Cartier divisors and geometry of normalG-varieties

Timashev, Dmitry A.

doi:10.1007/bf01236468

Cited by 18 publications

(26 citation statements)

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“…Similar results for the case of varieties with general reductive group actions of complexity one can be found in [Tim00]. It is well known for toric varieties that the anti-canonical divisor is always big.…”

Section: Proposition 35 ([Ps] Section 34)supporting

confidence: 76%

Contributions to Algebraic Geometry

Gatto¹,

Scherbak²

2012

EMS Series of Congress Reports

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Abstract. This is a survey of the language of polyhedral divisors describing T -varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.

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Section: Proposition 35 ([Ps] Section 34)supporting

confidence: 76%

Contributions to Algebraic Geometry

Gatto¹,

Scherbak²

2012

EMS Series of Congress Reports

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“…In this section we will summarise the theory of group actions of complexity one on normal varieties. This is work mainly completed by Timashev [Tim97,Tim00] and proofs of all results in this section can be found in [Tim11]. This theory is complicated and for reasons of space our coverage of it is quite brief.…”

Section: Actions Of Complexity Onementioning

confidence: 99%

$K$-polystability of smooth Fano $\text{SL}_2$-threefolds

Rogers¹

2022

Preprint

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We prove the K-polystability of all smooth complex Fano threefolds admitting an effective action of SL 2 but not of a 2-torus or 3-torus. In particular, the existence of Kähler-Einstein metrics on varieties in the families (1.10), (1.15), (1.16), (1.17), (2.21), (2.27), (2.32), (3.13), (3.17), (3.25) and (4.6) of the Mori-Mukai classification of smooth Fano threefolds is proved.Theorem 2.2. [CDS15] A smooth complex Fano variety X admits a Kähler-Einstein metric if and only if (X, K −1 X ) is K-polystable. Unfortunately, since there are generally infinitely many test configurations for a given polarised variety, it is very difficult to check K-(poly/semi)stability in the general case, even accounting for the Li-Xu theorem.The α-invariant of Tian [Tia87] was for a long time one of the only practical methods to check K-(poly)stability. Recently, work of Abban-Zhuang [AZ20,AZ21] has provided other more powerful methods of verification. Otherwise, most progress on this front has come from the equivariant perspective due to the work of Datar-Székelyhidi [DS15], which we summarise now.Definition 2.5. Let X be a G-variety, and let π : L → X be a line bundle on X. We say that L is G-linearised if there is a G-action on L such that π is G-equivariant and the map π −1 (x) → π −1 (g • x) induced on the fibres is linear for all g ∈ G and all x ∈ X.Definition 2.6. Let G be a reductive algebraic group and let (X, L) be a polarised variety with a G-action on X such that L is G-linearised. A test configuration (X , L) of exponent m is G-equivariant if there is a G-action on (X , L) which commutes with the C × action and such that the isomorphisms between (X, L ⊗m ) and (X t , L t ) for t = 0 are G-equivariant. Then (X, L) is equivariantly K-(poly/semi)stable if it is K-(poly/semi)stable with respect to G-equivariant special test configurations.The main result of Datar-Székelyhidi is the following: Theorem 2.3. [DS15] Let G be a reductive algebraic group and let X be a smooth complex Fano G-variety. Then (X, K −1 X ) is equivariantly K-polystable if and only if X admits a Kähler-Einstein metric.Remark. We should mention that the result of Datar-Székelyhidi has been generalised to the singular case when G is finite by Liu-Zhu [LZ20] and for general reductive groups by Zhuang [Zhu21]. Specifically, Zhuang uses a purely algebraic argument showing (among other things) that K-polystability of a log Fano pair (X, ∆) is equivalent to G-equivariant K-polystability when G is reductive.2.2. β invariant. Here we discuss an invariant introduced by Fujita [Fuj16] and Li [Li17] which they have shown to have an intimate connection to K-stability. We must first include some preliminary definitions.

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“…We denote the complexity by c. We enumerate the possible locations of the blocks X ij . If Lemmas 5,6,7,8 give an estimate c 2 for a given case, we shall not consider this case. We shall not consider cases, that are symmetrical to the cases already considered.…”

Section: Lemma 5 Consider An Action Obtained From the Original Actimentioning

confidence: 99%