Wells, Jr., R. O.

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AUTOMORPHIC COHOMOLOGY ON HOMOGENEOUS COMPLEX MANIFOLDS': by R. 0. Wells, JI*. Autolnorphic forms on the flpper half plane have been under investigation sillce the late nineteenth celltury (see the cknsslc text by Fricke-Klein [ ' i ] ,  for insta1lce). In several variables a~itomorphic fornls were studied by Blu- mentl~al, Hilbert, and Picard near the tun1 of the century and more recently by Siege1 [I41 011 the Siege1 Lipper llalf space. The study of automorphic for~ns on general Lie groups was introduced by Harish-Cliandra [I11 and has been a subject of considerable interest (cf. Bore1 [ d l  and Murakalni [12]). In particular, automorphic forlns on a bou~tded synlmetric dolnain can be co~lsidered as holo~norphic sections of ho~nogeneous vector bundIes illdilced by finite-dilfiensional representatio~is of a colnpact isotropy group, the classical case corresponding to powers of the ca~~ol l ica l  bilndle (see Bore1 [4]). A~~tomorph ic  forms on a dolllai~l D are defined with respect to a discrete transfor~natio~l r group acting 011 D, and the quotients of any two for~lls (of the same weight) are meromorphic fiinctions on the qr~otient space DIT. This quotient space lnay have singc~larities as a n  analytic space depending on the nature of T, and thus the classical str~dy of auto~norphic forms was a means of studying the fi~nction field on certain classes of allalytic spaces. This is the somewhat tenuous relation of the present note to the theme of this confere~lce on  "singularities." Suppose D is a bounded Hermitian sylnmetric domain of the form D = GIK where C is a setnisi~nple real Lie group and K is a maximal compact subgroup. Let p be a finite-dimensionaI unitary representation of K ,  and let E, be the associated h o n ~ o g e ~ ~ e o u s  vector bundle over D (cf. $2). If r is a discrete subgroup of C which acts properly d iscor~t in~~ously  on D, then the r-invariant sections of H' (D .E ,~ ) ,  denoted by H;(D,E,,), are the automorphic forms on D (with additional growth restrictiolls a t  the Cusps in the case dim,D = I). Generalizations of such Hermitian bounded " Research supported by NSF-GP-19011 at Rice University. 148 RICE UNIVERSITY STUDIES sy~ii~netric do~nains  arise in algebraic geometry as tlie classifying spaces for Hodge structures on conipact algebraic (more geiierally Kahler) manifolds (Grifiths [9j,  [lo]) .  In this situation such do~na ins  D are homogeileo~l~ complex ~nanifolds of the for111 G/V wliere G is a semisimple Lie grotip and T/ is a conipact (but not necessarily ~naxilnal compact) subgroup of G, If E is a lio~noge~icousvector b~liidle overDassociated to a finite-dimensional representation p of V, then Schmid has sliowrr that for most represe~itatiotls p in the case where V is not ~naxiinal  compact (so that D is rjot an Hermitian boii~lded symmetric domain, see Sclimid [13f, Griffiths-Schmid [ lo]) .  Thus there are 110 atltomorpllic forms on these generalizations of the upper half plane or  Siegel's upper half space (wl~icll are the classifyi~lg spaces for the Hodge structures on Rie~nai i~i  surfaces). In  general, Schmid has shown that H1l(D, E,,) is an  infinite-dimensional Frechet space which should be the atxalogue of HO(D, E,) in the classical case of bounded symmetric don~ains.  Here q = idim,K/V where K is t l~e ~naxiliial compact subgroup of G containing V (which tur~is  out to be tlie same as the coriiplex diniension of the riiaximal-dimensio~ial compact co~nplex submanifold of D).  The r-invariant colio~~iology classes H:(D,E,) are tIie ~ia tura l  candidates for "automorphic forms" on D. These are the nuto- nlorplric c o h o ~ ~ ~ o l o g j ~  clarses of Griffiths (Griffitlis [$I, [lo],  Griffiths- Schmid [ lo] ) .  A t  present very little is know11 about this particular versioil of "a~ho-  morpliic forms," except in the case where D / T  is compact (see Griffiths- S c l ~ ~ n i d  [lo]). In particular, if r is the llatilral ge~leralization of Siegel's modular group, then it is t~nkiiown whether N;?(D, E,,) is finite-dimensional. I11 this note we announce sonie new results concerni~lg the geonierry of these more general liomogeneous compiex ~liaiiifolds D = G/V. It 1s possible that the techniques developed here will be usefi~l in attacking tlie above proble~n of finite dimensionality of the auto~norpliic coliomology groups. Specifically, we show that for a sufficiently nonsingular repre- sentation p, the automorpliic coho~iiology classes on D can be represented as the r-invariant sections of a liolomorphic vector b~lndle over a Stein manifold M naturally and equivarialltly associated to D. Here M is essen- tially tlie space of compact q-dimensional subvarieties of  D. A consequence of t h ~ s  result is the convergence of the PoilicarC series AUTOMORPHIC COHOMOLOGY 149 here q5 e IY4(D, E,,) is a n  absolutely integrable cohornology class(with respect a suitable metric). Such a PoincarC series is an  automorphic col~omology lass, and the classical study of a ~ ~ t o ~ n o r p h i c  forms was very dependent 011 ~~nders t and i~ lg  such series. Many of the results anno~inced here were cot~jectured by Griffiths [.')1, [ lo] .  Details will appear elsewhere a t  a later date, ~ c t  G, be a connected complex se~nisin~ple Lie group and let P c G ,  be a Lic subgroup, i.e.. the coset space X = Gc.P is a compact l~omogeneo~is x manifold (which is kriown t o  be projective algebraic). Let G  c Gc a real for111 of G, such t l ~ a t  V = G n P is a compact snbgroup of G,. en the G-orbit of the coset ~ P E  Gc/P is a n  open subset D of the co~nplex ifold X. The ~natlifold D = G / V  is a homogeneous cornplex manifold, it is the geometry of such ~nauifolds in which we are principally rested. An example is given by the real Lorentz group G = SO(Zlr.X), here V = U(h)  x SO(/<), which arlses as the classifying space for periods lotornorphic 2-forms 011 a given projective algebraic manifold of cotnplex ~ s i o n  2 2 (cf. Griffiths [S]). In this case G  = SO(2h + 16, C), and P is en explicitly in Grifiths-Schmid [IO, p. 2621 (see this paper and also 17'1 for f~irtller backgroiund on this class of homogeneous co~nplex Ids). The subgroup P c Gc is parabolic if and only if P contains a el subgroup B (i,e., a lnaxirnal solvable subgroup), and there is a nat~l ra l  ose fibers are  coinpact cornplex subnlanifolds isomorphic to  PIB.  The nifolds G c / B  are flag rnailifolds and are somewlihat easier to study than y parabolic subgroup (cf. Bott [.5], Schmid [Is], ose now that K is a maximal colnpact subgroup of G  which contains 1 it is known that the irnage of the i~~ject ion nplex sub~nanifold Y of the colnplex mai~ifold D (cf. Bore1 131, s-Schmid [LO]). Let Y, denote the translate of the submanifold by g 6 G,, and let ?YX denote tile disjoint u~ l ion  of the complex ~ ~ i f o l d s  of X, each of whicll is of the form Y, for some g E Gc. The ig, admits a complex structure suc l~  that the mapping 150 RICE UNIVERSITY STUDIES i~idirced by tlze inclusion  napping on fibers is Iiolomorphic. Moreover, there is a psranletrization mapping where S is an algebraic homogeneo~~s  submatiifold of cN for some N .  Namely, the mapping n is a proper si~rjective li01o111orphi~ mapping of izlaxiinal rank such that n L 1 ( s ) ,  for s E S, is one of the fibers in the original disjoint ~rnioli !YX, i.e., n- ' ( .s)  = Y, for sorile B E G < :  (cf. Criffitlls [S], Wells [1:5], and Windham [IG]). Let !Y,, = r - ' ( D )  be the union of those translates of Y which are contained in  D, and let M = n(!Y,) c S be the parameter space for these translates. Theorem 1.  M is n Stein nzanifbld. The proof of this result uses the constr*uction of all appropriate exliaustion function for D due to Sclirnid [13] (cE Criffiths-Sclimid [10]), the soli~tion of the Levi problem for spaces of analytic cycles due to Andreotti-Norguet [ I ] ,  [S], and an exhaustion principle for Stein manifolds due to Docquier- Crauert [GI. A special case of this result was proven explicitly in Wells [l:i] for the case G = SO(2h ,  I ) ,  V = U(h). $3. I-lonzoyeneoltv Vector Bundles Let D = G / V  be a I~omogeneous complex manifold of the type considered in the previous section. S ~ ~ p p o s e  that [ I  : I.' G L ( E )  is a representation of V on the finite-din~ensional co~nplex vector space E. Consider tlie eclirivalence relation i n  the trivial vector bundle G x E + G :  i ( g , e ) ~ G  x E : ( y , e ) = ( g u , p ( u ) e ) ,  for U E V ) .  This equivalence relation induces in a natural manner a vector bundle E,, on the coset space G / V  = [gV),,,, which is called the hoinogetleous vector. bllrl~lle 0 1 1  D ir~cl~rcerl b y  the rep~.eser~tatiorz p (cf. Bott [;>I). Tlze adjective "homogeneo~~s" denotes tlie fact that the action of G on the liomogerzeous space G / V  lifts to an  action on the vector bundle E defined over G I V .  Suppose now that we are given a homoge~ieous vector bundle E,, -t 5 as above, and suppose that p is sufficiently non.sit~gulnr., i.e., the highest weight of the representation p is sufficiently bounded away from the boundary of tlie Weyl chamber of positive roots associated with tlie Lie AUTOMORPHIC COHOMOLOGY 151 algebra of G, and a maximal torus T c  V (note that the choice of Weyl &amber is dictated by the choice of the complex structure 011 D, which in our setting is given a priori; see Schmid [13], Griffiths-Schmid [10]). This colldition was used by Schtnid in his study of the infinite-dimensional represe~ltati~lls of G (Schmid [13]), and can be prescribed precisely in terms of the root structure of the Lie algebra in a given situation. In case P = B is a Bore1 subgroup of G,, then the condition above is satisfied by a suffi- cieiltly high power of the canonical bundle. Classically this would correspond to automorphic forms of high weight. I n  the case of a hoinogeneo~ts line bujldle, the highest weight is the differential of the (one-dimensional) representation and thus defines an element in the d~ia l  space to the Lie algebra. Let T*E be the p~~llback of E, to tVD, and let be the y th  direct image of z'$E on M, where q = dim,Y (we are  thinking of the direct image sheaves as vector bundles, since in this case the direct image sheaves are locally free). Then 4, is a vector bundle on M which has fibers isoinorphic to Hfl(Y, En,,). Moreover, the action of G on D induces an action on Hr(D, E,,) and on H1'(M, E,,), for all r 2 0. Schmid has shown that if p is sufficiently nonsingular, then and moreover, HYD, E,) has the natural structure of a Frechet space inherited from the n a t ~ ~ r a l  topology on spaces of C" differential forms and Dolbeault's isomorphism. The difficult part is to show that the range of 3 is closed so that the quotient space in the Dolbeault group is Hausdorff (see Schmid [13]). Theorem 2. Suppose the representation p is suficiently nonsingular. Then there is a G-equivariant topological injection By "topological" it1 the above theorem we mean that the algebraic injection is a closed subspace of the Frechet space H ' ( M , ~ ~ ) .  The proof of this theorem breaks down into two parts. First we see that the Leray spectral sequence relating cohomology on Y, to  cohomology on M degenerates, giving us an isomorphism 152 RICE UNIVERSITY STUDIES and thus giving a natural Frechet space structure to H4(gD,7*E,). This follows from the fact that M is Stein (Theorem 1) and Cartan's Theorem B on M (this doesn't use the homogeneous nature of the vector bundles). A different proof can be obtained as a consequence of the fact that the remaining direct image sheaves n',(z*E,) = 0 if j # q ,  which follows from the vanishing theorem of Bott [6] along with the local triviality of the fibra- tion GY, 4 M. The natural mapping r X :  H Y D ,  E,) + H4('YD, r*E,,) is then shown to be a topological i~ljection by using a "power series expan- sion" of the cohomology classes about the fiber Y in D due to Schmid [13]. Here the homogeneous nature of the vector bundles is used quite strongly, It seems likely that an alternative proof valid in greater generality can be constructed by tising an appropriate generalization of Grauert's direct image theorem to the case of a real-analytic fibering of compact complex manifolds (which structure D has in a natural way, namely the fibering G / V  -+ G /K).  Using the above results we obtain a representation for automorphic co- homology and the convergence of certain Poincare series. If T is a discrete subgroup of G which acts properly discontinuously on D by left translation, then I? induces an action on H Y D ,  E,) and on HO(D,&)  by Theorem 2. Let H:(D, E,) be the r-invariant subspace of H q ( D ,  E,) and let H;(M, $) be the I?-invariant subspace of H " ( M , ~ , , ) .  As an immediate consequence of Theorem 2 we have the following result. Theorem 3. T h e  rnappitzg 5 in (1) induces a topological injection Thus Theorem 3 tells us that automorphic cohomology can be interpreted as invariant sections of a vector bundle. This had been conjectured by Griffiths [8]. In particular we note that we do not lose any information about the automorphic cohomology as we pass to sections; however, we do not expect every r-invariant section of E, to come from an automorphic cohomology class, as simple examples show that this is highly unlikely in this particular type of geometric situation. AUTOMORPHIC COHOMOLOGY 153 Let E, 4 D be as above and Iet E, and the tangent bundle to D be equipped with G-invariant metrics. Using these metrics it is easy to make precise what i t  means for a (0, q) differential form with coefficients in E, to be in Li (absolutely integrable). We then say that a cohomology class in Hq(D,E,) is in L1 if it has a Dolbeault representative which is in L1. Let H:(D,E,) be the subspace of L1 coholnology classes in H Y D ,  E,) (cf. Griffiths [8]). As noted by Griffiths ([8], p. 619), the power series arguments of Schmid in [13] show that H:(D, E,) is none~npty (and infinite dimensional) provided that p is sufficiently nonsingular. We then have the following theorem conjectured by Griffiths in [8] ,  where p is as above. Lemma 4. Let 4 cHq1(D,E,,), a110 let ther~ (2) converges to at1 autonzor'phic cohornology class in H:(D, E,). This theorem was partially proved in Griffiths [8] where he showed that the associated PoincarC series on M converges in HO(M, E ~ ) ,  tlsitlg basic convergence arguments on this series of sections of a vector bundle stemming from the more classical theory. But it now follows easily from Theorem 2 and the convergence of (3) that the series in (2) converges. We strongly use the fact that the image of 4 in the mapping (1) is closed. If HYD, E,,) is finite dimensional as in the classical case, then, as was pointed out by PoincarC in his original paper on PoincarC series (which he called "theta-Fuchsian series"), it is necessary that almost all of the series of the form (2) vanish identically. Remark. In case the manifold D studied above fibers holomorphically over an Hermitian bounded symmetric domain (i.e., G / V  + GIK is a holomorphic mapping), then the above results are much easier as it is unnecessary to consider the deformation space nJ, introduced in $2. This is the case when D = G / V  admits a Kahler metric invariant under G and can be determined by the structure of the Lie algebras involved (Bore1 [3]). However, the classifying spaces for Hodge structures (Griffiths [8], [9]), for instance the example given in $2, do not have this property. RICE UNIVERSITY STUDIES REFERENCES [ I ]  ANDREOTTI, A., and F. NORGUET, "Problkme de LCvi et convexit6 holomorphe pour Ies classes de cohomologie," Ann. Sc~tola Norm. Sup. Pisa 20 (1966), 197-241. [Z] --, "La convexitt l~olomorphe dans I'tspace des cycIes d'une variCtt algtbrique," Ann. Scuola Norm. Sup. Pisa 21 (19671, 31-82. [3] BOREL, A., "Kahlerian coset spaces of semisimple Lie groups," Proc. Nat. Acad. Sci. U. S.  A. 40 (1 954), 1 147-1 151. [dl - , "Introduction to automorphic forms," Proceel-lings of Symposia in Pure Matl~ematics, Vol. IX (Providence, R. I.: Amer. Math. Soc., 1966), 199-210. [5] BOTT, R., "Homogeneous vector bundles," Ann. of Math. 66 (1957), 203-248. [6] DOCQUIER, F., and H. GRAUERT, "Levisclles Problem und Rungescl~er Satz fiir Teilgebeite Steinscher Mannigfaltigkeiten," Math. Ann. 140 (1960), 94-123. [7] FRICKE, R., and F. KLEIN, Vorlesungerz iiber die Theorie cler Automorplzen Funlctiorzen (Leipzig, 1877-1 9 12). [8] GRIFFITHS, P. A. "Periods of integrals on algebraic manifolds I," Amel.. J .  Math. 90 (1968), 568-626. [91 - , "Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems," Bull. Amer. Math. SOC. 76 (1970), 228-296. [lo] - ,and W. SCHMID, "Locally homogeneous complex manifolds," Acta Math. 123 (1969), 253-302. [ I l l  HARISH-CHANDRA, " utomorphic forms on a semisimple Lie group," Yroc. Nat. Acad. Sci. U. S.  A. 45 (1959), 570-573. [I21 MURAKAMI, S., L ' C ~ h o m ~ l o g i e ~  of vector-valued forms on compact, locally symmetric Riemann manifolds," P~~oceedings of Symposia in Pure Matlzematics, Vol. 1X (Providence, R. I.: Amer. Math. SOC,, 1966), 387-399. [I31 SCHMID, W. "Homogeneous Complex Manifolds and Representations of Semisimple Lie Groups" (Ph.D. Thesis), Univ. of California at Berkeley, 1967. [I41 SIEGEL, C. L. "Einfiihrung in die Theorie der Modulfunktionen n-ten Grades," Marlz. Ann. 166 (1939), 617-657. 1151 WELLS, R.  0. Jr., "Parametrizing the compact submanifolds of a period matrix domain by a Stein manifold," Symposium on Several AUTOMORPHIC COHOMOLOGY 155 Coinplex Variables, Park City, Utah 1970, Lecture Notes in Matlze- matics, Vol. 184 ( N .  Y.: Springer-Verlag), 121-150. [I61 WINDHAM, M., "Deforniation Spaces of Period Matrix Domains of Cot?7pacf Kci'hlcr Surfaces" (Ph  .D. Thesis), Rice University, 1970. [I71 WOLF, J., "The action of a real semisimple group on a complex flag manifold. 1:  orbit structure and holomorphic components," Bull. Amer. Math. Soc. 75 (1969), 1121-1237. 

Automorphic Cohomology on Homogeneous Complex Manifolds

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Automorphic Cohomology on Homogeneous Complex Manifolds

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