A fixed point formula for compact almost complex manifolds. (Q1872567)

Let \(M\) be a compact \(2m\)-dimensional almost complex manifold with the almost complex structure \(J\) and \(P\to M\) the associated principal \(\text{GL}(m;\mathbb{C})\)-bundle of \(M\). A diffeomorphism \(\psi:M\to M\) is said to be an automorphism of \(M\) if \(\psi\) commutes with \(J\). By \(A(M)\) the topological group consisting of all automorphisms of \(M\) is denoted. The main result of the paper is: Theorem. Let \(\ell= 0,1,2\) and \(\varphi\in S(n)\) (the set of symmetric homogeneous polynomials in \(x_1,\dots, x_n\) of order \(n\)) and \(\psi\) be any periodic element of \(A(M)\), and \(\gamma\) be any natural number which is prime to \(p\). Let \(\Omega(k)\) be the fixed point set of \(\psi^k\) \((1\leq k\leq p-1)\) consisting of compact connected almost complex manifolds \(N\), \(\nu(N,M)\) the normal bundle of \(N\) in \(M\) and \(N\) the fundamental cycle of \(N\). If \(\psi\) is of order \(p\), then the equality \[ \sum_{k=1}^{p-1} C_\ell(k,\gamma) \sum_{N\subset \Omega(k)} \text{Ch} (E_\varphi|_N, \psi^k) \text{Td}(TN) {\mathfrak U}(\nu(N,M),\psi^k) [N]\equiv O(\text{mod\,} p) \] holds for any \(n> m+\ell\), where \[ C_0(k,\gamma)=1, \quad C_1(k,\gamma)= \frac{1} {1-e^{-2n\sqrt{-1} \gamma^{k/p}}}, \quad C_2(k,\gamma)= \frac{1} {| 1-e^{-2n\sqrt{-1} \gamma^{k/p}}|^2}, \] \(\text{Td}(TN)= \prod_{k=1}^d \frac{x_k} {1-e^{-x_k}}\) (Todd class of \(TN\)), \((\dim N=2d)\); \(\text{Ch} (E_\varphi|_N,\psi^k)\) and \({\mathfrak U}(\nu(N,M),\psi^k)\) are characteristic classes defined by suitable formulas.