Residue formulas for logarithmic foliations and applications
HTML articles powered by AMS MathViewer
- by Maurício Corrêa and Diogo da Silva Machado PDF
- Trans. Amer. Math. Soc. 371 (2019), 6403-6420 Request permission
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 $\textrm {Sing}({\mathscr F}) \subsetneq {\mathcal D}$.References
- A. G. Aleksandrov, The index of vector fields, and logarithmic differential forms, Funktsional. Anal. i Prilozhen. 39 (2005), no. 4, 1–13, 95 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 39 (2005), no. 4, 245–255. MR 2197510, DOI 10.1007/s10688-005-0046-0
- Paolo Aluffi, Chern classes for singular hypersurfaces, Trans. Amer. Math. Soc. 351 (1999), no. 10, 3989–4026. MR 1697199, DOI 10.1090/S0002-9947-99-02256-4
- Elena Angelini, Logarithmic bundles of hypersurface arrangements in $\mathbf {P}^n$, Collect. Math. 65 (2014), no. 3, 285–302. MR 3240995, DOI 10.1007/s13348-014-0112-0
- Paul Baum and Raoul Bott, Singularities of holomorphic foliations, J. Differential Geometry 7 (1972), 279–342. MR 377923
- Jean-Paul Brasselet, Jose Seade, and Tatsuo Suwa, An explicit cycle representing the Fulton-Johnson class. I, Singularités Franco-Japonaises, Sémin. Congr., vol. 10, Soc. Math. France, Paris, 2005, pp. 21–38 (English, with English and French summaries). MR 2145946
- Jean-Paul Brasselet, José Seade, and Tatsuo Suwa, Vector fields on singular varieties, Lecture Notes in Mathematics, vol. 1987, Springer-Verlag, Berlin, 2009. MR 2574165, DOI 10.1007/978-3-642-05205-7
- F. E. Brochero Martínez, M. Corrêa, and A. M. Rodríguez, Poincaré problem for weighted projective foliations, Bull. Braz. Math. Soc. (N.S.) 48 (2017), no. 2, 219–235. MR 3654144, DOI 10.1007/s00574-016-0003-y
- Marco Brunella, Birational geometry of foliations, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2004. MR 2114696
- Shiing-shen Chern, A simple intrinsic proof of the Gauss-Bonnet formula for closed Riemannian manifolds, Ann. of Math. (2) 45 (1944), 747–752. MR 11027, DOI 10.2307/1969302
- Maurício Corrêa Jr. and Marcos Jardim, Bounds for sectional genera of varieties invariant under Pfaff fields, Illinois J. Math. 56 (2012), no. 2, 343–352. MR 3161327
- Maurício Corrêa Jr. and Márcio G. Soares, A note on Poincaré’s problem for quasi-homogeneous foliations, Proc. Amer. Math. Soc. 140 (2012), no. 9, 3145–3150. MR 2917087, DOI 10.1090/S0002-9939-2012-11193-1
- Maurício Corrêa Jr. and Márcio G. Soares, A Poincaré type inequality for one-dimensional multiprojective foliations, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 3, 485–503. MR 2833814, DOI 10.1007/s00574-011-0026-3
- Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970 (French). MR 0417174, DOI 10.1007/BFb0061194
- Igor V. Dolgachev, Logarithmic sheaves attached to arrangements of hyperplanes, J. Math. Kyoto Univ. 47 (2007), no. 1, 35–64. MR 2359100, DOI 10.1215/kjm/1250281067
- E. Esteves and S. Kleiman, Bounds on leaves of one-dimensional foliations, Bull. Braz. Math. Soc. (N.S.) 34 (2003), no. 1, 145–169. Dedicated to the 50th anniversary of IMPA. MR 1993042, DOI 10.1007/s00574-003-0006-3
- Xavier Gómez-Mont, An algebraic formula for the index of a vector field on a hypersurface with an isolated singularity, J. Algebraic Geom. 7 (1998), no. 4, 731–752. MR 1642757
- X. Gómez-Mont, J. Seade, and A. Verjovsky, The index of a holomorphic flow with an isolated singularity, Math. Ann. 291 (1991), no. 4, 737–751. MR 1135541, DOI 10.1007/BF01445237
- Shigeru Iitaka, Logarithmic forms of algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 525–544. MR 429884
- Nicholas M. Katz, The regularity theorem in algebraic geometry, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 437–443. MR 0472822
- Daniel Lehmann, Marcio Soares, and Tatsuo Suwa, On the index of a holomorphic vector field tangent to a singular variety, Bol. Soc. Brasil. Mat. (N.S.) 26 (1995), no. 2, 183–199. MR 1364267, DOI 10.1007/BF01236993
- Xia Liao, Chern classes of logarithmic vector fields, J. Singul. 5 (2012), 109–114. MR 2928937, DOI 10.5427/jsing.2012.5h
- Yoshiki Norimatsu, Kodaira vanishing theorem and Chern classes for $\partial$-manifolds, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 4, 107–108. MR 494655
- Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), no. 2, 265–291. MR 586450
- José A. Seade and Tatsuo Suwa, A residue formula for the index of a holomorphic flow, Math. Ann. 304 (1996), no. 4, 621–634. MR 1380446, DOI 10.1007/BF01446310
- Roberto Silvotti, On a conjecture of Varchenko, Invent. Math. 126 (1996), no. 2, 235–248. MR 1411130, DOI 10.1007/s002220050096
- Marcio G. Soares, The Poincaré problem for hypersurfaces invariant by one-dimensional foliations, Invent. Math. 128 (1997), no. 3, 495–500. MR 1452431, DOI 10.1007/s002220050150
- Tatsuo Suwa, Indices of vector fields and residues of singular holomorphic foliations, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1998. MR 1649358
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
- 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6403-6420
- MSC (2010): Primary 32S65, 32S25, 14C17
- DOI: https://doi.org/10.1090/tran/7584
- MathSciNet review: 3937330