The fixed point statements on Bergman type complex manifolds (Q1355268)

The author proves a number of meromorphic analogues of Cartan's fixed point theorem for holomorphic self-maps on bounded domains. Among them: Let \(f:M\to M\) be a meromorphic self-map of a connected complex manifold \(M\) having non-degenerate Bergman metric form \(b_M\), let \(N_{f(x)}= \#f^{-1}(x)\), and let \(\sigma(f)= ess-\sup_MN_f\). If \(f(x_0)= x_0\) for some \(x_0\in M\) with \(b_M (x_0)>0\), if \(F'(x_0)= \lambda Id\) for some \(\lambda \geq 1\), and if \(\text{det} f'(x_0) =\sqrt {\sigma (f)}\), then \(f\equiv Id\). An example is given of a holomorphic self-map \(f:Q\to Q\) such that \(f(x_0) =x_0\) and such that \(f'(x_0)\) is any given \(n \times n\) matrix. Consequently, the condition on \(\text{det} f'(x_0)\) is necessary.