Uniform stability of twisted constant scalar curvature K\"ahler metrics

By R. Dervan. 2014

148 citations recorded by the source corpus.

We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature K\"ahler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. We characterise the triviality of test configurations, by showing that a test configuration has zero minimum norm if and only if it has zero $L^2$-norm, if and only if it is almost trivial. We prove that the existence of a twisted constant scalar curvature K\"ahler metric implies uniform twisted K-stability with respect to the minimum norm, when the twisting is ample. We give algebro-geometric proofs of uniform K-stability in the general type and Calabi-Yau cases, as well as in the Fano case under an alpha invariant condition. Our results hold for line bundles sufficiently close to the (anti)-canonical line bundle, and also in the twisted setting. We show that log K-stability implies twisted K-stability, and also that twisted K-semistability of a variety implies that the variety has mild singularities.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9eac2353f881a2e02db55394bc2e494091621786

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Sasaki–Einstein metrics and K–stability (2015, unknown): “ −1 2 ˆr2(βτ−ωτ)∧ωn−1, recalling that βτ = √ −1∂∂logrψ and ωτ = √ −1∂∂log ˆr. As in [30] we use the twisted Futaki invariant to deﬁne a notion of (equivariant) twisted stability following also Dervan [35]. We will also choose a twisting form of a special form for which we will calculate an alternati”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, background): “ξ = Θ = 0), [33, 28, 9, 37, 72, 58, 80] for the twisted/conical Kähler-Einstein case (i.”
A non-Archimedean approach to K-stability (2018, methodology): “fact, we work on an arbitrary polarized smooth variety, using adjoint versions of the Ding and Mabuchi functionals, the latter inducing Dervan’s notion of twisted K-stability in the twisted Fano case [Der16]. The results here are used in [BBJ18], to give criteria for the existence of twisted Kahler{”
Kähler–Einstein metrics along the smooth continuity method (2015, methodology): “ 1−t V Z W θw(eθv−1)ωn φ+ n Z W θw(β−ωφ) ∧ ωn−1 φ . Recall here that (W,β) is the limit of the pair (M,α) under the C∗-action generated by w. The following result builds on work in [32] and Dervan [24]. Proposition 11. Using the same setup as above, we have the formula (22) 1 V Z W θw h (n+1)ωφ−nβ ”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, background): “eans that we reprove or mildly generalize some already known results [Oda12, Oda13b, OSa12, Sun13, OSu11]. During the preparation of the present work, we were informed of R. Dervan’s independent work [Der14a](see also [Der14b]), which has a substantial overlap with the present paper. First, test con”
K-stability of Fano varieties: an algebro-geometric approach (2020, background): “cal-to-global principle for K-stability as in many other higher dimensional geometry problems. In a diﬀerent direction, a uniform version of the deﬁnition of K-stability was introduced in [BHJ17] and [Der16] independently, which is more natural when one consider the space of all valuations instead o”
Geodesic rays and stability in the cscK problem (2020, methodology): “I would like to thank Chenyang Xu and Yuji Odaka for communications regarding Remark 6.14, Yuji Odaka for Remark 6.8 and bringing the paper [57] to his attention, Ruadháı Dervan for pointing out the reference [42] and Mingchen Xia for helpful comments.”