Digital Object Identifier (DOI) 10.1007/s002090100352 Math. Z. 242, 481–490 (2002) Residual indices of holomorphic maps relative to singular curves of fixed points on surfaces Filippo Bracci ⋆ , Francesca Tovena DipartimentodiMatematica,Universit` adiRoma“TorVergata”,ViadellaRicercaScientifica 00133 Roma, Italy (e-mail: fbracci@mat.uniroma2.it / tovena@mat.uniroma2.it) Received: 15 May 2000; in final form: 10 July 2001 / Published online: 1 February 2002 – c  Springer-Verlag 2002 Abstract. Let M beatwo-dimensionalcomplexmanifoldandlet f : M → M be a holomorphic map that fixes pointwise a (possibly) singular, com- pact, reduced and globally irreducible curve C ⊂ 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 . Introduction In [2], C. Camacho and P. Sad introduced the index of a holomorphic vector field relative to an invariant non-singular curve and proved an index for- mula. Their result was generalized by A. Lins Neto [6] to the case of an algebraic foliation and a (possibly) singular invariant curve in the complex projective plane. Finally T. Suwa [7] gave a definition of index and proved a formula when the invariant (singular) curve lies in a generic two dimen- sional complex manifold. Recently M. Abate [1] (cf. also Sect. 1), studying discrete dynamical systems, introduced an index for holomorphic self-maps ofatwodimensionalcomplexmanifoldfixingasmoothcompactcurve(and non-degenerate on it), proving an analogue of the Camacho-Sad Theorem. Here we generalize Abate’s result to the case of singular curves, finding an analogue of Suwa’s Theorem. ⋆ Partially supported by Progetto MURST di Rilevante Interesse Nazionale Propriet` a geometriche delle variet` a reali e complesse.