the r estriction of the linear G a -action on W to this A 7 h yp ersu rface is everywhere stable, b y u sing S L 2 -stabilit y for V × W as b efore. Note that f ( w 3 w 6 − w 4 w 5 , w 3 w 8 − w 4 w 7 , w 5 w 8 − w 6 w 7 ) is an S L 2 -in v arian t, so the asso ciated h yp ersu rface equation defining Y in V × W is uw 2 − v w 1 = 1+ f ( w 3 w 6 − w 4 w 5 , w 3 w 8 − w 4 w 7 , w 5 w 8 − w 6 w 7 ). Over an y field k , for generic f this describ es a sm o oth hyp ersurface in the S L 2 -stable locus of V × W , all of whose p oints hav e trivial isotropy in S L 2 ; we lea v e the details to the reader. In particular for generic f th e S L 2 action on Y is fr ee, and the qu otien t Y /S L 2 is smo oth. Sin ce the b ound ary B is d efined b y the sim ultaneous v anishing of u and v , it is a hyp ersurf ace in 0 × W and so is co dimension 2 in Y . Indeed, B consists of a rank 2 vec tor bund le o v er a principal S L 2 -bundle o v er a smo oth affine surf ace. The quotien t Z = B /S L 2 is thus co d imension 2 and a smo oth subv ariet y of the sm o oth Y /S L 2 . C on s equen tly if Y 1 and Y 2 (resp ectiv ely , Z 1 and Z 2 ) are the S L 2 -in v arian t v arieties (resp ectiv ely , b ound aries of the quotien ts) asso ciated with t w o different c hoices of f , say f 1 and f 2 , then any morphism from Y 1 /S L 2 to Y 2 /S L 2 extends to a morphism from Y 1 /S L 2 to Y 2 /S L 2 and vice-v ers a; so Y 1 /S L 2 ∼ = Y 2 /S L 2 ⇒ Y 1 /S L 2 ∼ = Y 2 /S L 2 ⇒ Z 1 ∼ = Z 2 . In p articular, if Z 1 6 ∼ = Z 2 then Y 1 /S L 2 6 ∼ = Y 2 /S L 2 . Th us the fact that there are arb itrary dimensional mo d uli of the surfaces 1 + f ( x, y , z ) = 0, and hence of the b oundaries Z , means there are arbitrary dimen sional mo duli of Y /S L 2 ∼ = A 7 / G a asso ciated with v arying the G a -action ( cf. [AD07] L emm a 5.5)). Since Z ∼ = B /S L 2 is a vec tor bund le o v er a smo oth affine su rface S , the map K 0 ( Z ) − → K 0 ( S ) is an isomorp hism. F u rthermore, the smo oth affin e su rface is defined as a hyp ersurface in A 3 . So for example, if we tak e a hyper s urface isomorp hic to a p ro du ct of a smo oth affine curve and th e affine line (the reader ma y c hec k that a family of examples in an y gen us ma y b e c hosen s o that Y is smo oth and so that B is con tained in the op en su bsc heme of 0 × W on wh ic h S L 2 acts freely , th us guarantee ing Z is smo oth), w e see that K 0 ( Z ) can b e made arbitrarily large by making the gen us of the cur v e high: sp ecifically , line bundles are cancellatio n stable so there is an injection from P ic ( Z ) into K 0 ( Z ), and affine curv es hav e mo d uli of line bun d les of dimen sion increasing with the genus. Because ev erything is s m o oth, the same argum ent as in th e previous Corollary no w imp lies the desired s tatemen t for G a -quotien ts of A 7 ; qu otien ts for larger dimensional A n ma y b e ac hiev ed b y taking other r epresen tations, e.g., W ⊕ k r for k the trivial represent ation and the d efi ning equation th e same as ab o v e. 13 4 Some comments on the ge neralized Se rre problem 4 Some commen ts on the generalized Serre p roblem Prop osition 4.1. Supp ose X is a top olo gic al ly c ontr actible smo oth c omplex variety of dimension ≤ 2 , then every ve ctor bu nd le on X i s isomorphic to a trivial bund le. 3 Pr o of. If X is a top ologically con tractible smo oth complex curve, then X is isomorphic to the affine line and the result follo ws from the Quillen-Suslin theorem. Therefore, w e can assu m e that X has dimension 2. Sup p ose therefore that X is a top ologically contract ible smo oth complex surface. By a L emm a of F ujita (see e.g., [Za ˘ ı99] Lemm a 2.1), w e kn o w that any suc h surf ace is affine. By a Th eorem of Gurjar-Sh astri, (see [Za ˘ ı99] Theorem 2.1) w e know that an y top ologically con tractible smo oth complex surface is rational. In particular, X admits a sm o oth pr o jectiv e compactificatio n ¯ X wh ic h is a smo oth p ro jectiv e rational su rface. By the classification of s urfaces ¯ X is birationally equiv alen t to a ru led surface. Murthy (see [Mur69] T h eorem 3.2) h as shown that ev ery vec tor bun dle on an y affine surface birationally equiv alen t to a ru led s urface is necessarily isomorphic to the direct sum of a trivial bu ndle and a line b undle. Thus, if P ic ( X ) is trivial, it follo w s that ev ery v ector bun d le on X is isomorp hic to a trivial b undle. T o see that P ic ( X ) is trivial for a sm o oth con tractible surface, c ho ose a compactification of ¯ X wh ose b oundary is a simple n ormal crossin gs d ivisor D . W e hav e an exact sequence for Chow groups C H i ( D ) − → C H i ( ¯ X ) − → C H i ( X ) − → 0 . In particular, taking i = 1 and using the fact that ¯ X and X are smo oth, we see that P ic ( ¯ X ) − → P ic ( X ) is s urjectiv e. By C orollary 2.2 of [Za ˘ ı99], we kno w th at P ic ( ¯ X ) is freely generated by the irr educible comp onen ts of D and thus P ic ( X ) is necessarily trivial. Corollary 4.2. If X i s any top olo gic al ly c ontr actible smo oth c omplex algebr aic surfac e, and p 1 , . . . , p n ar e finitely many p oints on X , then al l ve ctor bund les on X \ { p 1 , . . . , p n } ar e trivial. Pr o of. Indeed, if F is a lo cally free sheaf on X \ { p 1 , . . . , p n } , then there alwa ys exists a coheren t extension ¯ F of F to X . T he double dual ¯ F ∨∨ is a reflexiv e sheaf on X , whic h m ust b e locally free s in ce X is a sm o oth su rface; this provides a lo cally free extension of F . W e h a v e j ust sho wn that all v ector bu ndles on such an X are in f act trivial, and thus F must b e a trivial bun dle as w ell. Corollary 4.3. Ther e ar e p ositive dimensional mo duli of smo oth algebr aic surfac es which admit only trivial ve ctor bund les. These c an b e chosen so that they ar e affine and, as c omplex manifolds, c ontr actible or quasi-affine and non-c ontr actible. Pr o of. There are con tractible smo oth affine algebraic su rfaces of log Ko d aira dimension 1 that admit deformations (see [FZ94] Ex amp le 6.9). Up on remo ving fi nitely man y p oin ts, Corollary 4.2 fi nishes the r esult. 3 Added in pro of: Prop osotion 4.1 can b e found in Corollary 2 of [GS89] with a similar pro of. 14 REFERENCES R emark 4.4 . None of the examples menti oned in the pro of of the Corollary are A 1 -con tractible. In fact, cont ractible smo oth sur f aces of p ositiv e log K o daira dimension are kn o wn not to b e A 1 -con tractible (see [Aso]). Roughly sp eaking, this is th e case b ecause p ositiv e log Ko daira dimension s u rfaces do not h a v e “many” rational curves; in order f or a v ariet y to ev en b e A 1 - connected, on e exp ects that it should b e co v ered by c hains of A 1 s. R emark 4.5 . F or top ologically contrac tible smo oth complex affine v arieties of d imension n ≥ 3, the functor X 7→ V ( X ) b ecomes ev en more su b tle. Resu lts of Suslin imp ly that for pr o jectiv e mo dules of rank ≥ n stable isomorph ism implies isomorphism. I n this dir ection, results of Murthy imply(see [Mu r 02] Corollary 2.11) that if f , g are elemen ts of th e p olynomial rin g C [ x 1 , . . . , x n ] with g 6 = 0, then all stably free mo dules o v er C [ x 1 , . . . , x n , f /g ] of rank ≥ n − 1 are free. If X is a top ologically contrac tible smo oth complex 3-fold, of this form, then all v ector bun dles on X are trivial if and only if P ic ( X ) is tr ivial. In particular, Murthy ([Mur02] Theorem 3.6) uses this to deduce that all the Koras-Russell th reefolds, in particular the famous Russell cubic su r face x + x 2 y + z 2 + t 3 = 0, s atisfy the generalized Serre problem. A t the m oment, it is not known whether or n ot the Ru ssell cub ic is A 1 -con tractible. References [AD07] A. Asok and B. Doran. On unip otent quotients and some A 1 -contractible smo oth schemes. I nt. M ath. R es. Pap. , 5, 2007. art. id. rpm005. 2 , 3, 4, 5, 6, 8, 9 , 10, 11, 12 [Aso] A. Asok. A 1 -homotopy typ es and con tractible smooth schemes. Pr eprint, av aila ble at http://www .math.washin gton.edu/ ~ asok/ . 4, 5, 11 , 14 [DK07] B. Doran and F. Kirwan. T ow ards non- reductive geometric inv ariant theory . Quart. J. of Pur e and App. Math , 3(1), 2007. 4 , 10 [FZ94] H. Flenner and M. Zaidenberg. Q -acyclic surfaces and their d eformations. In Classific ation of algebr ai c varieties (L’A quila, 1992) , volume 162 of Contemp. Math. , pages 143–2 08. Amer. 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