On a character of the automorphism group of a compact complex manifold (Q1088173)

Let h be a hermitian metric on a compact complex m-dimensional manifold M and let \(\gamma_ h\) be its Ricci curvature. If G is the group of automorphisms of M the author defines \(f: G\to R\) by \[ f(\sigma)=\int_ M\log \det (\sigma^*h)\sum^{m}_{k=0}\gamma^ k_ h\wedge \gamma_ h^{m-k},\quad \sigma \in G. \] He shows that f is independent of h and is a character, i.e. \(f(\sigma \tau)=f(\sigma)+f(\tau)\). If \(c_ 1(M)\leq 0\) then \(f=0\). The author shows that f may be not trivial if \(c_ 1(M)>0\). Finally he gives a formula for f in terms of the differential graded algebra \(WU_ m\) used in the definition of the secondary characteristic classes.