Del Pezzo foliations with log canonical singularities
TL;DR: In this paper, the authors classify del Pezzo foliations of rank at least 3 on projective manifolds and with log canonical singularities in the sense of McQuillan.
About: This article is published in Journal of Pure and Applied Algebra.The article was published on 2022-05-01 and is currently open access. It has received 3 citations till now. The article focuses on the topics: Mathematics & Gravitational singularity.
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TL;DR: In this article, 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, was determined.
Abstract: 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.
8 citations
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TL;DR: In this paper, 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, was determined.
Abstract: 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.
5 citations
Cites background from "Del Pezzo foliations with log canon..."
...cases with rF = rankF −1 and rF = rankF −2 are called del Pezzo and Mukai foliations respectively, and these foliations are intensively studied in the above quoted papers [AD14, AD16, AD17]. See also [Fig19] for a study of del Pezzo foliations, and [Ara19] for an account of this topic on Fano foliations. From this point of view, the canonical foliations on horospherical manifolds give several new example...
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TL;DR: A survey of the theory of holomorphic foliations on complex manifolds can be found in this paper , where the authors briefly discuss some of the early works on this theory, mostly concerned with the local behavior of the leaves near the singularities.
Abstract: 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 C2. 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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Book•
01 Jan 1998TL;DR: In this paper, the authors introduce the minimal model program and the canonical class of rational curves, and present the singularities of the model program, as well as three dimensional flops.
Abstract: 1. Rational curves and the canonical class 2. Introduction to minimal model program 3. Cone theorems 4. Surface singularities 5. Singularities of the minimal model program 6. Three dimensional flops 7. Semi-stable minimal models.
1,754 citations
Book•
01 Jan 1996
TL;DR: It is shown here how the model derived recently in [Bouchut-Boyaval, M3AS (23) 2013] can be modified for flows on rugous topographies varying around an inclined plane.
Abstract: I. Hilbert Schemes and Chow Varieties.- II. Curves on Varieties.- III. The Cone Theorem and Minimal Models.- IV. Rationally Connected Varieties.- V. Fano Varieties.- VI. Appendix.- References.
1,560 citations
651 citations
Journal Article•
647 citations
TL;DR: In this article, the Ricci curvature of a Kdhler manifold has been characterized in terms of the first Chern class of a manifold, which is closely related to Ricci's curvature.
Abstract: Hirzebruch and K odaira [4] have given characterizations of the complex projective spaces. A similar characterization for the complex hyperquadrics has been given by B rie sk o rn [1 ]. (See also a recent paper o f Morrow [8 ] on these topics.) The purpose of the present paper is to give slightly different characterizations o f these spaces. Our motive is to give characterizations which will be useful in differential geometry of compact }Mier manifolds of positive curvature. Our results are expressed in terms of the first Chern class of a manifo ld . The first Chern class is closely related to the Ricci curvature of a manifold. We refer the reader to the paper [6] fo r an application of results of this paper to 3-dimensional compact Kdhler manifolds of positive curvature. A similar characterization has been used recently by Howard [ 5 ] in his work on positively pinched Kdhler manifolds. Results which can be found in Hirzebruch's book [3 ] are used freely often without explicit references. The cohomology o f M with coefficients in the sheaf a( F ) of germs of holomorphic sections of a line bundle (or a vector bundle) F will be denoted by H * (M ; F) instead o f H * (M ; n (F) ) . In
381 citations