Singular foliations and differential \(p\)-forms. (English) Zbl 0997.58001

The notion of integrable differential \(p\)-form is introduced in order to present an adequate analytic object to study the structure of the codimension \(p\) singular foliations. After a preliminary section, in section 2 there are described completely the foliations induced by linear integrable \(p\)-forms on \(K^n\) \((K=\mathbb{R},\mathbb{C})\) and in section 3 there are given the main results on the local structure of plane fields and on the local structure of foliations. Finally, the validity of the results established in sections 2 and 3 is discussed in the context of real analytic, and class \(C^r\), \(p\)-forms and the extension for holomorphic foliations of the complex projective space is presented.


58A10 Differential forms in global analysis
32S65 Singularities of holomorphic vector fields and foliations
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[1] de Medeiros, A.S.). - Structural Stability of Integrable Differential Forms. , 597 (1977), 395-428. · Zbl 0363.58007
[2] de Rham, G.). - Sur la division de formes et de courrants par une forme linéaire. Commentari Math. Hel., 28 (1954), 346-352. · Zbl 0056.31601
[3] Godbillon, C.). - Géométrie Différentielle et Méchanique Analytique. Hermann, Paris, (1969). · Zbl 0174.24602
[4] GOMEZ-MONT y L. ORTIZ-BOBADILLA, X.). - Sistemas Dinamicos Holomorfos en Superficies. Aportaciones Matematicas, Sociedad Matematica Mexicana, 3, , (1989). · Zbl 0855.58049
[5] Kupka, I.). - The singularities of integrable structurally stable Pfaffian forms. Proc. Nat. Acad. Sci., 52 (1964). · Zbl 0137.41404
[6] Malgrange, B.). - Frobenius avec singularités - 2. Le cas général. Inventiones Math., 39 (1977), 67-89. · Zbl 0375.32012
[7] Mattei, J.F.) et Moussu. - Holonomie et Intégrales Premières. Ann. Scient. Éc. Norm. Sup., 4e. Série, 13 (1980), 469-523. · Zbl 0458.32005
[8] Lins Neto, A.). - Holomorphic Rank of Hypersurfaces with an Isolated Singularity. Bol. Soc. Bras. Mat., 29, N. 1 (1998), 145-161. · Zbl 0912.32028
[9] Northcott, D.G.). - Lessons on Rings, Modules and Multiplicities. Cambridge University Press, (1968). · Zbl 0159.33001
[10] Whitney, H.). - Complex Analytic Varieties. Addison-Wesley; Reading, MA, (1972), 87. · Zbl 0265.32008
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