By D.C. Spencer

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**Extra resources for Partial Differential Equations: Proceedings of Symposia in Pure Mathematics**

**Example text**

Thus g1) which by definition is p,tr has of the form no pole at 0 or and hence is constant. -s Part (b) of the theorem is obtained by evaluating the constant at t = 1. The theorems above and methods of proof by Kosniowski and Lusztig suggested to Atiyah and Hirzebruch that they try the same idea on the Dirac operator where for spin manifolds. They found THEOREM [8]. fold of dim 4k. If A(M) + 0, then any compact subgroup of the group of diffeomorphisms of M is finite. METHOD OF PROOF. One shows that if S' acts nontrivially on M, then A(M) = 0.

When M is Kahler, the qth cohoC). In general H9M) = Hodge splitting of mology with coefficients in the sheaf of germs of forms of type (p, 0). Suppose now X is a holomorphic vector field with simple isolated zeros, so that g1 = e" is a one-parameter group of holomorphic transformations with simple points. Let lm(X) denote the linear transformation which X induces at the fixed point ni, so that dg1(m) = eUm (X) Then L(p,g1) = trace — m€Mg is the induced action of on Suppose are the eigenvalues — and L(p, g1) = of lm(X); then, tr ehlm(X) = is the pth elementary symmetric function.

7. M. F. Atiyah and J. L. Dupont, Vectorfields with finite singularities, Acta Math. 128 (1972), 1—40. 8. M. F. Atiyah and F. Hirzebruch, Spin manifolds and group actions, Essays on Topology and Related Topics (Memoirs dédiés a Georges de Rham), Springer, New York, 1970. 9. M. F. Atiyah and G. B. Segal, The index of elliptic operators. II, Ann. of Math. (2) 87 (1968), 531-545. MR38 #5244. 10. M. F. Atiyah and I. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484—530.