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K-stability for Kähler manifolds
(International PressMathematical Research Letters, 2017-09)
We formulate a notion of K-stability for Kähler manifolds, and prove one direction of the Yau–Tian–Donaldson conjecture in this setting. More precisely, we prove that the Mabuchi functional being bounded below (resp. ...
Non-reductive automorphism groups, the Loewy filtration and K-stability
(l'Institut FourierAnnales de l'Institut Fourier, 2016-03-18)
We study the K-stability of a polarised variety with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several ...
Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics
(Oxford University PressInternational Mathematics Research Notices, 2015-10-14)
We introduce a norm on the space of test configurations, called the minimum norm. We conjecture that uniform K-stability is equivalent to the existence of a constant scalar curvature Kähler metric. This uniformity is ...
Alpha invariants and coercivity of the Mabuchi functional on Fano manifolds
(Université Paul SabatierAnnales de la Faculté des Sciences de Toulouse, 2016)
We give a criterion for the coercivity of the Mabuchi functional for general Kàhler classes on Fano manifolds in terms of Tian's alpha invariant. This generalises a result of Tian in the anti-canonical case implying the ...
On K-stability of finite covers
(Oxford University PressJournal of the London Mathematical Society, 2016)
We show that certain Galois covers of K-semistable Fano varieties are K-stable. We use this to give some new examples of Fano manifolds admitting Kähler-Einstein metrics, including hypersurfaces, double solids and threefolds.
A finite dimensional approach to Donaldson's J-flow
Consider a projective manifold with two distinct polarisations $L_1$ and $L_2$. From this data, Donaldson has defined a natural flow on the space of Kähler metrics in $c_1$($L_1$), called the J-flow. The existence of a ...
Alpha Invariants and K-Stability for General Polarizations of Fano Varieties
(Oxford University PressINTERNATIONAL MATHEMATICS RESEARCH NOTICES, 2014-09-26)
We provide a sufficient condition for polarisations of Fano varieties to be K-stable in terms of Tian’s alpha invariant, which uses the log canonical threshold to measure singularities of divisors in the linear system ...