A note on M. Soares' bounds
The Poincare-Hopf index \(i\) of a holomorphic map germ \(f:(\mathbb{C}^n,0)\rightarrow (\mathbb{C}^n,0)\) is \[ \dim_{\mathbb{C}}\mathbb{C} [| x_1,\dots,x_n| ]/\langle f_1,\dots,f_n\rangle. \] In [Math. Nachr. 278, No. 6, 703--711 (2005; Zbl 1069.32014)] \textit{M. Soares} found some upper bounds for \(i\). He has constructed a foliation of \(\mathbb{C}\mathbb{P}^n\) with an isolated singularity at 0 of the same index as the original map germ. When all the singularities of the foliation are isolated, M. Soares deduced the bound using the Baum-Bott formula; in the general case he used an analytic deformation argument. In the present paper, the authors replace the deformation argument of M. Soares by a direct application of Fulton's intersection theory. They find a bound for the Poincaré-Hopf index of an isolated singularity of a foliation of \(\mathbb{CP}^n\). They also give special bounds for foliations that admit smooth invariant hypersurfaces not passing through the singularity.
scientific article; zbMATH DE number 5015308