A signature theorem for disk bundles and the eta invariant
The author applies the signature theorem for compact Riemannian manifolds with boundary [\textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer}, Math. Proc. Camb. Philos. Soc. 77, 43-69 (1975; Zbl 0297.58008) and \textit{P. B. Gilkey}, Adv. Math. 15, 334-360 (1975; Zbl 0306.53042)] to disc bundles over compact manifolds. Let \(E\) be an oriented real vector bundle of rank 2 over a compact oriented Riemannian manifold \(M\) of dimension \(4k-2\). It is assumed that the Riemannian metric on the total space of \(E\) is induced by a fibre metric \(h\) on \(E\) and a connection on \(E\) compatible with \(h\). The signature of the unit disc bundle of \(E\) is computed as a sum of an integral over \(M\) of a characteristic polynomial \(L_ k(E,M)\) and \(\lim_{r\to 0}\eta(S_ r(E))\). Here \(L_ k(E,M)\) is a homogeneous polynomial of degree \(2k-1\), which can be expressed explicitly in terms of the coefficients of the Hirzebruch \(L\)-polynomial, the Euler class of \(E\) and the Pontryagin classes of \(M\), and \(\eta(S_ r(E))\) is the eta-invariant of the circle bundle of radius \(r\). This result is used to compute the signature of \(D(E)\) of holomorphic vector bundles over compact complex manifolds. When \(M\) is an odd dimensional Hodge manifold and \(E\) is a positive holomorphic line bundle the signature of \(D(E)\) is expressed by the bigraded Betti numbers of the base.
scientific article; zbMATH DE number 440261
scientific article