Equivariant resolution of points of indeterminacy

Reichstein, Zinovy; Youssin, Boris

doi:10.1090/s0002-9939-02-06595-4

Cited by 18 publications

(9 citation statements)

References 14 publications

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“…Thus, by the equivariant version of the Hironaka resolution of indeterminacy [RY02] and by virtue of Condition 0.2, we can resolve the map f after blowing up X along Z. Now the first assertion is clear.…”

Section: This Implies Thatmentioning

confidence: 86%

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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Section: This Implies Thatmentioning

confidence: 86%

Fano manifolds and stability of tangent bundles

Kanemitsu

2019

Preprint

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“…We need also the elimination of indeterminacy for algebraic spaces. The following proof follows [RY02].…”

Section: Flatness Of Good Degenerationsmentioning

confidence: 98%

An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points

Hu¹

2021

Preprint

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“…Then Z is C * -invariant. This follows from the existence of an equivariant strong resolution of singularities (see [RY02,Ko07]).…”

Section: The Proofmentioning

confidence: 99%

Kaehler-Einstein Fano threefolds of degree 22

Cheltsov¹,

Shramov²

2018

Preprint

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We study the problem of existence of Kähler-Einstein metrics on smooth Fano threefolds of Picard rank one and anticanonical degree 22 that admit a faithful action of the multiplicative group C * . We prove that, with a possible exception of two explicitly described cases, all such smooth Fano threefolds are Kähler-Einstein.All varieties are assumed to be projective and defined over the field of complex numbers.

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Equivariant resolution of points of indeterminacy

Abstract: Abstract. We prove an equivariant form of Hironaka's theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro's equivariant Chow lemma.

Cited by 18 publications

References 14 publications

Fano manifolds and stability of tangent bundles

Fano manifolds and stability of tangent bundles

An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points

Kaehler-Einstein Fano threefolds of degree 22

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