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A note on M. Soares’ bounds. (English) Zbl 1089.32025

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)] 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.

MSC:

32S65 Singularities of holomorphic vector fields and foliations
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
37F75 Dynamical aspects of holomorphic foliations and vector fields

Citations:

Zbl 1069.32014
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References:

[1] Esteves, E., The Castelnuovo-Mumford regularity of an integral variety of a vector field on projective space, Math. Res. Lett., 9, 1-15 (2002) · Zbl 1037.14022
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[4] Soares, M., The Poincaré problem for hypersurfaces invariant by one-dimensional foliations, Invent. math., 128, 495-500 (1997) · Zbl 0923.32025 · doi:10.1007/s002220050150
[5] Soares, M., Bounding Poincaré-Hopf indices and Milnor numbers, Math. Nachrichten, 278, 6, 703-711 (2005) · Zbl 1069.32014 · doi:10.1002/mana.200310265
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[7] Vainsencher, I., Classes características em geometria algébrica (1985)
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