Codimension One Regular Foliations on Rationally Connected Threefolds

Figueredo, João Paulo

doi:10.1007/s00574-022-00298-5

Cited by 2 publications

(1 citation statement)

References 15 publications

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“…After a sequence of smooth blowdowns centered at smooth curves, 𝑋 → 𝑋 1 → 𝑋 2 → ⋯ → 𝑋 𝑛 , one reaches a smooth threefold 𝑋 𝑛 , together with a regular codimension one foliation ℱ 𝑛 on 𝑋 𝑛 , such that either 𝐾 ℱ 𝑛 is nef, or 𝑋 𝑛 → 𝑌 𝑛 is a Mori fiber space with fibers tangent to ℱ 𝑛 . In the latter case, the second named author showed in [Fig21] that ℱ 𝑛 is induced by a fibration by rational surfaces, and hence so is ℱ. This proves Touzet's conjecture for codimension one foliations on threefolds in some special cases, and reduces the problem to understanding regular foliations with nef canonical class.…”

Section: Regular Foliationsmentioning

confidence: 67%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

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In these notes we survey some aspects of the theory of holomorphic foliations on complex manifolds. The origins of the theory go back to works of Darboux, Poincaré and Painlevé, where it was developed to study solutions of ordinary differential equations on ℂ 2 . We briefly discuss some of the early works on this theory, mostly concerned with the local behavior of the leaves near the singularities. We then move the focus from local to global properties. Birational geometry has had a great influence on the development of a global theory of holomorphic foliations. After reviewing the Enriques-Kodaira classification of projective surfaces and explaining the general philosophy of the Mininal Model Program, we explore some of their recent counterparts for foliations.

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Section: Regular Foliationsmentioning

confidence: 67%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

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On codimension one holomorphic distributions on compact toric orbifolds

Rodríguez Peña

2024

Int. J. Math.

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The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we provide a classification for regular distributions on rational normal scrolls and weighted projective spaces. Additionally, under specific conditions, we prove that the singular set of a codimension one holomorphic foliation on a compact toric orbifold admits at least one irreducible component of codimension two, and we also present a Darboux–Jouanolou type integrability theorem for codimension one holomorphic foliations. Our results are exemplified through various illustrative examples.

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Codimension One Regular Foliations on Rationally Connected Threefolds

Cited by 2 publications

References 15 publications

Foliations on Complex Manifolds

Foliations on Complex Manifolds

On codimension one holomorphic distributions on compact toric orbifolds

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