On some integral invariants, Lefschetz numbers and induction maps (Q916113)

Let M be a compact complex manifold, G a compact subgroup of the complex Lie group of all holomorphic automorphism of M and \({\mathcal G}\) the Lie algebra of G. For \(g\in G\) the Lefschetz number, L(g) is by definition \(L(g)=\sum_{i}(-1)^ itr(g| H^ i),\) \(H^ i\) the ith Dolbeault cohomology group of M. The authors show that the second term of the Taylor expansion of L (exp tX), \(X\in G\), coincides up to a constant with a character f: \(G\to {\mathbb{C}}\) previously defined by the first author. f vanishes if M admits a Kähler-Einstein metric.