Deformations of nodal Kähler-Einstein Del Pezzo surfaces with discrete automorphism groups

Spotti, Cristiano

doi:10.1112/jlms/jdt076

Cited by 19 publications

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“…It is also worth noting that this kind of continuity at the boundary may be seen as an higher dimensional generalization in the KE case of the glueing results obtained by Spotti in [41] and Biquard-Rollin in [9], but our proof here follows a very different approach. Another important point to remark, especially in view of applications to moduli spaces, is that the converse of our theorem, that is the fact that weak KE Fano varieties are indeed K-polystable was proved (without 1 In general one can also define the notion of being smoothable, without the condition on the relative anti-canonical divisor.…”

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confidence: 71%

“…Let π : X → ∆ and π ′ : X ′ → ∆ be two Q-Gorenstein smoothings of Q-Fano varieties X 0 and X ′ 0 . Suppose X t and X ′ t are bi-holomorphic for all t = 0, and X 0 and X ′ 0 are both K-polystable with discrete automorphism group, then X 0 and X ′ 0 are isomorphic varieties.Notice that Corollary 1.2 would also follow from the separatedness of general polarized K-stable varieties, which is attempted purely algebraically by Odaka and Thomas [38].It is also worth noting that this kind of continuity at the boundary may be seen as an higher dimensional generalization in the KE case of the glueing results obtained by Spotti in [41] and Biquard-Rollin in [9], but our proof here follows a very different approach. Another important point to remark, especially in view of applications to moduli spaces, is that the converse of our theorem, that is the fact that weak KE Fano varieties are indeed K-polystable was proved (without 1 In general one can also define the notion of being smoothable, without the condition on the relative anti-canonical divisor.…”

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confidence: 76%

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Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties

Spotti¹,

Sun²,

Yao³

2016

Duke Math. J.

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In this article we prove the existence of Kähler-Einstein metrics on Q-Gorenstein smoothable, K-polystable Q-Fano varieties and we show how these metrics behave, in the Gromov-Hausdorff sense, under Q-Gorenstein smoothings. P N by sections of K −λ X . A weak Kähler-Einstein metric on X is a Kähler current in 2πc 1 (X) with locally continuous potential, and that is a smooth Kähler-Einstein metric on the smooth part X reg of X. Note there are different definitions in the literature, but they are all equivalent in this context [23]. A Q-Fano variety X is called Q-Gorenstein smoothable 1 if there is a flat family π : X → ∆, over a disc ∆ in C so that X ∼ = X 0 , X t is smooth for t = 0, and X admits a relatively Q-Cartier anti-canonical divisor −K X /∆ (in this case π : X → ∆ is called a Q-Gorenstein smoothing of X 0 ). It is well-known that, by possibly shrinking ∆, we may assume that X t is a Fano manifold for t = 0, and that there exists an integer λ > 0 such that K −λ Xt are very ample line bundles with vanishing higher cohomology for all t ∈ ∆. Moreover, the dimension, denoted by N (λ), of the corresponding linear systems | − λK Xt | is constant in t. Thus, when needed, we may assume that the family X is relatively very ample, i.e. there is a smooth embedding i :Xt . The main theorem of this paper is the following result, which extends the results of [12,13,14,15] to Q-Gorenstein smoothable Q-Fano varieties and simultaneusly gives some understanding on the way such singular metric spaces are approached by smooth KE metrics on the (analytically) nearby Fano manifolds.Theorem 1.1. Let π : X → ∆ be a Q-Gorenstein smoothing of a Q-Fano variety X 0 . If X 0 is K-polystable then X 0 admits a weak Kähler-Einstein metric ω 0 .Moreover, assuming that the automorphism group Aut(X 0 ) is discrete, X t admit smooth Kähler-Einstein metrics ω t for all |t| sufficiently small and (X 0 , ω 0 ) is the limit in the Gromov-Hausdorff topology of (X t , ω t ), in the sense of [21].Few remarks are in place. First, by the generalized Bando-Mabuchi uniqueness theorem [6] the above weak Kähler-Einstein metric ω 0 is unique up to Aut 0 (X 0 ), the identity component of Aut(X 0 ), thus can be viewed as a canonical metric on X 0 . Second, the above theorem does not just state that there is a sequence of nearby KE Fano manifolds which converges, in the Gromov-Hausdorff sense, to the weak KE metric, but that all the nearby KE Fano manifolds actually converge to the unique singular limit (X 0 , ω 0 ). Thus this property provides a good topological correspondence between complex analytic deformations (alias flat-families) and the notion of Gromov-Hausdorff convergence. For example, by the Bando-Mabuchi theorem we have the following immediate corollary: Corollary 1.2. Let π : X → ∆ and π ′ : X ′ → ∆ be two Q-Gorenstein smoothings of Q-Fano varieties X 0 and X ′ 0 . Suppose X t and X ′ t are bi-holomorphic for all t = 0, and X 0 and X ′ 0 are both K-polystable with discrete automorphism group, then X 0 and X ′ 0 are isomorphic varieties.Notice t...

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confidence: 71%

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confidence: 76%

Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties

Spotti¹,

Sun²,

Yao³

2016

Duke Math. J.

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“…To avoid such technical issues, and also to cover the general n-dimensional case, one can use a slightly different gluing construction due to Biquard-Rollin [11] and Spotti [46] in the surface case. In these papers, a flat holomorphic family is given, and the gluing is carried out purely at the level of Kähler potentials.…”

Section: Discussionmentioning

confidence: 99%

Calabi-Yau manifolds with isolated conical singularities

Hein

Sun

2017

Publ.math.IHES

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Let X be a complex projective variety with only canonical singularities and with trivial canonical bundle. Let L be an ample line bundle on X. Assume that the pair (X, L) is the flat limit of a family of smooth polarized Calabi-Yau manifolds. Assume that for each singular point x ∈ X there exist a Kähler-Einstein Fano manifold Z and a positive integer q dividing KZ such that − 1 q KZ is very ample and such that the germ (X, x) is locally analytically isomorphic to a neighborhood of the vertex of the blow-down of the zero section of 1 q KZ. We prove that up to biholomorphism, the unique weak Ricci-flat Kähler metric representing 2πc1(L) on X is asymptotic at a polynomial rate near x to the natural Ricci-flat Kähler cone metric on 1 q KZ constructed using the Calabi ansatz. In particular, our result applies if (X, O(1)) is a nodal quintic threefold in P 4 . This provides the first known examples of compact Ricci-flat manifolds with non-orbifold isolated conical singularities.

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“…(This can be defined, for instance, by flowing along the vector fields orthogonal to the fibres under the ω X metric. Or one can prescribe the diffeomorphism explicitly in the coordinate neighbourhood U 1 and try to extend it outside U 1 , similar to [11]. Many reasonable constructions will satisfy the desired estimates.)…”

Section: Cy Metrics On Smoothings Of the Nodal K3 Fibrementioning

confidence: 99%

A gluing construction of collapsing Calabi–Yau metrics on K3 fibred 3-folds

2019

Geom. Funct. Anal.

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Deformations of nodal Kähler-Einstein Del Pezzo surfaces with discrete automorphism groups

Abstract: In this paper, we prove that generic small partial smoothings of Kähler-Einstein (KE) Del Pezzo orbifolds with only nodal singularities, and with no non-zero holomorphic vector fields, admit orbifold KE metrics which are close in the Gromov-Hausdorff sense to the original KE metric.

Cited by 19 publications

References 28 publications

Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties

Existence and deformations of Kähler–Einstein metrics on smoothable Q-Fano varieties

Calabi-Yau manifolds with isolated conical singularities

A gluing construction of collapsing Calabi–Yau metrics on K3 fibred 3-folds

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