Showing papers on "Calabi conjecture published in 1983"

The -genus of complex hypersurfaces and complete intersections

Abstract: In this note, we classify the even-dimensional complex hypersurfaces and complete intersections which carry a metric of positive scalar curvature. This is done by computing the A-genus of these manifolds to eliminate all cases not known to carry such a metric. If M is a real 4n-dimensional manifold, then Hirzebruch [1] has defined the A-genus of M as a certain polynomial in the Pontrjagin classes of M. By a theorem of Lichnerowicz [5], if M is a spin manifold, then the nonvanishing of the A-genus is an obstruction to the existence of a metric of positive scalar curvature on M. In this paper, we consider the case when M is a complex hypersurface, or more generally a complete intersection. If we denote by M = V2n the complete intersection of hypersurfaces defined by homogeneous polynomials of degree di.... .,dr in CP(2n + r), then one knows in principle how to compute the A-genus of M in terms of dI,. . . ,dr; M is spin precisely when 2n + r + 1 Xid, is even. Our main observation here is that for complex hypersurfaces V2n the formula for A(VJ/n) is somewhat simpler than one might expect: THEOREM 1. A v2n 2. (d/2 + n)(d/2 + (n -1)) ..(d/2 -n) ^( d ~~~~(2n + 1)! The formula of Theorem I was also obtained in [4, p. 259]. Our proof given below extends readily to give a formula for complete intersections, which is however somewhat more cumbersome. Nonetheless, we are able to determine when A( VdT .dr) is zero, so that we can show our main result: THEOREM 2. (i) If 2n + r + I d, is even, then V2n carries a metric of positive scalar curvature if and only if Ed, 2n + r. (ii) If 2n + r + 1 d, is odd and n > 1, then VJ2n d always carries a metric of positive scalar curvature. Theorem 2(ii) follows immediately from a theorem of Gromov and Lawson [3]-any simply-connected manifold of dimension > 5 which is not spin carries a metric of positive scalar curvature. Received by the editors June 16, 1982. 1980 Mathematics Subject Classification. Pnrmary 57R20. 'Partially supported by NSF Grant MCS8102747. ?1983 American Mathematical Society 0002-9939/82/0000-0882/$02.00 528 This content downloaded from 157.55.39.247 on Wed, 27 Apr 2016 04:49:39 UTC All use subject to http://about.jstor.org/terms THE A -GENUS OF COMPLEX HYPERSURFACES 529 In addition, one direction of Theorem 2(i) is an immediate consequence of the Calabi conjecture, as proved by Yau [6]: if :di < 2n + r, then the first Chern class of Vd2n d is a positive scalar multiple of the Kahler cl.i.s in CP(2n + r), restricted to V]2,n . By Yau's theorem, we may then find a Kahler metric on V2n d whose first Chern form is equal to this multiple of the Kahler form. Since the first Chern form is essentially the Ricci tensor, we have constructed a metric of positive Ricci curvature (and hence positive scalar curvature) on V2n The rest of the paper is devoted to a proof of the other direction of Theorem 2(i). The plan is this: in ?1 we prove Theorem 1, along the well-known general lines of Hirzebruch [1]. One must use a little care to obtain the formula of Theorem 1. From this, Theorem 2(i) for hypersurfaces is obvious. In ?2, we then turn to the general case of complete intersections, where the number theory involved becomes somewhat more difficult. 1. Proof of Theorem 1. Let rq denote the Kahler class of H2(CP(2n + 1)). If we apply the Whitney product formula to (1) T*(V12n) N(Vd2n) T*(CP(2n + 1)) restricted to Vd2n where N(M) is the normal bundle, together with the identity (2) c(l(Vd2)) = d (see, for instance, [2, p. 146]), we get (3) (i + cI(Vd2n) + C2(Vd) + * * ?+C2f(Vd2n))(1 + d.'q) (1 + The corresponding formula in Pontrjagin classes is then (4) (1 + p1(V2n) + . . +p (V]2n))(I + d2 _r2) = (1 + n ) or, formally, (4') (1 + p1(V2n) + +p (V]2n)) (I + n2)2n?2/(l + d2 .,2). We now use the formal factorization of (4') to evaluate the A-genus of Vd]n using the fact that the A-genus is the multiplicative sequence given by the function (vz/2)/sinh(Vz/2) [1]. We find that LEMMA 1. (2 sinh(dFz /2)) Z)2?} d(V2f) {the coefficientofZn in( (2 sinh(_z ))2 (A PROOF. By the formula for A, applied to the factorization (4'), ( )2 I+ dq \ (5) A(M) = ____ _ _ _ 2 sinh(1/2) ) 2 sinh(dq/2)) and it is easily seen that the coefficient of q 2n in the right-hand side of (5) is 1 d times the right-hand side of Lemma 1. On the other hand, d2n[V]2] = d, proving Lemma 1. This content downloaded from 157.55.39.247 on Wed, 27 Apr 2016 04:49:39 UTC All use subject to http://about.jstor.org/terms