Pull-back components of the space of foliations of codimension ≥2

e Silva, W.; Neto, A.

doi:10.1090/tran/7245

Pull-back components of the space of foliations of codimension $\ge 2$
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by W. Costa e Silva and A. Lins Neto PDF

Trans. Amer. Math. Soc. 371 (2019), 949-969 Request permission

Abstract:

We present a new list of irreducible components for the space of k-dimensional holomorphic foliations on $\mathbb P^{n}$, $n\geq 3$, $k\ge 2$. They are associated to pull-back of dimension one foliations on $\mathbb P^{n-k+1}$ by non-linear rational maps.

References

César Camacho and Alcides Lins Neto, The topology of integrable differential forms near a singularity, Inst. Hautes Études Sci. Publ. Math. 55 (1982), 5–35. MR 672180, DOI 10.1007/BF02698693
Julie Déserti and Dominique Cerveau, Feuilletages et actions de groupes sur les espaces projectifs, Mém. Soc. Math. Fr. (N.S.) 103 (2005), vi+124 pp. (2006) (French, with English and French summaries). MR 2200857, DOI 10.24033/msmf.415
D. Cerveau and A. Lins Neto, Irreducible components of the space of holomorphic foliations of degree two in $\mathbf C\textrm {P}(n)$, $n\geq 3$, Ann. of Math. (2) 143 (1996), no. 3, 577–612. MR 1394970, DOI 10.2307/2118537
D. Cerveau, A. Lins Neto, and S. J. Edixhoven, Pull-back components of the space of holomorphic foliations on ${\Bbb C}{\Bbb P}(n)$, $n\geq 3$, J. Algebraic Geom. 10 (2001), no. 4, 695–711. MR 1838975
S. C. Coutinho and J. V. Pereira, On the density of algebraic foliations without algebraic invariant sets, J. Reine Angew. Math. 594 (2006), 117–135. MR 2248154, DOI 10.1515/CRELLE.2006.037
Fernando Cukierman and Jorge Vitório Pereira, Stability of holomorphic foliations with split tangent sheaf, Amer. J. Math. 130 (2008), no. 2, 413–439. MR 2405162, DOI 10.1353/ajm.2008.0011
F. Cukierman, J. V. Pereira, and I. Vainsencher, Stability of foliations induced by rational maps, Ann. Fac. Sci. Toulouse Math. (6) 18 (2009), no. 4, 685–715 (English, with English and French summaries). MR 2590385, DOI 10.5802/afst.1221
M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
Ivan Kupka, The singularities of integrable structurally stable Pfaffian forms, Proc. Nat. Acad. Sci. U.S.A. 52 (1964), 1431–1432. MR 173214, DOI 10.1073/pnas.52.6.1431
J. P. Jouanolou, Équations de Pfaff algébriques, Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979 (French). MR 537038, DOI 10.1007/BFb0063393
Alcides Lins Neto, Componentes irredutíveis dos espaços de folheações, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2007 (Portuguese). 26$^\textrm {o}$ Colóquio Brasileiro de Matemática. [26th Brazilian Mathematics Colloquium]. MR 2370063
A. Lins Neto, Germs of complex two dimensional foliations, Bull. Braz. Math. Soc. (N.S.) 46 (2015), no. 4, 645–680. MR 3436562, DOI 10.1007/s00574-015-0107-9
A. Lins Neto and M. G. Soares, Algebraic solutions of one-dimensional foliations, J. Differential Geom. 43 (1996), no. 3, 652–673. MR 1412680, DOI 10.4310/jdg/1214458327
Frank Loray, Jorge Vitório Pereira, and Frédéric Touzet, Foliations with trivial canonical bundle on Fano 3-folds, Math. Nachr. 286 (2013), no. 8-9, 921–940. MR 3066408, DOI 10.1002/mana.201100354
Airton S. de Medeiros, Structural stability of integrable differential forms, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 395–428. MR 0451274
Georges de Rham, Sur la division de formes et de courants par une forme linéaire, Comment. Math. Helv. 28 (1954), 346–352 (French). MR 65241, DOI 10.1007/BF02566941
Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 334, Springer-Verlag, Berlin, 2006. MR 2247603
W. Costa e Silva, Stability of branched pull-back projective foliations, Bull. Braz. Math. Soc. (N.S.) 48 (2017), no. 1, 29–44. MR 3623760, DOI 10.1007/s00574-016-0007-7