Del Pezzo foliations with log canonical singularities

Figueredo, João Paulo

doi:10.1016/j.jpaa.2021.106926

Cited by 2 publications

(1 citation statement)

References 17 publications

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“…Under restrictions on the singularities, there is a classification of del Pezzo foliations of rank 𝑟 ≥ 3, parallel to Fujita's classification of del Pezzo manifolds. We refer to [Fig22] for an account on the classification of del Pezzo foliations. The birational classification of codimension one foliations on surfaces can be summarized as follows (see [Bru15] and [McQ08]).…”

Section: Canonical Class Of Foliationsmentioning

confidence: 99%

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: Canonical Class Of Foliationsmentioning

confidence: 99%

Foliations on Complex Manifolds

Araujo¹,

Figueredo²

2022

Notices Amer. Math. Soc.

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Fano manifolds and stability of tangent bundles

Kanemitsu

2019

Preprint

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We determine the stability/instability of the tangent bundles of the Fano varieties in a certain class of two orbit varieties, which are classified by Pasquier in 2009. As a consequence, we show that some of these varieties admit unstable tangent bundles, which disproves a conjecture on stability of tangent bundles of Fano manifolds. 0.1. A Fano manifold X is, by definition, a smooth projective variety X whose anti-canonical divisor −K X is ample. From a differential geometric viewpoint, it is very important to detect which Fano manifolds admit Kähler-Einstein metrics and, after the celebrated works [Tia97,Don02,Ber16,CDS15a,CDS15b,CDS15c,Tia15], it has been accomplished that the existence of a Kähler-Einstein metric on a given Fano manifold X is equivalent to a purely algebraic stability condition for X, called K-polystability.As is well-known, not every Fano manifold admits a Kähler-Einstein metric. For example, Matsushima proved that the existence of a Kähler-Einstein metric on a Fano manifold X implies the reductivity of the automorphism group of X [Mat57]. The automorphism group of a Fano manifold X is, however, not always reductive. Also it is usually very difficult to determine whether or not a given Fano manifold X admits a Kähler-Einstein metric, though the existence of such a metric is rephrased by the K-polystability of the Fano manifold X in question.Therefore it would be useful to study several variants of stability conditions on Fano manifolds. As such a variant, stability of tangent bundles (in the sense of Mumford-Takemoto) has attracted several attention of researchers. By the Kobayashi-Hitchin correspondence, the polystability of the tangent bundle is equivalent to the existence of a Hermitian-Einstein metric on the bundle [Kob82, Lüb83, Don85, UY86, Don87], and hence the polystability of the tangent bundle is weaker than the existence of a Kähler-Einstein metric. Also, thanks to its simplicity, the stability of the tangent bundle is rather easy to handle in the framework of algebraic geometry. Moreover it is expected that, for a given Fano manifold X, the (in)stability of the tangent bundle reflects very well the geometry of X. For example, a folklore conjecture 1 claims the following: Conjecture 0.1 (Stability of tangent bundles). Let X be a Fano manifold. Assume that the Picard number ρ X of X is one. Then the tangent bundle Θ X is (semi)stable. Conjecture 0.1 has been confirmed in the following cases:(1) r X = 1 [Rei78, Theorem 3], where r X is the Fano index of X;

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Del Pezzo foliations with log canonical singularities

Cited by 2 publications

References 17 publications

Foliations on Complex Manifolds

Foliations on Complex Manifolds

Fano manifolds and stability of tangent bundles

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