Torus equivariant K-stability
Giulio Codogni,Jacopo Stoppa +1 more
TLDR
In this paper, it is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group.Abstract:
It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds.read more
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Regularity of weak minimizers of the K-energy and applications to properness and K-stability
TL;DR: In this article, it was shown that the existence of a cscK metric in the space of a compact Kahler manifold implies J-properness of the K-energy.
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A Guided Tour to Normalized Volume
Chi Li,Yuchen Liu,Chenyang Xu +2 more
TL;DR: A survey on the recent theory on minimizing the normalized volume function attached to any klt singularities is given in this article, where the authors present a survey of the literature on minimizing normalized volume functions attached to singularities.
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K-stability for K\"ahler Manifolds
Ruadhaí Dervan,Julius Ross +1 more
TL;DR: In this article, the authors formulate a notion of K-stability for Kahler manifolds, and prove one direction of the Yau-Tian-Donaldson conjecture in this setting.
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K-semistability is equivariant volume minimization
TL;DR: In this paper, it was shown that the K-semistability of a Fano variety is equivalent to the condition that the normalized volume is minimized at the canonical valuation of the cone associated to any positive Cartier multiple of the variety.
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On K-Polystability of cscK Manifolds with Transcendental Cohomology Class
TL;DR: In this paper, it was shown that the existence of a constant scalar curvature Kähler metric implies geodesic K-polystability, in a sense that is expected to be equivalent to K polystability in general.
References
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Geometric Invariant Theory
TL;DR: Geometric invariant theory for moduli spaces has been studied extensively in the mathematical community as mentioned in this paper, with a large number of applications to the moduli space construction problem, see, for instance, the work of Mumford and Fogarty.
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Kähler-Einstein metrics with positive scalar curvature
TL;DR: In this article, it was shown that the existence of Kahler-Einstein metrics implies the stability of the underlying Kahler manifold in a suitable sense, which disproves a long-standing conjecture that a compact KG admits KG metrics if it has positive first Chern class and no nontrivial holomorphic vector fields.
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Scalar Curvature and Stability of Toric Varieties
TL;DR: In this paper, a stability condition for a polarised algebraic variety is defined and a conjecture relating this to the existence of a Kahler metric of constant scalar curvature.
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Groupes Réductifs
Armand Borel,Jacques Tits +1 more
TL;DR: In this paper, the authors present conditions générales d'utilisation (http://www.numdam.org/conditions), i.e., Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
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Instability in invariant theory
TL;DR: Theorem 4-2 of Mumford and Raghunathan as discussed by the authors states that a vector v in a reductive group G is S-unstable if and only if there exists a one-parameter subgroup X of G such that v is Sunstable for the induced Gm-action on V. This result was given by Mumford in [11] for linearly reductive groups and conjectured by him to hold in general.