Local rigidity of quasi-regular varieties

Pasquier, Boris; Perrin, Nicolás

doi:10.1007/s00209-009-0531-x

Cited by 33 publications

(28 citation statements)

References 12 publications

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“…We refer to [30], [31] or [19] for more details on the geometry of these varieties and on the geometry of smooth Fano horospherical varieties of Picard rank one.…”

Section: 1mentioning

confidence: 99%

Sanya Lectures: Geometry of Spherical Varieties

Perrin¹

2017

Acta. Math. Sin.-English Ser.

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These are expanded notes from lectures on the geometry of spherical varieties given in Sanya. We review some aspects of the geometry of spherical varieties. We first describe the structure of B-orbits. Using the local structure theorems, we describe the Picard group and the group of Weyl divisors and give some necessary conditions for smoothness. We later on consider B-stable curves and describe in details the structure of the Chow group of curves as well as the pairing between curves and divisors. Building on these results we give an explicit B-stable canonical divisor on any spherical variety.

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“…We refer to [30], [31] or [19] for more details on the geometry of these varieties and on the geometry of smooth Fano horospherical varieties of Picard rank one.…”

Section: 1mentioning

confidence: 99%

Sanya Lectures: Geometry of Spherical Varieties

Perrin¹

2017

Acta. Math. Sin.-English Ser.

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“…If now X t is a contact manifold for t = 0, when is X 0 still a contact manifold? If b 2 (X t ) = 1, there is a counterexample due to [PP10], see also [Hw10].…”

Section: Deformations I: the Rational Casementioning

confidence: 99%

Contact Kähler Manifolds: Symmetries and Deformations

Peternell

Schrack

2014

Algebraic and Complex Geometry

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We study complex compact Kähler manifolds X carrying a contact structure. If X is almost homogeneous and b 2 (X) ≥ 2, then X is a projectivised tangent bundle (this was known in the projective case even without assumption on the existence of vector fields). We further show that a global projective deformation of the projectivised tangent bundle over a projective space is again of this type unless it is the projectivisation of a special unstable bundle over a projective space. Examples for these bundles are given in any dimension.

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“…These are two orbit varieties, an open orbit and a closed orbit, under the action of their automorphism groups which are non-reductive. See Theorem 2.10 for the precise statements and notation following [PP10]. We also write down some known interesting geometry of these nonhomogeneous horospherical manifolds in Subsection 2.3.…”

Section: Introductionmentioning

confidence: 99%

Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

Hwang¹,

Kim²,

Park³

2021

Preprint

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A horospherical variety is a normal G-variety such that a connected reductive algebraic group G acts with an open orbit isomorphic to a torus bundle over a rational homogeneous manifold. The projective horospherical manifolds of Picard number one are classified by Pasquier, and it turned out that the automorphism groups of all nonhomogeneous ones are non-reductive, which implies that they admit no Kähler-Einstein metrics. As a numerical measure of the extent to which a Fano manifold is close to be Kähler-Einstein, we compute the greatest Ricci lower bounds of projective horospherical manifolds of Picard number one using the barycenter of each moment polytope with respect to the Duistermaat-Heckman measure based on a recent work of Delcroix and Hultgren. In particular, the greatest Ricci lower bound of the odd symplectic Grassmannian SGr(n, 2n + 1) can be arbitrarily close to zero as n grows.

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Local rigidity of quasi-regular varieties

Cited by 33 publications

References 12 publications

Sanya Lectures: Geometry of Spherical Varieties

Sanya Lectures: Geometry of Spherical Varieties

Contact Kähler Manifolds: Symmetries and Deformations

Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one

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