Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One

Lee, Jae Hyouk; Park, Kyeong Dong; Yoo, Sungmin

doi:10.3390/math9010102

Cited by 5 publications

(2 citation statements)

References 35 publications

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“…In this case, the only possible lattice moment polytope is [0, 2α 1 ] = [0, 4̟ 1 ] from the inequality −̟ ∨ 1 , (x − 1)α 1 ≥ −1. Indeed, this is the moment polytope of the 3-dimensional projective space P 3 which is an example of wonderful compactifications constructed by De Concini and Procesi [DCP83] (see also [LPY21,Example 2.4]).…”

Section: Spherical Varieties and Colors Equivariant Compactifications...mentioning

confidence: 99%

K-stability of Gorenstein Fano group compactifications with rank two

Lee¹,

Park²,

Yoo³

2022

Preprint

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We give a classification of Gorenstein Fano bi-equivariant compactifications of semisimple complex Lie groups with rank two, and determine which of them are equivariant K-stable and admit (singular) Kähler-Einstein metrics. As a consequence, we obtain several explicit examples of K-stable Fano varieties admitting (singular) Kähler-Einstein metrics. We also compute the greatest Ricci lower bounds, equivalently the delta invariants for K-unstable varieties. This gives us three new examples on which each solution of the Kähler-Ricci flow is of type II.

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Section: Spherical Varieties and Colors Equivariant Compactifications...mentioning

confidence: 99%

K-stability of Gorenstein Fano group compactifications with rank two

Lee¹,

Park²,

Yoo³

2022

Preprint

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“…We also write down some known interesting geometry of these nonhomogeneous horospherical manifolds in Subsection 2.3. Contrary to smooth projective symmetric varieties of Picard number one (see [LPY21]), all nonhomogeneous projective horospherical manifolds of Picard number one admit no Kähler-Einstein metrics by a theorem of Matsushima in [Mat57] since their automorphism groups are not reductive.…”

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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