Residue formulas for logarithmic foliations and applications

Corrêa, Maurício; da Silva Machado, Diogo

doi:10.1090/tran/7584

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