Novikov's higher signature and families of elliptic operators

Abstract

Here sign (M t l... i r) denotes the usual Hirzebruch signature. (Note that our definition differs slightly from the one of W. C. Hsiang [6] by the presence of the factors 2.) It is easy to see that (*) is independent of the choice of basis. Novikov conjectured that the expression (*) is a homotopy invariant of the manifold X and provided evidence [11] in favor of this conjecture. Rohlin [14] has obtained further partial results. The proof in general has been obtained by W. C. HsiangFarrell [6] and Kasparov [7] using nonsimply connected surgery. One of the results of this paper is a new proof of Novikov's conjecture based on a completely different approach. For any Kahler manifold X there is an associated complex torus Pic (X) whose points are the isomorphism classes of holomorphic line bundles o n ! which are topologically trivial. Let Lp denote the holomorphic line bundle corresponding to p e Pic (X). Then there is a holomorphic line bundle L over X x Pic (X) such that for a given p € Pic (X), the restriction L\X x {p} is isomorphic to Lp. (See [8].) Let AT*X be the p-th exterior power of the holomorphic cotangent bundle of X, and let O(A*T*X®L) be the sheaf of holomorphic sections of ΛT*X(g)L