Hirzebruch, F.

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MATHEMATICS: F. HIRZEBRUCHthe Institute for Advanced Study Electronic Computer, and Mr. E. Hovmollerwho analyzed the charts used in the forecasts.* This work was sponsored jointly by O.N.R. and the Geophysics Researah Directorate of theAir Force Cambridge Research Center, under contract N6-ori-139 with O.N.R. It was presentedat a joint meeting of the American Meteorological Society and the Fluid Dynamics Section of theAmerican Physical Society in Washington, D. C., May 1953.1 Charney, J. G., and Phillips, N. A., "Numerical Integration of the Quasi-geostrophic Equationsfor Barotropic and Simple Baroclinic Flows," J. Meteorol., 10, No. 2, 71-99 (1953).2Charney, J. G., "On the Scale of Atmospheric Motions," Geofye. Publikcasjoner, 0810, 17,No. 2, 17 pp. (1948).8 This iteration method is essentially the extrapolated Liebmann method. See Frankel, S. P.,"Convergence Rates of Iterative-Treatments of Partial Differential Equations," Math. Tablesand Other Aids to Computation, 4, No. 30, 65-75 (1950).; and Young, D., "Iterative Methods forSolving Partial Difference Equations of Elliptic Type" (1951). Seminar given at NumericalAnalysis Seminar of Ballistic Research Laboratories,-Aberdeen Proving Ground, Md. (based onPh.D. thesis by author, Harvard University, Cambridge, Mass., 1950).ARITHMETIC GENERA AND THE THEOREM OF RIEMANN-ROCHFOR ALGEBRAIC VARIETIESBY FRIEDRICH HIRZEBRUCHTHE INSTITUTE FOR ADVANCED STUDYCommunicated by S. Lefschetz, December 21, 1963Introduction.-In a preceding note' we posed the problem whether the arithmeticgenus II(V.) of a (non-singular) algebraic variety2 coincides with the Todd genusT(V.). The purpose of the present note is to prove that this is actually the case.Moreover we prove a main theorem (M) which gives a formula for the Euler-Poincar6 characteristic of V, with respect to the cohomology' of V,, with coefficientsin the sheaf (faisceau) of local holomorphic cross-sections of any complex analyticbundle W over V, which has the complex vector space C, as fibre and the lineargroup GL(q, C) as structure group. The main theorem expresses this Euler-Poincar6 characteristic as a polynomial in the Chern classes of the tangential bundleof V. and in the Chern classes of the bundle W. As special cases one gets the Toddformula H(V.) = T(Vn) and the "Theorem of Riemann-Roch for arbitrary dimen-sions." The author wishes to extend his hearty thanks to Professors A. Borel, K.Kodaira, and D. C. Spencer, with all of whom he had many valuable discussions.The notes of Kodaira and Spencer4 are essentially used. The author also wants topoint out that the main theorem (in a slightly different formulation) was conjec-tured by J. P. Serre in a letter to Kodaira and Spencer. The proof of the maintheorem uses the index theorem of the author' which involves essentially the theoryof "cobordisme" due to R. Thom. Full details of the proof of the main theorem willappear elsewhere.1. The Main Theorem and Some of Its Consequences.-Let V- be an algebraicvariety (non-singularly imbedded in some complex projective space). Let W be acomplex analytic bundle over Vn with the complex vector space C, as fibre and thelinear group GL(q, C) as structure group. Denote by ci the Chern olasses of the110 PROC. X. A. S.LMATHEMA TICS: F. HIRZEBRUCHtangential bundle of V. and by dj the Chern classes of the bundle W. (ci e H2i(VnrZ), dj e H2j(V", Z);O _ i _ n,O _ j < q, co = do = 1.) We introduce the formalroots 7yL, 5j:n n q qE cxi =II (1 + 'yix), ZdjxJ=l (1 +5jx).i=O i j=O j= 1Every formal power series which is symmetric in the yf as well as in the 5j will beconsidered as a power series in the Chern classes ci and dj and hence as an element ofthe cohomology ring of V.. Denoting by Q the rationals we define the operator Knon the cohomology ring H*(V., Q) as follows: For every u e H*(V., Q) we take thecomponent of topological dimension 2n which we consider in the unique fashion as arational number. This number is denoted by K.[U].MAIN THEOREM (M). Using the above notations we putUx(Vn, W) = E (-1)' dim H1(VUn W),i =owhere HI(VU, W) is the i-dimensional cohomology group of V, with coefficients in thesheaf (faisceau) of local holomorphic cross-sections of W. We havex(VF, W) = Kn [(el, + e + * +Ae)1le--ic1/2X 74~~'y~2Kn [e (eai + e32 + .. + eaY) II 'i= 1 sinh -yj/2JRemark: x/sinh x is a power series in x2. Since the 'y can be regarded as theformal roots of the Pontrjagin polynomial' of Vn, we see that x(Vn, W) is a poly-nomial in cl, the Pontrjagin classes of Vn, and the Chern classes of the bundle W.Now let W be the trivial bundle over Vn with C, as fibre. The main theoremimplies'THEOREM 1. The Todd genus of an algebraic variety is equal to the arithmeticgenus:nII(Vn) = E (- 1)ig = T(Vn).i=OMore generally, let F be an arbitrary complex line bundle (fibre Cj) over Vn.Denote by f = c(F) its characteristic class (f e H2(V., Z)). The main theoremimpliesTHEOREM 2. We haveX(Vex F) = = sinh y j/2Hence x(Vn, F) is a polynomial inf + cl/2 and the Pontrjagin classes of V,.Theorem 2 can be considered as a generalization of the theorem of Riemann-Roch to arbitrary dimensions. It contains the known Riemann-Roch theorems forn = 1, 2, 3. In the case n = 2 the term dim H'(Vn, F) has to be identified withthe superabundance of F. A similar remark applies to dim H'(Vn, F) and dimH2(VU, F) in the case n = 3. In general, the term dim H ,(V, F) is equal toVOL. 40, 1954 IIIMATHEMATICS: F. HIRZEBRUCHdim H(Vn, K - F), where K is the canonical bundle of V,. The characteristicclass of K is equal to -cl. Kodaira6 proved that H'(Vn, F) = 0 for i > 0, if thecharacteristic cohomology class of the bundle F - K "contains" a closed positivedefinite Hermitian form. In this case F - K is called ample. An arbitrary divisorD of V,, represents a complex line bundle, which we also denote by D. Obviouslydim IDI + 1 = dim HO(V,, D). Denoting the cohomology class of D by d we getTHEOREM 2*. IfD - K is ample (in the sense of Kodaira), then we havedim DI +1= KS [ed+c'/2I /2(For n = 1, this is the classical formula: dim IDI = d - p, where cl = 2 - 2p.)Applying (M) to the case where W is the bundle of complex covariant p-vectors,yields5THEOREM 3. Let h" q be the number of harmonic forms of type (p, q) on Vn. Wehavex'(Vn) = x(V , W) = -1) = K [(Ee + iIje~ti-iiThe sum on the right side has to be extended over all the (n) possible combinations.We attach to every algebraic variety the polynomialnXv(Vn) = E X (Vn)YOp=Oand get by easy calculationsTHEOREM 3*. We haveXV(Vn) = KB [ II Q(/)]whereQl,(x) = -x(y + 1) .e x(y+1) -Y+Theorem 3* is for y = 0 a restatement of Theorem 1. For y = it gives that'S p~qhjqX-i(Vn) = E (-1) hp, q=Ois equal to the Euler-Poincar6 characteristic cn of V". For y = 1 we obtainTHEOREM 4. We haveX!(Vn) =A (1)P q = Kn HI.P, Q=O i=1 tgh JTheorem 4 is known even for Kihler manifolds, because the left side of the equa-tion is equal to the index r(V).T8 The right side is also equal to r(V.) by the indextheorem of the author.1 The main theorem admits many further applicationswhich we do not mention in this note.112 PRoc. N. A- S.40IATHEMATICS: F. HIRZEBRUCH2. A Sketch of the Proof of the Main Theorem.-In section 1 we derived theTheorems 1-4 from the main theorem (M). Actually the proof of (M) goes almostin the opposite direction and involves the followiiig steps.2.1. Proof of Theorem 1 by Means of the Known Theorem 4.-A complex manifoldMm is called a split manifold, if the structure group GL(m, C) of its tangential bundlecan be reduced complex analytically to the group A(m, C) of all triangular matrices(all entries above the diagonal are 0). Let Vn be an arbitrary (non-singular) alge-braic variety, the tangential bundle has GL(n, C) as structure group. Hence wecan construct the associated bundle with GL(n, C)/A(n, C) as fibre. We obtainin this way an algebraic variety V* of dimension m = n + '/2n(n - 1). The factthat V* is algebraic can be proved, for example, by using the general theorem ofKodaira9 that every complex projective bundle over an algebraic variety is analgebraic variety. It can be proved that V* is a split manifold. It is almostobvious that ll(V*) = ll(V^). On the other hand, it follows from Theorem 4.2in the preceding note' that T(V*) = T(Vn). Hence it suffices to prove Theorem 1for all algebraic split varieties. Now let V* be an arbitrary algebraic split variety.Then we have over V* an increasing sequence of bundles Wi (in the sense of section1), where Wi has Ci as fibre and where Wm is the tangential bundle of V* (0 _i . m). We denote the complex line bundle Wj1Wj_, by Ai (1 < i . m) and itscohomology class by aj. We use the notation of the preceding note and obtainreadily the formulaem2mT(Vm) = E E r(a,arI, .., a,) (1)j=The inner sum has to be extended always over all the possible (m) combinations.By the four-term formula of Kodaira-Spencer" we define in the faisceau theoryvirtual indices xi. This can be done for all algebraic varieties. (It can be donenot only for xi, but even for xv ) We know that xI = r for all non-singular alge-braic varieties, (Theorem 4). Since XI and r both fulfill the "functional equation"3.3 of the preceding note,' we get by an induction argument on the dimension thatXi and r also coincide in the virtual case.'2By some rather complicated calculations based on exact sequences of sheaves wecan prove the following formula which corresponds to (1)m2m'I(V,n) = Z x(AroAr2o... oAr,) (2)j=0Here the terms xi denote virtual indices.13 Formulae (1) and (2) imply Theorem Ifor split varieties, and hence Theorem 1 is proved in general.2.2. Proof of Theorem 2.-The virtual II and the virtual T both fulfill the"functional equation" 4.3 of the preceding note. Hence the usual induction argu-ment already used for xi in 2.1 proves that II and T also coincide in the virtual case.This proves Theorem 2 (i.e., the main theorem for q = 1). Moreover, the maintheorem follows immediately for all cases in which the bundle W admits the tri-angular group A(q, C) as structure group.2.3. Proof of the Main Theorem.-We use the notation of the first section. Weconstruct over V,, the bundle V*, with fibre GL(q, C)/A(q, C) which is associatedN 01.. 40, 1954 113MATHEMATICS: F. HIRZEBRUCHwith W. The manifold V* is, according to Kodaira's theorem,9 algebraic (m =n + '/2q(q - 1)). We lift the bundle W up to V* and call the lifted bundle W*.The bundle W* splits, i.e., admits A(q, C) as structure group. Hence Theorem 2implies that x(V*, W*) is given correctly by the formula of the main theorem. Aspectral sequence argument shows that x(V*, W*) = x(V", W). On the otherhand, we can prove by a slight generalization of the Theorem 4.2 of the precedingnote' that the formula of the main theorem gives for x(V*, W*) and x(Vn, W) thesame values. This concludes the proof of the main theorem.Remark.-The formulae (1) and (2) are special cases of more general formulaevalid for xv with y regarded as indeterminate. The "functional equations" forII = xo = T and for xi = r are special cases of a "functional equation" valid for X..1 These PROCEEDINGS, 39, 951-956 (1953). We use the notations of the preceding note.2 Complex dimensions are indicated by a subscript.3 All the cohomology groups occurring in this note have finite dimensions. See Cartan, H.,and Serre, J. P., Compt. rend. acad. sci., Paris, 237, 128-130 (1953), and Kodaira, K., these PRO-CEEDINGs, 39,.865-868 (1953).4 Kodaira, K., and Spencer, D. C., these PROCEEDINGS, 39, 641-649, 868-877 (1953). K.Kodaira, loc. cit. in ref.- 3.A Here we use the theorem of Dolbeault (Compt. rend. acad. sci., Paris, 236, 175-177 (1953))which states that H'(V5, W(")) - HP" I, where W(P) is the bundle of covariant p-vectors and whereH", e is the linear space of all harmonic forms of type (p, q) on V"..6 Kodaira, K., "On a Differential-Geometric Method in the Theory of Analytic Stacks," thesePROCEEDINGS, 39, 1268-1273 (1953).7 Conversely every complex line bundle can be represented by a divisor. See Kodaira, K., andSpencer, D. C., loc. cit. in ref. 4, p. 874.8 Hodge, W. V. D., Proc. Int. Congr. Math., I, 182-192 (1952), and Proc. Lond. Math. Soc. (3),1, 104-117 (1951).9 Kodaira, K., not yet published.10 The sum on the right side starts for j = 0 with r(V). The formula (1) was used by theauthor to prove Theorem 4.1 of the preceding note (loc. cit. in ref. 1). See also Hirzebruch, F.,"The Index of an Oriented Manifold and the Todd Genus of an Almost Complex Manifold,"Notes, Princeton University, 1953 (mimeographed).11 Kodaira, K., and Spencer, D. C., "On a Theorem of Lefschetz and the Lemma of Enriques-Severi-Zariski," these PROCEEDINGS, 39, 1273-1278 (1953), formula (14).12 The fact that x, fulfills the functional equation can be obtained from the four-term formulaby some calculations.13 The small circle o denotes "virtual intersection."114 PRoc. N. A. S.

Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties

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Arithmetic genera and the theorem of Riemann-Roch for algebraic varieties

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