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We give a survey of the recent progress on the study of K-stability of Fano varieties by an algebro-geometric approach.

We develop a general approach to prove K-stability of Fano varieties. The new theory is used to (a) prove the existence of Kähler-Einstein metrics on all smooth Fano hypersurfaces of Fano index two, (b) compute the stability thresholds for hypersurfaces at generalised Eckardt points and for cubic surfaces at all points, and (c) provide a new algebraic proof of Tian’s criterion for K-stability, amongst other applications.

We show that in any Q -Gorenstein flat family of klt singularities, normalized volumes are lower semicontinuous with respect to the Zariski topology. A quick consequence is that smooth points have the largest normalized volume among all klt singularities. Using an alternative characterization of K-semistability developed by Li, Liu, and Xu, we show that K-semistability is a very generic or empty condition in any Q -Gorenstein flat family of log Fano pairs.

We show that the anti-canonical volume of an$n$-dimensional Kähler–Einstein$\\mathbb{Q}$-Fano variety is bounded from above by certain invariants of the local singularities, namely$\\operatorname{lct}^{n}\\cdot \\operatorname{mult}$for ideals and the normalized volume function for real valuations. This refines a recent result by Fujita. As an application, we get sharp volume upper bounds for Kähler–Einstein Fano varieties with quotient singularities. Based on very recent results by Li and the author, we show that a Fano manifold is K-semistable if and only if a de Fernex–Ein–Mustaţă type inequality holds on its affine cone.

We show that any $n$ -dimensional Fano manifold $X$ with $\\unicode[STIX]{x1D6FC}(X)=n/(n+1)$ and $n\\geqslant 2$ is K-stable, where $\\unicode[STIX]{x1D6FC}(X)$ is the alpha invariant of $X$ introduced by Tian. In particular, any such $X$ admits Kähler–Einstein metrics and the holomorphic automorphism group $\\operatorname{Aut}(X)$ of $X$ is finite.

We show that Calabi–Yau fibrations over curves are uniformly K-stable in an adiabatic sense if and only if the base curves are K-stable in the log-twisted sense. Moreover, we prove that there are cscK metrics for such fibrations when the total spaces are smooth.

In the probabilistic construction of Kähler–Einstein metrics on a complex projective algebraic manifold X —involving random point processes on X —a key role is played by the partition function. In this work a new quantitative bound on the partition function is obtained. It yields, in particular, a new direct analytic proof that X admits a Kähler–Einstein metrics if it is uniformly Gibbs stable. The proof makes contact with the quantization approach to Kähler–Einstein geometry.

We show that complete intersection log del Pezzo surfaces with amplitude one in weighted projective spaces are uniformly$K$-stable. As a result, they admit an orbifold Kähler–Einstein metric.

We prove that all smooth Fano threefolds of rank 4 and degree 24 are K-stable.

We prove that constant scalar curvature Kähler (cscK) manifolds with transcendental cohomology class are K-semistable, naturally generalising the situation for polarised manifolds. Relying on a recent result by R. Berman, T. Darvas and C. Lu regarding properness of the K-energy, it moreover follows that cscK manifolds with discrete automorphism group are uniformly K-stable. As a main step of the proof we establish, in the general Kähler setting, a formula relating the (generalised) Donaldson–Futaki invariant to the asymptotic slope of the K-energy along weak geodesic rays.