Novikov's higher signature and families of elliptic operators
Novikov's higher signature and families of elliptic operators
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DOI：
10.4310/jdg/1214430829
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发表时间：
1972
影响因子：
2.5
通讯作者：
G. Lusztig
中科院分区：
文献类型：
--
作者：
G. Lusztig
在这里签名（M t l…）ir)表示通常的Hirzebruch签名。（请注意，由于因子2的存在，我们的定义与W. C. xiang[6]的定义略有不同。）很容易看出（*）与基的选择无关。Novikov推测表达式（*）是流形X的同伦不变量，并提供了支持该猜想的证据[11]。Rohlin[14]获得了进一步的部分结果。一般证明已由W. C. HsiangFarrell[6]和Kasparov[7]用非单连通手术得到。本文的结果之一是基于一种完全不同的方法对诺维科夫猜想进行了新的证明。对于任意Kahler流形X，存在一个伴生的复环面Pic (X)，其点是全纯线束的同构类。它们在拓扑上是微不足道的。设Lp表示ppe (X)对应的全纯线束。那么存在一个全纯线束L / X X Pic (X)，使得对于给定的p€Pic (X)，约束L\X X {p}与Lp同构。(参见[8])。设AT*X为X的全纯余切束的p次外幂，设O（A*T*X®L）为ΛT*X(g)L的全纯截面集
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