The Calabi conjecture and K-stability

By Y. Odaka. 2010

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We algebraically prove K-stability of polarized Calabi–Yau varieties and canonically polarized varieties with mild singularities. In particular, the “stable varieties” introduced by Kollar–Shepherd-Barron [“Threefolds and deformation of surface singularities.” Inventiones Mathematicae 91 (1988): 299–338] and Alexeev [“Moduli spaces M g,n (W) for surfaces.” Proceeding of “Higher-dimensional complex varieties (Trento, 1994)”, de Gruyter, Berlin, 1996, 1–22], which form compact moduli space, are proven to be K-stable although it is well known that they are not necessarily asymptotically (semi) stable. As a consequence, we have orbifold counterexamples to the folklore conjecture “K-stability implies asymptotic stability”. They have Kahler–Einstein (orbifold) metrics, so the result of Donaldson [“Scalar curvature and projective embeddings. I.” Journal of Differential Geometry 59, no. 3 (2001): 479–522] does not hold for orbifolds.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/27f61a511f307fe9c5ea94403b0c379521df5daf

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Citations in the corpus

Geodesic rays and stability in the cscK problem (2020, background): “(13) For any φ = φ(X ,L) ∈ HNA(L), we follow the notations used in [16] (see also [57]) to define: E(φ) := 1 n+ 1 L̄·n+1 =: E(X ,L) (14) Λ(φ) := L̄ · L·n P1 =: Λ (X ,L) (15) J(φ) := Λ(φ)−E(φ) =: J(X ,L) (16) I(φ) := L̄ · L·n P1 − L̄ ·n+1 + L̄·n · LP1 (17) R(φ) := K log XP1/P(1) · L̄·n =: R(X ,L) (18”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “— In the terminology of [61], HNA B (φ) + V −1 · ((X0 −X0,red) · L) coincides (up to a multiplicative constant) with the ‘discrepancy term’ of the Donaldson–Futaki invariant, while S̄B ENA(φ) + RNA B (φ) corresponds to the ‘canonical divisor part’.”
Testing log K-stability by blowing up formalism (2011, methodology): “We review these as the eﬃciency of that theory in the study of stability turned out in [16] and developed in [17], [18], [14] among others which we follow.”
On the moduli of Kähler-Einstein Fano manifolds (2012, background): “re mainly from the following backgrounds. • The inﬁnite-dimensional moment map picture on the space of complex structures by Fujiki, Donaldson ([Fuj90], [Don97], [Don04]). • The observations ([Od08], [Od11]) that the projective moduli of general type varieties with semi-log-canonical singularities, ”
Existence and deformations of Kähler–Einstein metrics on smoothable $\mathbb{Q}$-Fano varieties (2014, result): “1 should be also compared with the result on the existence of KE metrics and K-polystability on semi-log-canonical varieties with ample canonical Q-bundle obtained respectively by Berman-Guenancia in [7] and by Odaka in [34] and [35] (although the reason for the KE/K-polystability correspondence in ”
Compact Moduli Spaces of Del Pezzo Surfaces and K\"ahler-Einstein metrics (2012, methodology): “ce of smoothable Q-Fano varieties to have log terminal singularities, see [23]. This type of singularities also appear to be the worst singularities allowed for K-semistability of Fano varieties, see [58]. We ﬁnish this subsection by a discussion on the surfaces parametrized by the ox and cateyes, w”
Complex optimal transport and the pluripotential theory of K\"ahler-Ricci solitons (2014, background): “ = 0 the extension of the previous theorem to singular Fano varieties seems challenging. Indeed, the corresponding existence statement may then even turn out to be false as indicated by an example in [37], which reveals that the K-stability of a singular polarized variety (X,L) does not imply the ex”
Alpha invariant and K-stability of Q-Fano varieties (2010, unknown): “We remark that the first author ([24]) also found similar proofs for the finiteness of Aut(X) by using K-stability in the case of general type varieties, which is well known, and of polarized Calabi-Yau varieties.”
A generalization of the Ross--Thomas slope theory (2009, background): “3 (ii), nor can extend (i) to higher dimensional varieties with semi-log-canonical singularities, as we will show explicit counterexamples in [17].”
K-stability of Fano varieties: an algebro-geometric approach (2020, background): “X,L) with a cscK metric and ﬁnite automorphism group is asymptotically Chow stable, however, this is known to not hold for either Xhas an inﬁnite automorphism group (see [OSY12]) or Xis singular (see [Oda12a]). Remark 2.5. The notions of K-stability can be deﬁned for a log Fano pair (X,∆) (See [Don1”
Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “In Section 3 we also provide several geometric criteria for a variety to be uniformly K-stable with respect to the minimum norm, strengthening work of Odaka [33] and Odaka-Sano [33].”