Singular Foliations

There are many aspects to singular foliations, and many open questions and problems. Here I am collecting references on the subject, divided into topics. Maybe one day I’ll write a book about singular foliations :)

The topics include:

* Examples and constructions

* “Proper” singular foliations

Camacho, César; Azevedo Scárdua, Bruno On codimension one foliations with Morse singularities on three-manifolds. Topology Appl. 154 (2007), no. 6, 1032–1040.
– Scárdua, Bruno
; Seade, José Codimension 1 foliations with Bott-Morse singularities II. J. Topol. 4 (2011), no. 2, 343–382.

* Baum-Bott residues

– M. Abate, F. Bracci and F. Tovena, Index theorems for holomorphic maps and foliations , Indiana Univ. Math. J. 57 (2008), 2999-3048
– P. Baum and R. Bott, Singularities of holomorphic foliations , J. Differential Geom. 7 (1972), 279-342
– Bracci & Suwa, Perturbation of Baum-Bott residues, 2010
Camacho, César; Lehmann, Daniel Residues of holomorphic foliations relative to a general submanifold. Bull. London Math. Soc. 37 (2005), no. 3, 435–445.
– S Sertoz, Residues of singular holomorphic foliations, 1989
– T. Suwa, Indices of Vector Fields and Residues of Singular Holomorphic Foliations, Actualit ́es Math ́ematiques, Hermann, Paris, 1998
Seade, José; Verjovsky, Alberto Invariance topologique de la classe d’Euler pour des feuilletages singuliers. (French) [Topological invariance of the Euler class for singular foliations] C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 6, 645–648.

* Codimension 1 foliations

Cerveau, Dominique; Lins-Neto, Alcides; Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric Algebraic reduction theorem for complex codimension one singular foliations. Comment. Math. Helv. 81 (2006), no. 1, 157–169.
Cano, Felipe Reduction of the singularities of codimension one singular foliations in dimension three. Ann. of Math. (2) 160 (2004), no. 3, 907–1011.
Casale, Guy Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières. (French) [Singular foliations of codimension one, Galois groupoids and first integrals] Ann. Inst. Fourier (Grenoble) 56 (2006), no. 3, 735–779.
Cerveau, Dominique; Lins-Neto, Alcides; Loray, Frank; Pereira, Jorge Vitório; Touzet, Frédéric Complex codimension one singular foliations and Godbillon-Vey sequences. Mosc. Math. J. 7 (2007), no. 1, 21–54, 166.

* Local normal forms and moduli

* Ergodic theory of singular foliations

* The leaf space

– Pradines, How to define the differentiable graph of a singular foliation, 1988

* Holonomy

Debord : Holonomy Groupoids of Singular Foliations
Iakovos Androulidakis, Georges Skandalis, The holonomy groupoid of a singular foliation, 2007
Moussu, R. Sur l’existence d’intégrales premières holomorphes. (French) [On the existence of holomorphic first integrals] Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 4, 709–717.

* Tangential (leaf-wise) structures

* Transverse structures

* Integrable differential forms & Nambu structures

Farber, Michael Topology of closed one-forms. Mathematical Surveys and Monographs, 108. American Mathematical Society, Providence, RI, 2004. xii+246 pp.
de Medeiros, Airton S. Singular foliations and differential $p$-forms. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), no. 3, 451–466.

* Riemannian foliations

Lytchak, Alexander Geometric resolution of singular Riemannian foliations. Geom. Dedicata 149 (2010), 379–395.
– Alexandrino, Marcos M. Singular Riemannian foliations with sections. Illinois J. Math. 48 (2004), no. 4, 1163–1182
– P. Molino, Orbit-like foliations. Geometric study of foliations (Tokyo, 1993), 97–119, World Sci. Publ., River Edge, NJ, 1994.

* Actions of Lie groups and Lie algebras

Álvarez López, J. A.; Arraut, J. L.; Biasi, C. Foliations by planes and Lie group actions. Ann. Polon. Math. 82 (2003), no. 1, 61–69.

* Lie groupoids and Lie algebroids

Crainic, Marius; Fernandes, Rui Loja Integrability of Lie brackets. Ann. of Math. (2) 157 (2003), no. 2, 575–620.

– Jotz, The leaf space of a multiplicative foliation

* Low dimensions

Mendes, Luís Gustavo Kodaira dimension of holomorphic singular foliations. Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 2, 127–143.
Fenley, Sérgio R. Foliations with good geometry. J. Amer. Math. Soc. 12 (1999), no. 3, 619–676.
Le Floch, Laurent Rigidité générique des feuilletages singuliers. (French) [Generic rigidity of singular foliations] Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 6, 765–785.

* C* algebras of singular foliations

– Sheu, Singular foliation C* algebras, 1988
Androulidakis, Iakovos; Skandalis, Georges Pseudodifferential calculus on a singular foliation. J. Noncommut. Geom. 5 (2011), no. 1, 125–152.

* Various / related

РLevitt, 1-formes ferm̩es singuli̬res et groupe fondamental, Invent Math 1987
– F Loray, Singular foliations with trivial canonical class, 2011
– Rajeevsarathy, The equivalence of measured foliations and measured laminations
Scárdua, Bruno Differential algebra and Liouvillian first integrals of foliations. J. Pure Appl. Algebra 215 (2011), no. 5, 764–788.
Wolak, Robert A. Connections for singular foliations on stratified manifolds. Publ. Math. Debrecen 63 (2003), no. 4, 623–633.
Rebelo, Julio C. Meromorphic vector fields and elliptic fibrations. Michigan Math. J. 52 (2004), no. 1, 33–59
– C. F. B. Palmeira, Open manifolds foliated by planes , Ann. of Math. 107 (1978), 109–131.
Malgrange, Bernard Le groupoïde de Galois d’un feuilletage. (French) [The Galois groupoid of a foliation] Essays on geometry and related topics, Vol. 1, 2, 465–501, Monogr. Enseign. Math., 38, Enseignement Math., Geneva, 2001.
Brunella, Marco Birational geometry of foliations. Available electronically at Monografías de Matemática. [Mathematical Monographs]
Mendes, Luís Gustavo Kodaira dimension of holomorphic singular foliations. Bol. Soc. Brasil. Mat. (N.S.) 31 (2000), no. 2, 127–143

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