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

By Robert J. Berman. 2012

271 citations recorded by the source corpus.

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 $$\mathbb {Q}$$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 $$\lambda $$λ-entropy functional on K-unstable Fano manifolds are also given.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/babb817410f56a150865057383f1d163ab69a20e

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Citations in the corpus

Minimizing normalized volumes of valuations (2015, unknown): “y out calculations to show that there is a close relation between minimization of vol with Fujita’s divisorial stability ([c Fuj15a]) which is a consequence of K-stability ([Tia97], [Don02], see also [Ber12]). The calculations will essentially show that the derivative of vol at the canonial valuatio”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “For background material and further references on geodesic rays, their regularity and relation to test configuration and K-stability and we refer the reader to [3, 4, 5, 11, 12, 15, 20, 21, 23, 40, 32, 33, 34, 35, 36, 37, 38, 39, 41].”
On K‐stability and the volume functions of ℚ‐Fano varieties (2015, background): “In particular, if X admits Kähler–Einstein metrics, then (X,−KX) is divisorially semistable by [2, 15].”
Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, background): “rol on the Ricci tensor, the injectivity radius and with bounded diameter) are indeed Q-Fano varieties, i.e., they have at worst log-terminal singularities, and moreover they must be K-polystable, by [11]. Summing up, a central motivation of the present work was to investigate how Ka¨hler-Einstein m”
On the moduli of Kähler-Einstein Fano manifolds (2012, methodology): “y expected, the existence of Ka¨hler-Einstein metrics on a Fano manifold is proved to be equivalent to the K-stability. The proof of stability of KE Fano manifolds is established by [Mab08], [Mab09], [Ber12] and for the converse direction, please consult [CDS12a], [CDS12b], [CDS13], [Tia12b] which d”
Uniform K-stability and asymptotics of energy functionals in Kähler geometry (2016, unknown): “ and F= M B. In addition, Berman proved that in the log Fano case, the Ding functional D B admits DNA B as non-Archimedean limit. Indeed, the following result follows from Proposition 3.8 and x4.3 in [Berm16]. Theorem 4.3. Let (X;B) be a subklt pair with L:= K (X;B) big and nef, (X;L) a test congura”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “It is known, for example, that the existence of such a metric implies K-stability when the manifold has discrete automorphism group [15, 44], and that the two are equivalent when L = −KX [2, 9, 7, 8, 51], so that the manifold is Fano and the metric is Kähler-Einstein.”
On the Yau‐Tian‐Donaldson Conjecture for Singular Fano Varieties (2017, background): “It can be seen as the generalization of Berman’s expansion of log-Ding energy in [1] along test configurations to the case of any subgeodesic rays.”
Explicit Gromov-Hausdorff compactifications of moduli spaces of K\"ahler-Einstein Fano manifolds (2017, methodology): “By [6] we know Z is K-polystable, so Z is a GIT polystable cubic hypersurface.”
A non-Archimedean approach to K-stability (2018, background): “ Ent() := R X Adis the entropy of , dened as the integral of the log discrepancy function A: X!R + [f+1g. Another functional, the (non-Archimedean) Ding functional is also useful. First dened in [Berm16], it can be written as D= L E, where Eis the Monge{Ampere energy, and L is the Legendre transform”
Kähler–Einstein metrics and volume minimization (2016, methodology): “In particular, the Yau–Tian–Donaldson correspondence has been established for smooth Fano manifolds (see in particular [57], [3], [16,58]).”