Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces

Abstract.

Let M be a two-dimensional complex manifold and let $f:M\to M$ be a holomorphic map that fixes pointwise a (possibly) singular, compact, reduced and globally irreducible curve $C\subset M$. We give a notion of degeneracy of f at a point of C. It turns out that f is non-degenerate at one point if and only if it is non-degenerate at every point of C. When f is non-degenerate on C, we define a residual index for f at each point of C. Then we prove that the sum of the indices is equal to the self-intersection number of C.