Chern numbers and the indices of some elliptic differential operators (Q539024)

Given an \(n\)-dimensional compact complex manifold \(M\) and a partition \(\lambda=\lambda_1\cdots \lambda_l\) of weight \(n\), it is interesting to know if the Chern number \(c_{\lambda_1}\cdots c_{\lambda_l}[M]\) can be determined by \(\chi^p(M)\) or by the indices of some other elliptic differential operator. The simplest cases \(c_n[M]\) and \(c_1c_{n-1}[M]\) are due to [\textit{F. Hirzebruch}, Topological methods in algebraic geometry. Berlin-Heidelberg-New York: Springer-Verlag. (1966; Zbl 0138.42001)] and by \textit{A. S. Libgober} and \textit{J. W. Wood} [J. Differ. Geom. 32, No.~1, 139--154 (1990; Zbl 0711.53052)]. The author of the present paper, inspired by Libgober and Wood's proof, shows that on compact, spin, almost-complex manifolds the Chern numbers \(c_n\), \(c_1c_{n-1}\), \(c_1^2c_{n-2}\) and \(c_2c_{n-2}\) can be determined by the indices of twisted Dirac operators and signature operators. In this way, he gets a new direct proof of Libgober and Wood's result and a divisibility result of certain characteristic numbers for such manifolds.