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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Residue formulas for logarithmic foliations and applications


Authors: Maurício Corrêa and Diogo da Silva Machado
Journal: Trans. Amer. Math. Soc. 371 (2019), 6403-6420
MSC (2010): Primary 32S65, 32S25, 14C17
DOI: https://doi.org/10.1090/tran/7584
Published electronically: February 1, 2019
MathSciNet review: 3937330
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Abstract: In this work we prove a Baum-Bott type formula for noncompact complex manifold of the form $ \tilde {X}=X-{\mathcal D}$, where $ X$ is a complex compact manifold and $ {\mathcal D}$ is a normal crossing divisor on $ X$. As applications, we provide a Poincaré-Hopf type theorem and an optimal description for a smooth hypersurface $ {\mathcal D}$ invariant by an one-dimensional foliation $ {\mathscr F}$ on $ \mathbb{P}^n$ satisfying $ {\rm Sing}({\mathscr F}) \subsetneq {\mathcal D}$.


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Additional Information

Maurício Corrêa
Affiliation: Departamento de Matemática, Universidade Federal de Minas Gerais, Avenida Antônio Carlos 6627, 30123-970 Belo Horizonte, Minas Gerais, Brazil
Email: mauriciojr@ufmg.br

Diogo da Silva Machado
Affiliation: Departamento de Matemática, Universidade Federal de Viçosa, Avenida Peter Henry Rolfs, s/n—Campus Universitário, 36570-900 Viçosa, Minas Gerais, Brazil
Email: diogo.machado@ufv.br

DOI: https://doi.org/10.1090/tran/7584
Keywords: Logarithmic foliations, Poincar\'e--Hopf type theorem, residues
Received by editor(s): November 4, 2016
Received by editor(s) in revised form: November 29, 2017
Published electronically: February 1, 2019
Additional Notes: This work was partially supported by CNPq, CAPES, FAPEMIG, and FAPESP-2015/20841-5.
Article copyright: © Copyright 2019 American Mathematical Society