Affine Manifolds and Orbits of Algebraic Groups Author(s): William M. Goldman and Morris W. Hirsch Source: Transactions of the American Mathematical Society, Vol. 295, No. 1 (May, 1986), pp. 175-198 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2000152 Accessed: 17/06/2010 04:01
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http://www.jstor.org crossed homomorphismu: Aff(E) ) E, g g(0).
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 295, Number 1, May 1986
AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS1 WILLIAMM. GOLDMAN AND MORRIS W. HIRSCH Dedicated to the memory of Jacques Vey
ABSTRACT.This paper is the sequel to The radiance obstruction and par- allel forms on a;ffne manifolds (lYans. Amer. Math. Soc. 286 (1984), 629 649) which introduceda new family of secondarycharacteristic classes for affine structures on manifolds. The present paper utilizes the representation of these classes in Lie algebra cohomology and algebraic group cohomologyto deduce new results relating the geometric propertiesof a compact affine mani- fold Mn to the action on Rn Ofthe algebraichull A(r) of the affine holonomy group r c Aff(Rn). A main technical result of the paper is that if M has a nonzero cohomology class representedby a parallel k-form,then every orbit of A(r) has dimension > k. When M is compact, then A(r) acts transitively provided that M is complete or has parallel volume; the converse holds when r is nilpotent. A 4-dimensional subgroup of Aff(R3) is exhibited which does not contain the holonomy group of any compact affine 3-manifold. When M has solvableholonomy and is complete, then M must have parallel volume. Conversely,if M has parallel volume and is of the homotopy type of a solvmanifold, then M is complete. If M is a compact homogeneous affine manifoldor if M possesses a rational Riemannianmetric, then it is shown that the conditions of parallel volume and completeness are equivalent.
Thispaper is the sequelto ourprevious paper [22]. In that paperwe exploited certaincharacteristic classes (exterior powers of the radianceobstruction) to obtain relationshipsbetween various properties of affinemanifolds. Our results supported the conjecture(first made by L. Markus[36]) that a compactaffine manifold is completeif and only if it has parallelvolume. We shall refer to this as the main conjecture. Let M be a compactaffine manifold with developingmap dev:M > E and affineholonomy representation h: 7r > h(7r)= r c Aff(E). Here1r is the groupof decktransformations of the universalcover M, E is the vectorspace Rn, and Aff(E) is the groupof affineautomorphisms of E. The map dev is a locally affineimmersion which is equivariantrespecting h. The linearholonomy homb morphismis the compositionA: 1r > GL(E)of h with the naturalhomomorphism L:Aff(E) > GL(E). The obstructionto r fixinga pointin E is a 1-dimensional cohomologyclass CM E H1(M;Ex) with coefficientsin E twistedby A. It comes froma universalclass in the groupcohomology H1 (Aff(E); EL) whichcontains the
Received by the editors December 12, 1984 and, in revised form, May 29, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary57R99, 53C05;Secondary 53C10, 55R25. 1Research partially supportedby fellowshipsand grants from the National Science Foundation.
(a)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page
175 W. M. GOLDMAN AND M. W. HIRSCH 176 The resultsin our previouspaper were obtained by calculatingCM in various cohomologytheories singular,tech, andde Rham.In the presentwork we obtain furtherinformation by relatingCM to the cohomologyof the algebraichull A(r) of r in Aff(E). Ourgeneral theme is to relategeometrical properties of M to algebraic propertiesof A(r). Forexample the mainconjecture states that completeness,a geometricalproperty of M, is equivalentto the conditionthat A(r) preservevolume in E. Manyother propertiesof M can be describedin termsof A(r). Thus r is solvablewhen the algebraicgroup A(r) has the sameproperty. It turnsout that this conditionbrings us closerto affirmingthe mainconjecture. PROPOSITIONS. LetM be a compactaffine manifold with solvable holonomy groupr. Then: (a) If M is complete,then M has parallelvolume. (b) If M hasparallel volume and the solvable rank of r is less thanthe dimension of M, thenM is complete. (The solvablerank is the minimumsum of the ranksof the abelianquotients in a compositionseries. Forexample, the hypothesesof (b) are satisfiedif M is homotopy-equivalentto a solvmanifold.) Oneof the maincomponents of the proofof PropositionS is the followingresult: PROPOSITIONT. LetM be a compactaffine manifold. If M is complete,or if M has parallelvolume, then A(r) acts transitivelyon E. As an immediatecorollary we obtain PROPOSITIONF. If M is as in PropositionT and is connected,then every rationalfunction on M is constant. By a rationalfunction on M we meana functiondefined in a denseopen set, whichappears rational in everyaffine coordinate chart. Equivalently, such a func- tion correspondsvia the developingmap (fromthe universalcovering of M to E) to a rationalfunction on E whichis fixedby A(r). In a similarway one defines polynomialfunctions (and more generally tensors) on M. It is unknownwhether any compactaffine manifold can supporta nonconstantpolynomial function. In dimensionthree this cannothappen (D. Fried[9], W. Goldman[19]). Forcertain classes of affinemanifolds, transitivity of A(r) is sufficientfor com- pleteness. Theseinclude homogeneous affine manifolds, manifolds admitting ra- tionalRiemannian metrics, and, more generally, affine manifolds whose developing mapsare covering maps onto semialgebraic open sets. Ourmethods give a geometricproof of the followingresult of J. Helmstetter[25], whichis relatedto the mainconjecture: PROPOSITIONH. A left-invariantaffine structure on a Lie groupG is complete if andonly if everyright-invariant volume form on G is parallel. Ourresults also applyto manifoldswhich are not assumedto be completeor to haveparallel volume. The basic strategyis to derivea lowerbound for the dimensionsof the orbitsof A(r) fromknowledge of somecohomologically nontrivial exteriorform on the manifold;a parallelvolume form is a specialcase. Whenr is nilpotentwe use this methodto strengthensome of the resultsof Fried,Goldman andHirsch [13]: AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 177
PROPOSITIONN. LetAI be a compactaffine manifold with nilpotent holonomy groupr. Then: (a) Thehighest degree of a nonvanishingexterior power of the radianceobstruc- tion equalsthe smallestdimension of an orbitof A(r). (b) M is completeif andonly if A(r) acts transitivelyon E. (c) If M is incomplete,then the orbitsof A(r) havingsmallest dimension lie outsidethe developingimage (in fact thereis onlyone). It seemsreasonable to conjecturethat a similarpicture holds more generally, say for r solvable.This has been verifiedfor the specialcase that M has dimension threeand solvable fundamental group (Goldman [19]). By explicitexample we showthat condition(a) of PropositionN doesnot hold for arbitraryalgebraic subgroups of the affinegroup. In this way condition(a) canbe usedto excludecertain subgroups of Aff(E)from being holonomy groups of compactaffine manifolds. More precisely we exhibitalgebraic subgroups of Aff(E) whichact transitivelyon E anddo not containany such holonomy groups. The proofsof theseresults heavily use the structuretheory of nilpotentaffine groupswhich is developedin ourprevious paper [13]. The uniquesmallest-dimen- sionalorbit of the algebraichull is provedto be an affinesubspace Eu, coinciding with what we calledthe Fittingsubspace in [13]. This subspaceis characterized as the uniquer-invariant affine subspace upon which r acts unipotently(i.e. the linearpart L(r) c GL(E)is a groupof unipotentlinear transformations). The dimensionof Eu is a measureof completenessof the affinestructure, because the correspondingpower of the radianceobstruction is a cohomologicalinvariant whose nonvanishingexpresses an algebraiccondition akin to parallelvolume. We shall call the restrictionof r to Eu the Fittingcomponent of the holonomygroup of M. A necessarycondition that a unipotentaffine action of a finitelygenerated group G be a Fittingcomponent of the holonomyof a compactmanifold is that the actionbe syndeticon all of E, i.e. that thereexists a compactset K c E suchthat GK = E. Not everysyndetic unipotent affine action can be realizedas the Fittingcomponent of a nilpotentholonomy group of a compactaffine manifold: PROPOSITIONE. Let G be the subgroupof Aff(R3) comprisingthe following affineautomorphisms: 1 t u s O 1 2v t+v2 O 0 1 v (wherethe matricesrepresent respectively the linearand translationalparts of an affineautomorphism of R3). ThenG acts transitivelyon R3 and doesnot con- tain the Fittingcomponent of a nilpotentholonomy group of any compactaffine 3-manifold.In particularG doesnot containthe holonomygroup of any compact affine3-manifold. (It followsthat the finitelygenerated discrete subgroup consisting of integral matricesdoes not containthe holonomygroup of anysuch manifold.) The outlineof this paperis as follows.§1 is an expositionof the cohomology theoryof Lie algebras,Lie groupsand algebraicgroups as it relatesto affineac- tionsand radiance. The mainresult we use is a theoremof Hochschild[29] which phism,1.2 Letdenoted g beL: a Lieaff(E) algebra. gl(E),Any whosehomomorphism kernel consists p: g gl(E)precisely givesof E the the parallel struc-
W. M. GOLDMAN AND M. W. 178 HIRSCH identifiesthe algebraic cohomologyof an algebraicgroup as the spaceof (undera Levisubgroup) in the invariants Lie algebracohomology of its unipotentradical. Fromthis we deduceour main technical result: PROPOSITIONG. Let G be an algebraicsubgroup of Aff(E). For x E dimensionof theG-orbit of x E, the boundsfrom above the degrees of nonvanishing of the radianceobstruction of G. powers In an appendixto §1 we give an accountof left-invariant Lie groups. While affinestructures on most of the resultsdetailed there are knownin contextsby the workof moregeneral Helmstetterand others, our treatment is considerably geometric.We have included these more resultssince they provide beautiful examples of affinestructures, whose homogeneity makes them §2 particularlyunderstandable. resumesthe studyof compactaffine manifolds. we showhow UsingHochschild's theorem, parallelvolume implies the transitivityof the this we deduceseveral algebraichull. From corollaries:Markus' conjecture for affinestructures are homogeneousor have rational which Riemannianmetrics, nonexistence of rational functions,upper bounds on the degreesof we polynomialtensors, etc. The basicfact use is the followingcorollary of PropositionG: PROPOSITIONM. Let M be an affinemanifold. If thereexists a formon M whichhas nonzero parallelk- cohomologyclass, then every orbit of the actionof thealgebraic hull A(r) on E has dimension> k. §§3and 4 discuss affinemanifolds with solvableand nilpotent spectively.The partialresults on holonomyre- Markus'conjecture use a well-known(but not as well-documented)lemma on representationsof solvable in[11] (whichis groups,which may be found basedupon the treatmentin Raghunathan[42]). §4on nilpotentholonomy The resultsin dependon previousresults proved in [13 and23]. ACKNOWLEDGEMENTS.This paperhas its rootsin our collaborationwith D. Fried,which in turnis basedupon unpublished fulto workof J. Smillie.We are grate- them for manyhelpful conversations. Discussions C.Moore on with G. Hochschildand algebraicgroups have been crucialto this work. toJ. Helmstetter,and We are grateful the late J. Vey for helpfulsuggestions and for theirresults to us. Conversations describing with G. Levitton foliationshave also played an importantrole in the developmentof the ideashere. 1. Affine representations of Lie groups, groups. Lie algebras and algebraic 1.1As usualE is a finite-dimensionalreal vector space and GL(E) Aff(E))is the groupof all (respectively linear(resp. affine) automorphisms of E. TheLie gl(E)of GL(E)consists of all linear algebra vectorfields E aijxi@/0xj,while the Liealge- braaff(E) of Aff(E)consists of affine vectorfields ,(aijxi + bj)a/0xj (aij,bj E Thecanonical homomorphism L: R) . Aff(E) ) GL(E)induces a Liealgebra homomor- vectorfields E bj@/@xj.If X E aff(E) as above,then the translationalpart isthe unique parallel vector field u(X) whichagrees with X at 0, namelyE bj0/@xj. tureof a g-module.To any g-moduleE areassociated cohomology groups H* (SeeCartan and Eilenberg [6], (g, E) . Chevalleyand Eilenberg [7], and Koszul [33] for more 1.3 Let oe: g aff(E) be an affine representationof the Lie algebrag. The
AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROtJPS 179 detailson Lie algebracohomology.) If G is a Lie groupwith Lie algebra, then the cohomologyof g is the cohomologyof the complexof left-invariantdifferential formson G. In particularthe exterioralgebra A*(g*) on the dual of g may be identifiedwith the differentialgraded algebra of all left-invariantexterior forms on G; the resultingcohomology is H*(g). Nowsuppose that p:g ) gl(E) definesa g-moduleE (orEp). Considerthe trivialE-bundle over G witha left-invariantflat connectiongiven by Vp:E ) g* X E, Vp(v):x | ) p(x)(v)) wherev- E, x E g. This map extendsto a covariantdifferential operator from exteriorpforms with valuesin E to exterior(p + 1)-formswith valuesin E. The cohomologyH*(g,E) is definedas the cohomologyof the subcomplexcompris- ing left-invariantE-valued differential forms. One can identifythis complexwith A*(g)X E, therebyobtaining a differentialgraded structure on A*(g)X E, with the usualformula for the differential. compositionL o oeturns E into a g-moduleand u o oeis an E-valuedcocycle on , whereu: aff(E) ) E is the translationalpart (a Lie algebracocycle on aff(E)). It is easy to see that conjugatingoe by a translationdoes not changethe structure of E as a g-modulebut altersu o oeby a coboundary.Thus the cohomologyclass of u o oein H1(g;E) is invariantunder translational conjugacy and is calledthe radianceobstruction of oe,denoted ca E H1(g; E). Evidentlyca = Oprecisely when thereexists v E E suchthat oe(X)(v)= Ofor all X E g. In that caseoe is conjugate (by translationby v) to a linearrepresentation oe: 0 ) gl(E) c aff(E) andwe say that oeis radiant. 1.4 Let oe:g ) aff(E) be an affinerepresentation. We shall alwaysconsider aff(E)as a subalgebraof the Liealgebra gl(E@R) using the embeddingJ: aff(E) gl(E @ R) definedby
J(X)= L(o ) u(X)
whereX E aff(E). We defineOgJ to be the compositionof oewith J. (Forthe definitionof OgJin the caseof a group,see [22].) Weshall say that oeis reductiveif andonly if the associatedlinear representation O8J:g ) gl(E @ R) is fully reducible.(We makethe same definitionfor affine representationsof groups, semigroups, algebras, etc. Forexample if g is semisimple, thenany finite-dimensional affine representation is reductive.) The followinglemma is usefuland well known (see Milnor [40, 2.3]): LEMMA.(i) A reductiveaffi;ne representation is radiant. (ii) A radiantaffiine representation oe is reductiveif and onlyif its linearpart L o oeis fullyreducible (i.e. reductive). 1.5 Givenan affinerepresentation oe: g ) aff(E) we will considerthe exterior powersAk(c) E Hk(g;AkE) of its radianceobstruction. Any pairingB:E1 x E2 ) E3 of g-modulesinduces a pairingof the Lie algebracohomology groups HP(g;E1)x Hq(g;E2) ) HP+q(g;E3).If E is a g-moduleand c E H1(g;E) is a cohomologyclass (i.e. a radianceobstruction of an affinerepresentation), then W. M. GOLDMAN AND M. W. 180 HIRSCH we inductivelydefine An(c) to be the imageof (c,An-1 c) underthe cohomology pairingH1(g;E) x Hn-1(g;An-1E) ) Hn(g;AnE) inducedby the pairingof g-modulesE x An-1 E ) An E. 1.6Everything we have saidconcerning Lie algebras is an infinitesimal analogousstatements about Lie versionof groups.An alternativeapproach is to workin the differentiablecohomology of G (whichis the sameas of G) and the continuouscohomology then pass to the Lie algebraby differentiating is easierto compute cocycleson G. Sinceit with Lie algebracohomology, instead we define obstructionca of an affine the radiance representationca of a connectedLie group G to be the radianceobstruction of the correspondingaffine (the representationof its Lie algebra latterrepresentation being also denoted by oe). Thefollowing basic fact (whichmotivated this ofusing the section)illustrates the usefulness cohomologyof g to linearizequestions about G: THEOREM. Let (:X:G ) Aff(E) be an affinerepresentation of a connectedLie groupG. LetcO E H1(g;E) bethe radiance of obstructionof (:x.If oe(G)has an orbit dimension< k, thenAm cO= Ofor all m > k. PROOF. Sincethe cohomologyclass ca is invariant gation,we shall undertranslational conju- assumethat the G-orbitof Ohas dimension< denotethe evaluation k. Let fO:G ) E map of oeat 0, i.e. fO:g | ) (:x(g)(O).Then the dfoTeG ) ToE- E satisfiesdfo o j differential = u, whereu: g ) aff(E) is the translational partof oe:g ) aff(E) and j: g ) TeGis the isomorphismdefined by restricting left-inllariantvector fields on G to the identitye. Sincedimoe(G)(O) < k, the imageu(g) has represented dimension< k. Now Am(c) is by the cocycleAm(u) whichtakes values in zerosince dimu(g) = Amu(g), whichequals dimoe(G)(O)< k < m. HenceAm(u) = Oand inHk (g; Am (E)) . Q.E.D. Am(c) is zero 1.7 C OROLLARY. If n = dimE andAn (ca ) 78O, then cx . (G) acts transitivelyon 1.8Next we shall examinethe radianceobstruction and its powersas classesin a cohomologytheory cohomology foralgebraic groups. The cohomology theory isdue to Hochschild[29] (see also we use Mostow[41]), and a mainresult of [29]reduces itscalculation to Lie algebracohomology. Let G be a linearalgebraic group overR andsuppose that p:G ) GL(V)is an algebraicrepresentation of G on a realvector space V. arational (Inthat case we say that V is G-module.)Let g denotethe Liealgebra of G. ofGX denoted The algebraiccohomology Ha*lg(G;Vp) (or Ha*lg(G; V) whenp is understood) cohomologyof the is definedto be the complexof Eilenberg-MacLane cochains f: G x onG whichare regular G x x G ) V functionson G x G x x G. It is easyto see that the braiccochains form a subcomplexof alge- the analytic(resp. differentiable, continuous, Eilenberg-MacLane) cochains on G. Moreoverthe onG at derivativeof algebraiccocycle (e,e,...,e) givesaLie algebracocycleg x g giventhe x x g ) V, whereV is inducedg-module structure. In this way thereis whichinduces a a naturalchain map cohomologyhomomorphism Ha*lg(G; V) ) H*(g;V). It is wellknown (see e.g. Humphreys[31]) that everylinear algebraic group decomposesas a semidirectproduct G U > AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 181 largestunipotent normal subgroup of G and R is a maximalreductive subgroup of G (uniqueup to conjugacyin G). Weshall denote the unipotentradical of an algebraicgroup G by UG; it playsan essentialrole throughoutthis paper. If G is an algebraicsubgroup of Aff(E), then the reductivesubgroup R is a reductive groupof affinetransformations and is thereforeradiant and conjugate in Aff(E)to a fullyreducible linear representation. Thebasic result on calculationof algebraiccohomology is dueto Hochschild[29] whichwe summarizeas follows: THEOREM(HOCHSCHILD). LetG be a linearalgebraic gro?lp over R andV a rationalG-mod?>le. (i) If G is ?anipotent'then Halg(G; V) mapsisomorphically onto the Lie algebra cohomologyH* (g; V) . (ii) If G = U X R as above,then the restrictionmap Halg(G; V) ) Halg(U;V) whereE1 is the Lie algebraof U. 1.9 Weapply this cohomologytheory to affinerepresentations as follows.Let E be, as usual,a vectorspace. The group Aff(E) is a linearalgebraic group via the em- beddingJ: Aff(E) > GL(E@ R). Thetranslational part u: Aff(E) > E is a regular mapwhich is a cocyclewith values in the linearrepresentation L: Aff(E) ) GL(E); henceu is an algebraic1-cocycle on G with valuesin the Aff(E)-moduleE. More generally,if G c Aff(E)is an algebraicsubgroup, its translationalpart u: G ) E is an algebraiccocycle. We call its cohomologyclass [u]E H1lg(G;E) the (alge- braic)radiance obstr?%ction CGlgof G. By takingexterior powers we obtainalgebraic classesAk cGlg E Hklg(G;Ak E). It is clearthat underthe naturalhomomorphism Hklg(G;AkE) ) Hk(g;AkE) the algebraiccohomology class AkcGlgis mapped to the Lie algebracohomology class Ak cs,. THEOREM. Let G c Aff(E) be an algebraics?>bgro?>p. S?>ppose that A CGgE Hklg(G;Ak E) is nonzero.Then the ?>nipotentradical UG cannothave an orbitin E of dimension< k. (Notethat this theoremimplies that everyorbit of G in E musthave dimension > k. VVehave chosen to state this theoremin its strongerform.) PROOF. BY Hochschild'sTheorem 1.8(ii) the restrictionhomomorphism Hklg(G; A kE) ) H1Clg(UG; A kE) is injective;thus AkcUgG 7& 0. BY 1.8(i) the associatedLie algebracohomology classAk ca 7&0, where1 is the Lie algebraof UG. Nowapply 1.6. Q.E.D. 1.10 COROLLARY.Let G c Aff(E) be an algebraicsubgroup. S?lppose n = dimE. ThenAn CGg 7&0 impliesthat G acts transitivelyon E. In 02 theseresults will be appliedto compactaffine manifolds. 1.11 Equivalentconditions for transitivityof an algebraicagne group.One of the maintechniques of this paperis to deducetransitivity of an algebraicsubgroup of Aff(E)from various hypotheses. For this reasonwe collecthere a numberof 182 W. M. GOLDMAN AND M. W. HIRSCH conditionson groupr of affinetransformations equivalent to the transitivityof its algebraichull. A semialgebraicset is a nonemptyset definedby finitelymany polynomial equa- tionsand inequalities; if there are no inequalities,such a set is algebraic,or a variety. Alsonote that if G c Aff(E)is an algebraicsubgroup, then its complexificationGc is naturallycontained in the groupAffC(E @ C) of complex-affineautomorphisms of the complexifiedvector space E X C. THEOREM. Letr be a subgroupof Aff(E). Thenthe following conditions are equivalent: (a) Thealgebraic hall A(r) of r acts transitivelyon E; (b) Theanipotent radical uA(r) acts transitivelyon E; (c) r preservesno properalgebraic set; (d) r preservesno propersemialgebraic set; (e) Theanipotent radical uA(r) of the algebraichall of r preservesno proper semialgebraicset; (f) Thecomplexification Ac(r) acts transitivelyon E X C; (g) Thecomplexification uA(r)c of the anipotentradical of A(r) acts transi- tivelyon E X C. PROOF. The implications(b)>(a), (b)>(e), (d)>(c), and (g)$(f) are all ob- vious. Weprove (a)>(b). DecomposeA(r) = uA(r) > AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 183 orbitsbeing open. Choose x E E andlet J uA(r)> E andfC uAc(r) > EXC be the evaluationmaps at x for UA(r) and UAc(r), respectively.Clearly the differentialof fCat e E UAC(r) is the complexificationof the differentialof J at e E UA(r). Thus UA(r) acts transitively¢} df(e) is surjective¢} dfc(e) is surjective¢} uAC(r) acts transitivelyon EXC. The proofof 1.11 is now complete. Q.E.D. 1.12Dimensions of orbits and powers oJ the radianceobstruction. Theorems 1.6 and1.9 show a relationbetween the dimensionsof orbitsof an affineaction and the maximumnonvanishing power of the radianceobstruction. We may rewrite 1.6 as an inequality (*) max{k: A kC0 7&o} < min{dimGx: x E E} whichrelates an algebraicinvariant to a geometricinvariant of an affineaction. A naturalquestion is underwhat circumstances these two invariantsare equal. Theclass c0 is zeroprecisely when G has a stationarypoint, i.e. a O-dimensional orbit. Evidentlyif eitherside of the inequality(*) is zero, then both are zero. Moreover,if G has an orbitof dimension< 2, then (*) is actuallyequality. If G possessesa subgroupwhich acts simply transitively on E, thenequality holds in (*), i.e. /\nc0 7&0 wheren-dimE. We shallprove this in two steps. First, whenG acts simplytransitively, it will be provedin 01A.6below that An c0 7&O. Usingthis result,we provethat forany group G suchthat H c G c Aff(E),where Liealgebras, then /\n C0 = Oimplies An cb = i* /\n C0 = 0) a contradiction. Usinga similarargument, if G is a connectedk-dimensional subgroup of a simply transitiveaffine action H c Aff(E), then equalityholds in (*), i.e. /\kc0 7&O; howeverwe shallnot needthis result. In general,however, the inequality(*) willbe strict,even for transitive unipotent affineactions on R3. 1. 13 EXAMPLE.Let E = R3 andlet 0 be the Liesubalgebra of aff(E)consisting of affinemaps of the form O t v s O O u t . O O O u The correspondingLie groupG = exp(0)comprising affine maps 1 t v+te/2 s+(t2+ev)/2+t82/6) O 1 u t+82/2 O O 1 u acts transitivelyon E. Let ,r,v,v denoteelements of 0* correspondingto the variabless, t, a, and v. As Lie algebracochains (with real coefficients)we have dr = dv = O,dv = r/\v, d¢ = v/\v. SinceL(0) c sl(E), the 0-module/\3 E is one- dimensionalwith trivialaction of 0. Thusthe parallelvolume form determines an 0 > E definedin coordinatesby (s, t, a, v) § t (S, t, a). Thusthe 3-cochainff /\ r /\ v representsthe elementof H3(0) correspondingto /\3 c0. But ff /\ r /\ v = d(v /\ v) whence/\3 c0 = O. 19:immersion GG (g Eevp: G)is G Ean and automorphismthe action:Gof isthe Elocally structure. simplyIn thistransitive. case dev Thedetermines unique W. M. GOLDMAN AND M. W. HIRSCH 184 Weclaim A2 c0 7&O. The exterior2-form w = dx A dy + (y - z2/2)dy A dz is G- invariantand pulls back by the evaluationmap G > E to an algebraiccohomology class on G. Underthe Hochschildisomorphism 1.8(i) this cohomologyclass is representedby the Lie algebracocycle ff /\ v, whichis nonzero.Thus max{k: A kC0 + o} = 2 < 3 = min{dimGx:x E E} provingthat this inequalityis sharperthan inequality (*) of 1.12. 1A. Appendix: Left-invariantaffine structures on Lie groups. lA.1 Let G be an n-dimensional1-connected real Lie groupwith Lie algebra0 andlet E = Rn. An affinestructure on G determinesa developing map dev: G > E, uniqueup to compositionwith an elementof Aff(E). This affineimmersion serves as localaffine coordinates in anyopen set whereit is injective. 1A.2An affinestructure on G is left-invariant if eachleft-multiplication map a uniquehomomorphism oe:G > Aff(E)such that the diagram 19 t 1a(9) commutes. Lettinge E G be the identityelement, we see that devcoincides with the evalua- tionma? of oeat dev(e). Sincedev is an openmap, the developing image dev(G)is an openset. Evidentlyoe(G) acts transitively on dev(G),so that dev(G)is an open orbitof oe(G).Since dimG = dimE, the isotropygroup at any pointof dev(G) is discrete.We shallsay that an actionis locally simply transitive if thereexists an openorbit with discreteisotropy groups. Thus the actionoe arising from a left- invariantaffine structure is locallysimply transitive. Clearly the affinestructure on G is completeif andonly if oe(G)acts simplytransitively on E. Thusto everyleft-invariant affine structure on G therecorresponds an affine actionoe of G on E witha distinguishedopen orbit oe(G)p where the actionis locally simplytransitive. Conversely suppose p: G > Aff(E) is an affinerepresentation with dimG = dimE andp E E has an openorbit. Then evaluation of: at p is an affinestructure on G forwhich evp is a developingmap is left-invariant(because p is a homomorphismto Aff(E)). 1A.3Thus we mayidentify left-invariant affine structures on a Liegroup G with pairs(oe, Q), whereoe is a locallysimply transitive affine action of G and Q is an openorbit. Two left-invariantaffine structures are isomorphicif and only if the correspondingrepresentations are conjugatein Aff(E) by a conjugacytaking one orbitto another. Fix a left-invariantaffine structure on G anda correspondingpair (oe, Q). Com- mutativityof the diagramabove shows that left-invarianttensor fields on G corre- spondbijectively via dev to oe(G)invarianttensor fields on the openorbit U. In particularG has a left-invariantparallel volume form if and only if oe(G)is con- tainedin the special agne group SAff(E)comprising all volume-preservingaffine automorphismsof E. SimilarlyG has a left-invariantradiant vector field if and onlyif oe(G)fixes a pointof E. AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 185 If r c G is a discretesubgroup, then the spacer\G of rightcosets inherits an affinestructure from the left-invariantaffine structure on G and dev: G > E is a developingmap for the affinemanifold r\G. Any left-invarianttensor field on G descendsto a left-invarianttensor field on r\G. Let G be a Liegroup with a left-invariantaffine structure. By abuseof language we shallsay that G is radiant(respectively G has parallelvolume) in caseG has a left-invariantparallel volume form (resp. has a left-invariantradiant vector field). If oe:G > Aff(E)is an affinerepresentation which is locallysimply transitive at x E E, thenthe associatedaffine representation of Liealgebras, also denoted oe: 0 > aff(E),has the propertythat its translationalpart at x, givenby Y | > oe(Y)(x),is a linearisomorphism 0 > E . The imageoe(0) consists of affinevector fields on E. Thesevector fields integrate to givethe actionof G on E correspondingunder dev to left-multiplication;thus these vector fields correspond to right-invariantvector fields on G. In summary:on a Lie groupwith left-invariant agne structure,the right- invariantvector f elds are affinevector f elds. (On the otherhand, leJt-invariant vectorfields are generally not evenpolynomial vector fields; see [13].) 1A.4 Unlikeaffine structures on compactmanifolds, it is possiblefor a left- invariantaffine structure on a Liegroup to bothbe radiantand have parallel volume. Hereis the simplestexample. Let G1 be the subgroupof SL(2,R) comprising matricesof the form eS t O e-S fi ThenG1 acts simplytransitively on the half-plane{(x,y) E R2:y > 0} andthere- fore inheritsa left-invariantaffine structure, which is both radiantand volume- preserving. Thisexample also shows that in contrastto affinestructures on compactmani- folds,where it is conjecturedthat parallel volume implies completeness, it is possible for a left-invariantaffine structure to haveparallel volume and be incomplete.In the otherdirection, a left-invariantaffine structure which is completedoes not nec- essarilyhave parallel volume. Let G2 be the subgroupof Aff(R2)consisting of affinemaps of the form es 0 t 0 1 s s ThenG2 acts simply transitively on theentire plane and therefore the corresponding left-invariantaffine structure is complete;however every G2-invariant area form on R2 mustbe a constantmultiple of e-Y dx A dy so the left-invariantstructure does not haveparallel volume. Formore examples and a morethorough discussion of left-invariantaffine struc- tureson Lie groups,the readeris referredto Auslander[1], Boyom [4], Fried [10], Friedand Goldman[11], Fried,Goldman and Hirsch[13], Helmstetter[25, 26], Kim[56], Medina [38, 39], Milnor[40], and Vinberg [53]. 1A.5The followingtheorem is dueto J. Helmstetter[26]: THEOREM. LetG be a Lie groupwith left-invariant aff ne structure.Then the afWnestructure is completeif andonly if right-invariantvolume forms are parallel 0-moduleby R)i. Koszulproved that the cohomologyHn(0;R,) R. (Koszul W. M. 186 GOLDMAN AND M. W. HIRSCH COROLLARY. SupposeG is a unimodularLie groupwith left-invariant affine structure.Then this structureis completeif andonly if it has parallelvolume. Weshall give one and a halfproofs of Helmstetter'stheorem: one proofof the "if"assertion based on 1.6, anda moregeometric version of Helmstetter'soriginal proof. 1A.6Suppose that G is a Lie groupwith left-invariantaffine structure. Using 1.6,we showthat if right-invariantvolume forms are parallel, then G is complete. Wemay assumethat oe:G > Aff(E) is the affinerepresentation corresponding to left-multiplicationand that oeis locallysimply transitive at the origin0. Let WE denotea parallelvolume form on E; the hypothesisthat "right-invariantvolume forms areparallel" is equivalentto the assertionthat the volumeform WG = dev*wE is right-invarianton G. Wemust show that G acts simplytransitively on E. Since G alreadyacts locallysimply transitively on E, it will sufficeto showthat G acts transitively on E. By 1.6 it is enoughto showthat the top power/\n C; of the radianceobstruction is nonzero,where n = dimG. The proofthat /\n C,; 780 uses the classicalwork of Koszul[33] on Lie algebra cohomology.Koszul defines the analogof a fundamentalcohomology class in Lie algebra cohomology.Let n = dim0 and considerthe 0-module/\n 0 givenby the top exteriorpower of the adjointrepresentation. It is easy to see that this moduleis one-dimensionaland the representationof 0 on it is givenby the modular representationA: 0 > R, A(X) = tr ad(X). Weshall denote this one-dimensional goeson to usea generatorof this groupto definea Poincareduality in H*(0) butwe shallnot needthis.) Just as elementsof /\n 0* correspondto left-invariantvolume formson G, elementsof /\n 0* (8)R) correspondto right-invariantvolume forms on G. Koszul'stheorem asserts that a right-invariantvolume form has a nonzero cohomologyclass when suitably interpreted as a left-invariantform in a certainline bundle. To show that /\nc0 is nonzero,consider the pairingH°(0;/\nE* X R)i) x Hn(0; /\n E) > Hn(0;R)i) inducedby the coefficientpairing (/\n E* X R), x /\n E > R)i. Thenthe imageof (WE) /\n C0) is the cohomologyclass of the pull- backAJG in Hn(0;R)i). By Koszul'stheorem, this class is nonzero;hence /\n C0 7&O, whenceG acts transitively.Thus G has a completeaffine structure. 1A.7 One defectof this proofis that it givesno informationin the case that theaffine structure is incomplete.In Helmstetter'soriginal proof Theorem 1A.5 is deducedfrom a resultshowing a closerelationship between the growthof volume underright-multiplication and the geometryof the frontierof the developingimage. Ourefforts to understandfIelmstetter's proof led us to the 1. . followinggeometric elscusslon. Ourgeometric approach to Helmstetter'stheorems begins with the representa- tionof right-invariantvector fields as affinevector fields. Choose a basisX1 , . . ., Xn Eoe(0) of affinevector fields on E representinga basis of right-invariantvector fields onG. Thenthe exteriorproduct ,u = X1/\ /\Xn is a polynomialexterior n-vector fieldwhich represents a right-invariantvolume current on G. Thuswe maywrite ,u= f(x)a/@x1/\ /\ @/@xnwhere f: E > R is a polynomialof degree< n in theaffine coordinates (x1, . . ., xn) of E. It followsthat any right-invariantvolume formon G is a constantmultiple of f(x)-1dx1 /\ /\ dxn. °ec: GcAffC(E X C) is also locally simplytransitive at 0 (see 1.13). Suppose AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 187 Thefollowing theorem is Helmstetter'smain result from which he deduces1A.5: 1A.8 THEOREM. The developingmap dev: G > E is a coveringmap onto a connectedcomponent of the set {x E E: f(x) 7&O}. PROOF. Let 9G be a right-invariantRiemannian metric on G . Since9G is invariantunder the transitiveaction of G on itself by right-multiplications,the metric9G is complete. Furthermorethere is a symmetric2-form 9E on E with rationalcoefficients such that 9G = dev*gE.For let X1,. . ., Xn be an orthonormal basisof right-invariantvector fields; then the cometric9E dualto 9E is the tensor (X1)2+ + (Xn)2which is polynomialof degree< 2 in affinecoordinates, i.e. n g* = E gii(z)0/0Xi t3/t3X i,j=l whereeach gi; (X) is polynomialof degree< 2. Moreoverdet [gi; (X)] = X(x)2; and the matrixrepresenting 9E is [9ij (X)] = [9ij (X)] - 1 = det[9ij (X)] - 1adj [9i; (X)] and has rationalentries. Specifically, the symmetric2-form f(x)2gE(x) is a poly- nomialtensor field. Thus a right-invariantRiemannian metric on a Liegroup with left-invariantaffine structure must be rational(compare [20]). The developingmap dev: G > E is a localisometry from G to the opensubset Q of E where9E is Riemannianmetric. Since 9G is complete,dev is a coveringmap ontoa connectedcomponent of Q. Sincethe tensorfield f(x)2gE(x) is polynomial, it is everywheredefined. It followsthat if x E Aw,then either9E blowsup at x or 9E(X) is degenerate.In eithercase f(x) = O. Thusdev is a coveringmap onto a connectedcomponent of {x E E: f(x) 7z:O}. Thiscompletes the proofof 1A.8. lA.9 To deduce1A.5 from 1A.8 we proceedas follows. We first show that right-invariantparallel volume forms $ completeness.Observe that right-invariant volumeforms are parallelif and only if f(x) is constant.By 1A.8it followsthat dev is a coveringmap (and hence a diffeomorphism)onto E. HenceG is complete. In the conversedirection it will be usefulto complexify.Namely if oe:G > Aff(E) is locallysimply transitive at 0 E E, then the complexifiedrepresentation G acts simplytransitively on E. Thenby 1.11, Gc acts transitivelyon E X C. In particularGc must act simplytransitively on E X C. Applying1A.8 to the inducedcomplete left-invariant complex affine structure on Gc, we see that the polynomialfc: E X C > C has no zeros,and hence is constant.The proofof 1A.5 is complete. Q.E.D. 2. The algebraic hull of the holonomy group of a compact affine man- ifold. 2.1 Let M be a compactaffine manifold with a fixeduniversal covering p:M M, and let gr= gr1(M)be the groupof decktransformations. Let dev: M > E be a fixeddeveloping map with holonomyhomomorphism h: Tr > Aff(E). Let r = h(;r)c Aff(E) be the affineholonomy group and A(r) its algebraichull in Aff(E). Wedenote the unipotentradical of A(r) by UA(r). The mainresult of this sectionapplies Theorem 1.6 to affinestructures on com- pactmanifolds. andproperly discontinuous. M> Bgr >The Brmap BAff(E)6UA(r) E >is BAff(E).r-equivariant. Thereis a W. 188 M. GOLDMAN AND M. W. HIRSCH 2.2The affine tangent bundle Taff(M) of M (asdescribed in Goldmanand Hirsch [22])is classifiedby a sequenceof maps: (HereAff(E)6 denotes the groupAff(E) with the discretetopology.) Taff(M) is classiSedas a topologicallyflat bundleby the Srstmap, as a r-bundleby the next map,then as a flat affinebundle, and Snally as an affinebundle by the last map. 2.3 THEOREM(2.6 OF [22]). LetM be a compactaffine manifold which pos- sesses a parallelk-form S whichhas nonzerocohomology class [S] E Hk(M;R). Then/\ CM + ° Combiningthis theoremwith 1.6,we obtain 2.4 THEOREM. LetM be a compactaffine manifold which possesses a parallel k-formwhich has nonzerocohomology class in Hk(M: R). Thenevery orbit of uA(r) has an orbitof dimension> k. 2.5 COROLLARY.Let M be a compactaffine manifold with parallel volume. ThenA(r) acts transitively. (Observethat the transitivityof A(r) is equivalentto the transitivityof UA(r) andmany other algebraic and geometric conditions; see 1.11.) In 02.7Corollary 2.5 will be usedto proveseveral cases of the mainconjecture (that parallelvolume X completenessfor compactaffine manifolds). Before ex- ploringthe consequencesof 2.5, we observethat its conclusionis validunder the (conjecturallyequivalent) hypothesis of completeness. 2.6 THEOREM. LetM bea compactcomplete affine manifold. Then A(r) acts transitivelyon E. PROOF. DecomposeA(r) as a semidirectproduct uA(r) > UA(r)/H f E 1 1 r\ (uA(r)lH) ft Elr = M Thevertical arrows are covering maps and the mapf inducesan isomorphismon fundamentalgroups. Since UA(r)/H andE areboth contractible,it followsthat f' is a homotopyequivalence. Since both r\ (UA(r)/H) andM aremanifolds and Mis compact,f' is surjective.It followsthat dimUA(r)b= dimE. AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 189 Sinceevery orbit of a connectedunipotent affine action is closed(see Hochschild [30],Rosenlicht [43]), the orbitUA(r)b is openand closedin E and thus equals E. HenceUA(r), andequivalently A(r), acts transitivelyon E. Q.E.D. 2.7 Nowwe deducea fewimmediate corollaries of the transitivityof the algebraic hull. PROPOSITION.Let M be a compactaffine manifold having parallel volume. Thenthe developingimage dev(M) c E is not a propersemialgebraic set. PROOF. BY 2.5 A(r) acts transitivelyon E. By 1.11 it followsthat r pre- servesno propersemialgebraic set. Sincedev(M) is r-invariant,the proposition follows. Q.E.D. 2.8 COROLLARY.Suppose that M is a compactaffine manifold satisfying the hypothesis (**) dev:M ) E is a coveringmap onto a semialgebraicopen set. If M has parallelvolume, then M is complete. 2.9 REMARKS.(a) Wedo not knowif thereexists a compactincomplete affine manifoldwhose developing map is surjective.On the otherhand there are many examplesof compactaffine manifolds whose developing maps are not coverings. (SeeSmillie [45, 47], Sullivanand Thurston [49], and Goldman[14, 18] for these examples.)For example the productof anysurface of genus> 1 witha circleadmits an affinestructure whose developing map is not a coveringmap. Indeed if the gen;us is > 1, thenaffine structures exist whose developing images are the complementof theorigininR3 andwhoseholonomyisadensesubgroupofGL(3;R). In [13]it is shownthat if M is a compactincomplete affine manifold with nilpotent holonomy, thenthe developingmap is not surjective. (b) In generalthe developingimage is quitefar from being semialgebraic; in fact the boundaryof dev(M)need not be smoothor evenrectiSable. For example there exist convexcones Q c R3 coveringcompact affine 3-manifolds (homeomorphic to a productof a surfacewith a circle)whose boundaries are not c2 (Kacand Vinberg[32], Goldman[14]). In dimension4, there are domainsin R4 = c2 with nonrectiSableboundary, covering complex affine structures on S1 x T1(F), whereF is a compactsurface of genus> 1. (Both of these types of structures areaffine structures induced from projective structures; see Sullivanand Thurston [49],Benzecri [31], or Goldman[14, 18].) Thereare examples where the developing imagesare boundedby coneson circlebundles over limit sets of Kleiniangroups. SeeSullivan and Thurston [49] for more details. 2.10 THEOREM. LetM be a connectedcompact affine manifold. Assume that M is completeor hasparallel volume. Then: (i) M admitsno nonconstantrational function M ) R. (ii) Everyrational tensor feld of type(p,q) on M is polynomialof degree< (p+ q)Ann whereAn = (2n - 2) !/(2n-1 (n - 1)!) . polynomialexp:w:denotefunction2.12E PROOF. Jq{tensors the ELEMMA. and F:(i) mappinggroup Mg# onLetRThe denotes E} of such f: is compositionaffineE1M thatRa E2 the becompositionwhich automorphismsathemap rational inducedisdiagram g bijective0 ofep function onpolynomial G M the E M. and is tensor on Then such a M.polynomialmaps Aff(M) thatThere algebraand f-lexists is is by is a isomorphism polynomial. g polynomial. Lie aE rationalG. group Thus W. M. 190 GOLDMAN AND M. W. HIRSCH M f E pl 1 M F R commutes.The Euclideanclosure of eachlevel set F-l(c) whichmeets dev(M) is a r-invariantproper algebraic subset; this set is also A(r)-invariant.It fol- lowsfrom 2.5-2.6 and 1.11,however, that A(r) acts transitivelyon E; so this is impossible. Q.E.D. (ii) Let OM be a rational(p, q)-tensor on M. Thereis a uniquerational (p, q)- tensorfiJE such that dev*(E) = £ projectsto fiJM. SinceSE iS rationalit is UA(r)-invariant.Since UA(r) acts transitivelyon E by 2.5-2.6 it sufficesto provethe following: 2.11 LEMMA. Let G c Aff(E) be a unipotentsubgroup acting transitively on E. Thenevery G-invariant tensor fi;eld X of type (p,q) ts polynomialof degree < (p + q)An PROOF. Let g c aS(E) be the Lie algebraof G andlet b be the Lie algebraof the isotropygroup H = G n GL(E).Let Jqbe anylinear subspace complementary to b in 0, i.e. b d3¢4 = g. Recallthat a polynomialisomorphism of vectorspaces is a Theproof of the followinglemma is identicalto the proofin the case G simply transitive(Theorem 8.3 of [13])and is thereforeomitted. ok > E of degree< An By Lemma2.12, £(X) = (eXPf(X))#SOXwhere f: E ) A is the inverseto u o Sincedeg f < Anit followsthat degx < (p + q)An. Q.E.D. 2.13 Homogeneousaffine structures. Let M be an affinemanifold and let Aff(M) ofdiffeornorphisms of M. Wesay M is homogeneousif Aff(M) acts transitively on M. Forexample, a Lie groupwith left-invariantaffine structure is a homogeneous affinemanifold. Homogeneous aSne manifoldshave been studied by Koszul[34, 35],Matsushima [37], Shima [44], Yagi [54] and others. Weshall prove the mainconjecture for compact homogeneous aine manifolds: THEOREM. LetM bea compacthomogeneous affine manifold. Then M is com- pleteif andonly if M has parallelvolume. 2.14The firststep in the proofof Theorem2.13 is the followinglemma: LEMMA. Let M be a homogeneouscomplete affine manifold. Then the atfine holonomyr is unipotentand M has parallelvolume. PROOF. SinceM is complete,M E/r wherer C Aff(E) is a discretesub- group. Aff(M)is identifiedwith Nlr whereN c Aff(E) is the normalizerof r. proofsimmersionare dev: available.) ME (any It coveringfollows thatspace dev: of Ma Ehomogeneous is a coveringaffine map ontomanifold an openis AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 191 Sincer is discretethe identitycomponent N° centralizesr. It is easy to see that homogeneityof M impliesthat N° is transitiveon E. Let eyE r be any element. Sinceey is a decktransformation it has no Sxedpoint, which implies that 1 is an eigenvalueof the linearpart of ey.Therefore the Fittingcomponent of eyis nontriv- ial. (The Fittingcomponent is the largestey-invariant affine subspace upon which ? acts unipotently.)The centralizerof eyin Aff(E)preserves Eu; thereforeEu is invariantunder N°. But N° is transitive,so Eu = E. Q.E.D. REMARKS.The sameproof shows that if M is a completeaffine manifold such that the centralizerof r in Aff(E)leaves invariant no properaffine subspace (i.e. r acts irreduciblyon E), then r is unipotent.Using a similartechnique to the proof of 2.14, Auslander[1] provesthat the groupAff(M)°, where M is as above,is a nilpotentLie group. 2.15 PROPOSITION.Let M be a homogeneousaffine manifold. Then its devel- opingmap is a coveringonto a semialgebraicopen set. Theorem2.13 follows from Proposition 2.15, together with Corollary2.8. PROOF OF PROPOSITION2.15. Pass to the universalcovering space of M in orderto assumethat M has a homogeneousaffine structure induced by an still homogeneous).If g E Aff(M) then g o dev is anotherdeveloping map for M so thereexists (9) E AS(E) so that dev o g = (9) o dev. It is easy to see that : Aff(M) ) Aff(E)is a homomorphismwith respectto whichdev is equivariant. SinceAff(M) acts transitivelyon M, the Aff(M)-equivariantimmersion dev is a coveringmap onto its image.The developing image dev(M) is an openorbit of the identitycomponent G of (Aff(M)) andfor any connectedsubgroup G c Aff(E), the unionof openG-orbits is a Zariski-opensubset of E. (Fora proofof this fact, let g c aff(E) be the Lie algebraof G and for X E g let L(X) and ll(X) be the linearpart and the translationalpart of X, respectively.Then the differential of the evaluationmap of G at y E E is the linearmap uy: 0 ) E givenby X L(X)y+u(X). Thecondition that y E E haveG-orbit of dimension< k is precisely the conditionthat uy haverank < k. This is evidentlya polynomialcondition on y.) Thus the unionof open G-orbitsin E is Zariskiopen and in particular, sincea connectedcomponent of a Zariskiopen set is semialgebraic,any one open G-orbitis semialgebraic.(Indeed, a connectedcomponent of a semialgebraicset is itself semialgebraic;one may provethis using the existenceof semialgebraic triangulationsof semialgebraicsets (see Hironaka[58]) although more elementary semialgebraicset. Q.E.D. 2.16 RationalRiemanniarl metrics. Let M be an affinemanifold. A rational Riemannianmetric on M is a Riemannianmetric (deSned everywhere on M) whose coefficientsin local affinecoordinates are locallydeSned rational functions. For furtherdiscussion and examples of rationalRiemannian metrics, see Goldmanand Hirsch[20]. THEOREM. Let M be a compactafWne manifold with a rationalRiemannian metric9. Thefollowing conditions are equivalent: (a) M is complete; (b) M has parallelvolume; 192 W. M. GOLDMAN AND M. W. HIRSCH (c) g is a polynomialRiemannian metric; (d) M is finitelycovered by a completeaffine nilmanifold. PROOF. Suppose9M is a rationalRiemannian metric on M. Thereexists a r-invariantrational 2-tensor 9E on E suchthat dev*gE= P*9M (wherep: M ) M is the projection).Evidently 9E is invariantunder A(r), because9E is 0-invariant and rational. SinceA(r) acts transitivelyon E, it followsby 2.10(ii)that 9 is polynomial.Thus (a)r(c) . Since9E is deSnedon a Zariskiopen subset Q c E, it followsfrom Proposition 3 of [20] that dev: M ) Q is a coveringonto a connectedcomponent Q0. By 2.15,Q0 is semialgebraic;therefore 2.8 showsthat (b)>(a). By [13, Theorem8.4], (d)>(a). By [20, Theorem1], (c)X(d); and (d)>(a) is clear. Q.E.D. 3. Parallel volume and completeness for affine manifolds with solvable holonomy. 3.1 One obtainsinteresting classes of affinemanifolds by imposingalgebraic assumptionson the affineholonomy group r. A naturalassumption is that r be solvable,owing to the abundanceof examples:all knowncomplete affine manifolds have virtuallysolvable holonomy group. Manyinteresting compact incomplete affinemanifolds have solvable holonomy; see [13, 16, 19] forexamples. The equivalenceof parallelvolume and completenesswas provedin [13] for compactaffine manifolds with nilpotentholonomy; and it immediatelyfollows for virtuallynilpotent holonomy as well. We provethat for manifoldswith virtually solvableholonomy, completeness > parallelvolume, but we knowonly partialre- sultsin the otherdirection. 3.2 THEOREM. Let M be a compactcomplete affine manifold. Suppose that X = 1r1(M)is virtuallysolvable. Then M has parallelvolume. PROOF. BY [11, 1.5],M has a Snitecovering M whichis affinelyequivalent to a completeaffine solvmanifold, i.e. an affinemanifold of the formr\G where G is a Lie groupwith left-invariantcomplete affine structure and r c G a lattlce subgroup.Equivalently M = Elr wherer c G c Aff(E) with G actingsimply transitivelyon E. SinceG admitsa lattice,it is unimodular,so by Corollary1A.5 G is volume-preserving.Thus r is volume-preservingand M has parallelvolume. It followsthat M itselfhas parallelvolume. Q.E.D. 3.3 COROLLARY.Let M be a compact3-dimensional affine manifold which is complete.Then M has parallelvolume. PROOF. Theproof follows from 3.2 and [11, 2.12]where it is shownthat 7r1(M) is virtuallysolvable. Q.E.D. 3.4 Now we turn to the conversestatement. Let r be a solvablegroup and considerthe derivedseries r = rl D r2 D * * * D rs = {1}, whererk+1 is the derived groupof rk. The rankof r is deSnedto equalthe sum s-1 E dimR(R X ri/ri+l) i=l whichmay be infinite(even if r is Snitelygenerated). If r is polycyclic(see Milnor [40]or Raghunathan[42] for basicproperties of polycyclicgroups), then eachri projectionisr Aff(E)LEMMA.reductive.factors Let map p Let r4> throughGL(E) @:A(r)A(r) be)a RsolvableuA(r)a belinear the by Lieprojection representatzong g-1@(9)* groupof homomorphism dimensionOne of can a solvableprove< dimand (followingM. define group ther. AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 193 is finitelygenerated and rank(r)is finite. Furthermorerank(r) equals the rank of r as a polycyclicgroup, which is the virtualcohomological dimension of r. Solvableholonomy groups of completeaffine manifolds are known to be polycyclic, but Smillie(unpublished) has constructedcompact affine manifolds with solvable but not virtuallypolycyclic holonomy. 3.5 THEOREM.Let M bea compactafWne manifold with affne holonomygro?lp r c Aff(E)which is solvableand rank(r) < dimM. If M hasparallel volume, then M is complete. REMARK.It followsfrom [11, §1],that M is finitelycovered by a solvmanifold. The conditionrank(r) < dimM impliesthat the affineholonomy homomorphism 3.6 Thefollowing lemma is provedin [11, 01](which is basedupon Raghunathan [42, 4.28]). Thereexists a solvablesubgroup G c GL(V)having finitely many connected com- ponentssuch that p(r) c G c A(r) anddim(G) < rank(r). 3.7 PROOF OF 3.5. ApplyLemma 3.6 to the inclusionr c A(r) to obtaina subgroupG with r c G c A(r) with dimG < rank(r)< dim(E). Weclaim that G c Aff(E)acts simplytransitively on E. DecomposeA(r) = A(G) in the usualway, A(r) = uA(r) X R, whereR Raghunathan[42, Theorem4.28]) that for a connectedsolvable Lie groupG con- sistingof matrices,4> mapsG ontoUA(G). SinceuA(r) = UA(G), we see that 4S mapsG ontouA(r). SinceM is compactwith parallelvolume, 2.5 impliesthat UA(r) acts transi- tively.We now show G actstransitively: let y E E be a fixedpoint of the reductive groupR; since q>(g)y= g-ly for g E G, Gy = uA(r)y = E. It followsthat dimG = dimE. SinceE is simplyconnected, each isotropy group is trivial.Thus G acts simplytransitively. It is easy to see that M must now be complete. For example,we may find a G-invariant,and hence r-invariant, Riemarlnian metric 9E on E whichdefines a Riemannianmetric 9M whichis complete.Since dev: M ) E is a localisometry be- tweenP*9M and9E, it is a coveringmap onto E. ThereforeM iS complete. Q.E.D. 3.8 COROLLARY.Let M be a compactagne manifoldwhich is asphericaland has virtuallypolycyclic fundamental group (for example,if M is homeomorphicto a solvmanifold).Then M is completeif andonly if it has parallelvol?lme. PROOF. Thehypotheses on M implythat 7ris virtuallypolycyclic of rankequal to dimE. 3.8 followsimmediately from 3.2 and3.5. Q.E.D. Thereare manyinteresting examples of compactaffine manifolds with solvable holonomywhich are convex,i.e. dev: M ) E is a bijectiononto a convexopen subsetof E; see Koszul[34, 35], Goldman[16], and Vey [50, 5l, 52] fordiscussion. 3.8 impliesthat parallelvolume is equivalentto completenessfor this classof affine structures. 194 W. M. GOLDMAN AND M. W. HIRSCH 3.9 COROLLARY.Let M3 bea compact3-dimensional affine manifold with vir- tuallysolvable fundamental group. Then M is completeif andonly if it hasparallel volume. PROOF. It is provedin Evansand Moser [8] that 7r1(M)is virtuallysolvable of rank< 3. Nowapply 3.2 and3.5. Q.E.D. 4. Affine manifolds with nilpotent holonomy. 4.1 In whatfollows M will be a compactaffine manifold whose affine holonomy groupr is nilpotent.The class of affinemanifolds with nilpotent holonomy is a rich classof structures(see [13]and Smillie[45, 47, 48]). The equivalenceof parallel volumeand completenessis provedin [13],as well as severalother conditions for this classof manifolds.Combining this with resultsproved above we obtain THEOREM. Let M be a compactn-dimensional affine manifold with nilpotent affineholonomy group r c Aff(E) anddeveloping map dev: M ) E. Thefollowing conditionsare equivalent: (a) M is complete; (b) M has parallelvolume; (c) dev is surjective; (d) the affineaction of r on E is irreducible; (e) r is unipotent; (f) M is a completeaffine nilmanifold; (g) I\ CM + O; (h) A(r) acts transitively. PROOF. The equivalenceof (a), (b), (c), (d), (e), and (f) was provedin [13]. The implication(b)a(g) (whichholds without nilpotence) follows from Theorem 2.3, and (g)a(h) by 1.10. Weprove the contrapositiveof (h)a(d). Supposethat r is reducible,that is, r leavesinvariant an affinesubspace F c E. The affine subspaceF is A(r)-invariant,contradicting transitivity of A(r). Q.E.D. 4.2 The mainresult of this sectionis that inequality(*) of 1.12is an equality forthe algebraichull of a nilpotentholonomy group of a compactaffine manifold: THEOREM. LetM bea compactaffine manifold with nilpotent affine holonomy groupr c Aff(E). Thenthe largestpower k suchthat Ak CM is nonzeroequals the minimumdimension of an A(r)-orbitin E. Furthermoreif M is incomplete,the uniquek-dimensional orbit lies outsidethe developingimage. We conjecturethat Theorem4.2 holdsmore generally, e.g. for compactaffine manifoldswith solvableholonomy. 4.3 Weshall deduce 4.2 by showingthat the Fittingsubspace of the affineholon- omyis the miniinum-dimensionalA(r)-orbit, and a parallelvolume on the Fitting subspacehas nonzerocohomology class. Muchof the proofsof thesefacts are con- tainedin [13]and ourjoint workwith G. Levitt [23]. Wesummarize the results fromthese papers which we needas follows: PROPOSITION.Let M be a compactaffine manifold with nilpotent holonomy r c Aff(E) (i) Thereexists a uniquer-invariant affine subspace Eu C E uponwhich r acts unipotentlyand a uniqueaffine projection 7r: E ) Eu commutingwith r. tional 71: E Rsuch that for eachg E G, the expressionv(g) = 71(g(x))- 71(x)is AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 195 (ii) ro dev:M ) Eu is a surjectivefbration and Eu is disjointfrom the devel- Optngzmage. (iii) Thereexists oy E r whoselinear part L(-y) restricted to a fber of r is an expanston. 4.4 The affinesubspace Eu is calledthe Fittingsubspace. Let k = dimEu. Sincer acts unipotentlyon Eu it preservesa parallelvolume form gu on Eu. The pullbackfiJE = T*bJU iS a r-invariantparallel k-form on E. Henceit definesa parallel(and hence closed) k-form fiJM on M. PROPOSITION. Thecohomology class [(JJM] E Hk(M;R) is nonzero. For the proofsee Goldman,Hirsch and Levitt [23]. The followingcorollary statesthat the dimensionof the Fittingsubspace Eu is the maximumexponent of a nonvanishingpower of the radianceobstruction: 4.5 COROLLARY.A CM is a nonzerocohomology class in Hk(M;/\ E), but Ak+lcM iszeroinHk+l(M;tkE), wherek=dimEu. PROOF OF 4.5. That /\ CM 780 followsfrom 2.3. That /\ + CM = Ofollows from1.6. Q.E.D. 4.6 We shall deducethat the Fittingsubspace Eu is a minimum-dimensional orbitof A(r) fromthe following: THEOREM. (i) A(r) acts transitivelyon Eu; (ii) Everyr-invariant Zariski-closed nonempty subset of E containsEu. Sincean A(r)-orbitof minimumdimension is Zariski-closed,4.6 impliesthat the Fittingsubspace is an A(r)-orbitof minimumdimension. Thus for the proof of 4.2 it sufficesto prove4.6. 4.7 The proofof 4.6(i) is basedon the notionof a syndeticaction. We shall say that a groupG acts syndeticallyon a spaceX if and only if thereexists a compactK c X with GK = X. If M is a compactaffine manifold, then the affine holonomyaction on the developingimage is syndetic.If M hasnilpotent holonomy r, thenr acts syndeticallyon Eu becauser: dev(M) ) Eu is surjective[13, 6.9]. Furthermoresince the algebraichull of rlEU is the restrictionof A(r) to Eu, 4.6(i) followsfrom LEMMA. Let G c Aff(E) act unipotentlyand syndeticallyon E. ThenA(G) acts transitivelyon E. PROOF OF LEMMA 4.7. If G is unipotentthen so is A(G). If G acts syn- deticallyso does any groupcontaining G. Thuswe mayreplace G by A(G) and assumeG is an algebraicunipotent subgroup of Aff(E). Wemust show that G acts transitively. If dimE = 1, then G is a nontrivialgroup of translationsso A(G) is the full groupof translations.Hence the lemmais truewhen E is 1-dimensional. Inductivelyassume that 4.7 hasbeen proved for E havingdimension less than n. Supposethat E hasdimension n andG c Aff(E)acts unipotently and syndetically. SinceG actsunipotently it preservesa parallel1-form, i.e. thereexists a linearfunc- A(v(G))E1moreindependent= v:71-1(0) Gacts R isof Choosetransitively surjective:x E E.a Oneright-inverse sees foron R. otherwiseeasily Now,u: the that 71R exact G v:wouldto G v:sequenceR be is G G-invariant,) a R.homomorphism. G1 G ) Rcontradicting determinesFurther- W. M. 196 GOLDMAN AND M. W. HIRSCH syndeticity.The kernelKerv = G1 is a subgroupwhich acts on the hyperplane LetK be a compactfundamental set forthe actionof G on E, i.e. E = GK. Let K1 = U{U(t)Kn E1:t E R}. Weshow that K1 is compactand GlK1 = E1. Since K is compactthe set of t suchthat K intersects,u(-t)El is a compactset of real numbers;thus the set of all t E R suchthat ,u(t)Kn E1 is nonemptyis compact. HenceK1 is compact.Furthermore GlK1 = G1,u(R)Kn E1 = GK n E1 = ThereforeG1 acts syndeticallyon E1 with fundamentalset K1. By the inductionhypothesis, A(G1) acts transitivelyon each71-1(p). There is a commutativediagram: E 71 R 1 1 E/G 7 R/v(G) Since71 is surjective,71 is surjective,implying that R/v(G) is compact. Hence an exactsequence A(G1) > A(G) ) A(v(G)). SinceA(G1) acts transitivelyon v-1(p) and A(v(G)) acts transitivelyon R, it followsA(G) acts transitivelyon E. Q.E.D. PROOF OF LEMMA4.6(ii). Let V c E be a r-invariantsubset. Thenbeing Zariski-closed,V is A(r)-invariant.Write E = Eu q3F, whereF = Ker: E ) Eu. By 4.3(iii)there exists ty E r suchthat L(-y)restricted to F is a linearexpansion. Let ty(8)be the semisimplepart ofty; by [31], ty(8)E A(r). Then Eu is the stationaryset of ty(8)and ty(8) preserves each coset of F in E. Choosea cosetx + F whichintersects V, wherex E Eu. ThenV n (x + F) is a closed subsetof x + F invariantunder ty(8). Since for every y E (x + E), tyny > x E Eu, we havex E V. SinceA(r) acts transitivelyon Eu, A(r)x E Eu. Since V is A(r)- invariant,V 2 Eu. Q.E.D. 4.8 We haveseen that a compactaffine manifold with nilpotentholonomy r hasassociated with it a certainunipotent affine action, namely the restrictionof r to the Fittingsubspace. We call this unipotentaction the Fittingcomponent of theaffine holonomy action. It is naturalto askfor criteriathat a unipotentaffine actionbe a Fittingcomponent for the holonomyof a compactaffine manifold with nilpotentholonomy. Obvious necessary conditions are that G be finitelygenerated andthat G act syndeticallyon E. By combining1.13 with 4.2 we see that these conditionsare not sufficient: THEOREM. LetM be a compactafWne manifold with nilpotent holonomy r c AS(E).Suppose that the Fitting subspace Eu is three-dimensional.Then the action ofr restrictedto Eu cannotlie in the groupof all afWnetransformations of the form 1 t-u2/2 v s 0 1 u t . O 0 1 u AFFINE MANIFOLDS AND ORBITS OF ALGEBRAIC GROUPS 197 PROOF . By 4.2, A3CM + 0. 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Yagi, Hessian structures on afyine manifolds, Manifolds and Lie Groups, Papers in honor of Yozo Matsushima,Progress in Math., vol. 14, Birkhauser,Boston, Mass., 1981. 55. , On compact homogeneous affine manifolds, Osaka J. Math. 7 (1970), 457-475. 56. H. Kim, On complete left-invariant affine structures on 4-dimensional nilpotent Lie groups, Thesis, Univ. of Michigan, 1983. 57. H. Shima, On certain locally flat homogeneous manifolds of solvable Lie groups, Osaka J. Math. 13 (1976), 213-229. 58. H. Hironaka, Triangulations of algebraic sets, Algebraic Geometry, Arcata 1974, Proc. Sympos. Pure Math., vol. 29, Amer. Math. Soc., Providence, R.I., 1975, pp. 165-186. MATHEMATICALSCIENCES RESEARCH INSTITUTE, BERKELEY, CALIFORNIA 94720 DEPARTMENTOF MATHEMATICS,2-281, MASSACHUSETTSINSTITUTE OF TECHNOL- OGY, CAMBRIDGE,MASSACHUSETTS 02139 (Currentaddress of W. M. Goldman) DEPARTMENTOF MATHEMATICS,UNIVERSITY OF CALIFORNIA,BERKELEY, CALIFOR- NIA94720 (Currentaddress of M. W. Hirsch)