Singularities of holomorphic foliations

“…This means that, whenever U ⊂ M is an open subset and v is a holomorphic section of |U such that v x ∈ T x ∀ x ∈ U ∩ (M \ S), then v x ∈ T x holds also for x in U ∩ S. We remark that, given an involutive sheaf T , which induces a foliation with singular set S, there is a unique sheaf T that is both full and involutive, and such that T |M\S = T |M\S . We can therefore restrict our attention to full involutive sheaves as a way to avoid artificial singularities (see Baum andBott 1972, Suwa 1998).…”

Section: Rogério S Molmentioning

confidence: 99%

Flags of holomorphic foliations

Mol

2011

An. Acad. Bras. Ciênc.

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A flag of holomorphic foliations on a complex manifold M is an object consisting of a finite number of singular holomorphic foliations on M of growing dimensions such that the tangent sheaf of a fixed foliation is a subsheaf of the tangent sheaf of any of the foliations of higher dimension. We study some basic properties of these objects and, in P n C , n ≥ 3, we establish some necessary conditions for a foliation of lower dimension to leave invariant foliations of codimension one. Finally, still in P n C , we find bounds involving the degrees of polar classes of foliations in a flag.

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“…When Sing(P) has "good" dimension (that is, m = r − 1), the number of singularities is given by the classical formula of Baum, Bott, Kempf, Laksov ( [2], [4]). So Theorem 2.1 extends this classical formula for the case of "bad" dimension of the singularities.…”

Section: Higher Dimensional Foliations On a Projective Varietymentioning

confidence: 99%

“…, C l are the irreducible components of Sing(D). By Baum-Bott-KempfLaksov theorem (see [2] for the integrable case or [4] for the general case),…”

Section: One Dimensional Foliations On P Nmentioning

confidence: 99%

Number of singularities of a foliation on ${\mathbb P}^n$

Salas

2001

Proc. Amer. Math. Soc.

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Abstract. Let D be a one dimensional foliation on a projective space, that is, an invertible subsheaf of the sheaf of sections of the tangent bundle. If the singularities of D are isolated, Baum-Bott formula states how many singularities, counted with multiplicity, appear. The isolated condition is removed here. Let m be the dimension of the singular locus of D. We give an upper bound of the number of singularities of dimension m, counted with multiplicity and degree, that D may have, in terms of the degree of the foliation. We give some examples where this bound is reached. We then generalize this result for a higher dimensional foliation on an arbitrary smooth and projective variety.

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Singularities of holomorphic foliations

Cited by 165 publications

References 18 publications

On non-Kupka points of codimension one foliations on ℙ3

On non-Kupka points of codimension one foliations on ℙ3

Flags of holomorphic foliations

Number of singularities of a foliation on ${\mathbb P}^n$

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