Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics

Odaka, Yuji; Spotti, Cristiano; Sun, Song

doi:10.4310/jdg/1452002879

Cited by 110 publications

(170 citation statements)

References 110 publications

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“…In this section, we give a brief account on how Theorem 1.3 implies Theorem 1.1. Such an argument for surface appeared in [OSS16], and was also sketched in [SS17] for cubic hypersurfaces.…”

Section: Effective Bounds On Local Fundamental Groupsmentioning

confidence: 92%

“…This idea is successfully used to find out all smooth del Pezzo surfaces with a KE metric in [Tia90] and then extended to all (not necessarily smooth) limits of quartic del Pezzo surfaces in [MM93] which also gives an explicit construction of the compact moduli space. Later with a more focus on the stability study, the work of [MM93] was further extended to limits of all smooth KE surfaces in [OSS16].…”

Section: Introductionmentioning

confidence: 99%

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K-stability of cubic threefolds

Liu¹,

Xu²

2019

Duke Math. J.

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We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three dimensional canonical and terminal singularities, which was established during the study of the explicit three dimensional minimal model program.

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“…In this section, we give a brief account on how Theorem 1.3 implies Theorem 1.1. Such an argument for surface appeared in [OSS16], and was also sketched in [SS17] for cubic hypersurfaces.…”

Section: Effective Bounds On Local Fundamental Groupsmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

K-stability of cubic threefolds

Liu¹,

Xu²

2019

Duke Math. J.

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“…This is related to the expectation that one may be able to form compact moduli spaces of K-polystable Fano varieties if singular ones are included, or more precisely those with log terminal singularities; compare the discussions in [55,56] (where the surface case is considered).…”

Section: Theorem 11 Let X Be a Fano Variety Admitting A Kähler-einstmentioning

confidence: 98%

K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

2015

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It is shown that any, possibly singular, Fano variety X admitting a Kähler-Einstein metric is K-polystable, thus confirming one direction of the Yau-Tian-Donaldson conjecture in the setting of Q-Fano varieties equipped with their anti-canonical polarization. The proof is based on a new formula expressing the Donaldson-Futaki invariants in terms of the slope of the Ding functional along a geodesic ray in the space of all bounded positively curved metrics on the anti-canonical line bundle of X . One consequence is that a toric Fano variety X is K-polystable iff it is K-polystable along toric degenerations iff 0 is the barycenter of the canonical weight polytope P associated to X . The results also extend to the logarithmic setting and in particular to the setting of Kähler-Einstein metrics with edge-cone singularities. Applications to geodesic stability, bounds on the Ricci potential and Perelman's λ-entropy functional on K -unstable Fano manifolds are also given.

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“…with −K X − D − p * H ample. In the absolute case, the moduli space of K-stable Fanos can often be constructed quite explicitly [67], thus it seems much more reasonable that one could compute the analogous invariants in the Fano case.…”

Section: 2mentioning

confidence: 99%

Stable maps in higher dimensions

Dervan¹,

Ross

2018

Math. Ann.

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We formulate a notion of stability for maps between polarised varieties which generalises Kontsevich's definition when the domain is a curve and Tian-Donaldson's definition of K-stability when the target is a point. We give some examples, such as Kodaira embeddings and fibrations. We prove the existence of a projective moduli space of canonically polarised stable maps, generalising the Kontsevich-Alexeev moduli space of stable maps in dimensions one and two. We also state an analogue of the Yau-Tian-Donaldson conjecture in this setting, relating stability of maps to the existence of certain canonical Kähler metrics.

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Compact moduli spaces of Del Pezzo surfaces and Kähler–Einstein metrics

Cited by 110 publications

References 110 publications

K-stability of cubic threefolds

K-stability of cubic threefolds

K-polystability of Q-Fano varieties admitting Kähler-Einstein metrics

Stable maps in higher dimensions

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