Fórmulas residuais de tipo Bott e invariante de Futaki para orbifolds complexos

Abstract

This work is divided int two parts. In the first part of the work, in Theorem 2.1, we give a first version of Botts fórmula for a compact complex orbifold with isolated singularities. In Theorem 2.4, using a good resolution and the local Chern class "top" of the orbifold, we give a second version ot theorem 2.1. An interesting consequence of theorem 2.1 is its application to weighted projective spaces Pn ! = P(w0, ...,wn). For example, since we guarantee the existence of nontrivial holomorphic foliations in Pn ! (proposition 2.12), we deduce a formula for the sum of the Milnor numbers of an orbifold of a holomorphic foliation with isolated singularities in Pn ! (corollary 2.13); in particular such a formula is an obstruction to a non-singular foliation in Pn ! (corollary 2.14). In corollary 2.15, we present some relations between the singular set of the foliation and the singular set of P2 !. We end with some examples. As another application of theorem 2.1, similar to the smooth case, we introduce the Baum-Bott numbers associated with the singularities of a foliation with isolated singularities in a compact orbifold surface with isolated singularities. Then, in the Theorem 2.19, we deduce a formula for the sum of the Baum-Bott numbers orbifold for a foliation with isolated singularities in P2 !. Consequently, in the corollaries 2.22 and 2.24, we give a characterization of the foliations with radial singularities in P2 !. We end this part with some examples. As a final application of the theorem 2.1, we have theorem 2.25. This theorem, thogether with corollary 2.13, allows us to give a limit to the degree of an almost smooth curve irreducible invariant by a foliation, as a function of the degree of the foliation in P2 !; this is done in corollary 2.28, which is Poincaré Problem in P2 !, was given for the first time in [12] (we generalize this result: The hypothesis Sing(F) \ Sing(P2 !) = ? is not necessary). More generally, in theorem 2.29, we give a limit to the degree of an almost smooth hypersurface irreducible invariant by a foliation, as a function of the degree of the foliation in Pn ! . This last theorem is a work in collaboration with Fabio E. Brochero and Maurício Corrêa Jr. (see [7]). In the theorem 2.4, under certain hypotheses, we give a second version of Botts formula in a compact orbifold with isolated singularities, as a function of a good resolution of the orbifold and the local Chern class "top", which is defined locally around each singularity of the orbifold. We give a geometric interpretation of the local Chern class "top" for foliations in corollary 2.5. It is well known that the kth surface Hirzebruch Hk, for k > 1, is a good resolution of P(1 : 1 : k) and from this we derive theorem 2.17. In the second part of the work, following W. Ding and G. Tian [15], we give in Theorem 3.5 a demonstration of the location formula for the Calabi-Futaki invariant for a compact complex orbifold with isolated singularities (the annulment of the invariant is a necessary condition for the existence of Kähler-Einstein metrics on the orbifold). As an application of this theory, we study in theorem 3.8 the non-existence of Kähler-Einstein metrics in well-formed singular weighted projective spaces. We end is part with some examples.