A short proof of Rohlin’s theorem for complex surfaces

We prove that the signature of a two dimensional compact complex spin manifold is divisible by 16. Rohlin [R] showed that the signature of a smooth closed oriented spin 4manifold is divisible by 16. We give a proof for those manifolds which carry a complex structure. Recall that a manifold is spin if and only if the StiefelWitney class w2(X) vanishes. The key observation we need is that a complex manifold is spin if and only if its canonical bundle (the determinant of the holomorphic cotangent bundle) has a square root [A, 3.2]. By the dimension of a complex manifold, we mean its complex dimension. Theorem. If X is a two dimensional compact complex spin manifold then the signature o(X) is divisible by 16. Proof. Choose a holomorphic line bundle L such that L 0 L K, where K is the canonical bundle. Denote the holomorphic euler characteristic x(Ox(L)) by A. By Serre duality [S], the cup product gives a perfect pairing II'(X, Ox(L)) x H (X, Ox(L)) --H (X, Ox(K)) = C. Consequently H (X,Ox(L)) carries a nondegenerate skew-symmetric form, hence it is even dimensional. Therefore A = 2dimH0(X,Ox(L))-dimH1(X,Ox(L)) _0 (mod 2). On the other hand, by the Riemann-Roch and Hirzebruch signature theorems [AS], A= (cl (L)2 + c1(L). cl (X))/2 + (cl(X)2 + c2(X))/12 = (C1 (X)2 2c2(X))/24 =6r(X)/8 which shows that v(X) has the required divisibility. Received by the editors March 13, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57N 13, 32J 15. This work was done at MSRI where the author was supported by NSF grant DMS-8505550. ( 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page 1143 This content downloaded from 157.55.39.72 on Wed, 14 Sep 2016 04:19:52 UTC All use subject to http://about.jstor.org/terms