dimension two, so is a surface of general type, one gets convergence to the Kähler–Einstein metric on . Later, Jian Song and Tian studied the case where the projective variety has log-terminal singularities. Kähler–Ricci flow and existence of Kähler–Einstein metrics It is possible to give an alternative proof of the Chen–Donaldson–Sun theorem on existence of Kähler–Einstein metrics on a smooth Fano manifold using the Kähler-Ricci flow, and this was carried out in 2018 by Chen–Sun–Wang. Namely, if the Fano manifold is K-polystable, then the Kähler-Ricci flow exists for all time and converges to a Kähler–Einstein metric on the Fano manifold. Generalizations and alternative notions Constant scalar curvature Kähler metrics When the canonical bundle is not trivial, ample, or anti-ample, it is not possible to ask for a Kähler–Einstein metric, as the class cannot contain a Kähler metric, and so the necessary topological condition can never be satisfied. This follows from the Kodaira embedding theorem. A natural generalisation of the Kähler–Einstein equation to the more general setting of an arbitrary compact Kähler manifold is to ask that the Kähler metric has constant scalar curvature (one says the metric is cscK). The scalar curvature is the total trace of the Riemannian curvature tensor, a smooth function on the manifold , and in the Kähler case the condition that the scalar curvature is constant admits a transformation into an equation similar to the complex Monge–Ampere equation of the Kähler–Einstein setting. Many techniques from the Kähler–Einstein case carry on to the cscK setting, albeit with added difficulty, and it is conjectured that a similar algebro-geometric stability condition should imply the existence of solutions to the equation in this more general setting. When the compact Kähler manifold satisfies the topological assumptions necessary for the Kähler–Einstein condition to make sense, the constant scalar curvature Kähler equation reduces to the Kähler–Einstein equation. Hermite–Einstein metrics Instead of asking the Ricci curvature of the Levi-Civita connection on the tangent bundle of a Kähler manifold is proportional to the metric itself, one can instead ask this question for the curvature of a Chern connection associated to a Hermitian metric on any holomorphic vector bundle over (note that the Levi-Civita connection on the holomorphic tangent bundle is precisely the Chern connection of the Hermitian metric coming from the Kähler structure). The resulting equation is called the Hermite–Einstein equation, and is of special importance in gauge theory, where it appears as a