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 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society. 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