appear as the string background in superstring theory in 10 dimensions. Mathematically, this corresponds to the case where , that is, when the Riemannian manifold is Ricci flat. The existence of a Kähler–Einstein metric was proven in this case by Yau, using a continuity method similar to the case where . The topological assumption assumption introduces new difficulties into the continuity method. Partly due to his proof of existence, and the related proof of the Calabi conjecture, Yau was awarded the Fields medal. Theorem (Yau): A compact Kähler manifold with trivial canonical bundle, a Calabi–Yau manifold, always admits a Kähler–Einstein metric, and in particular admits a Ricci-flat metric. The case c1(X)>0 When the anticanonical bundle is ample, or equivalently , the manifold is said to be Fano. In contrast to the case , a Kähler–Einstein metric does not always exist in this case. It was observed by Akito Futaki that there are possible obstructions to the existence of a solution given by the holomorphic vector fields of , and it is a necessary condition that the Futaki invariant of these vector fields is non-negative. Indeed, much earlier it had been observed by Matsushima and Lichnerowicz that another necessary condition is that the Lie algebra of holomorphic vector fields must be reductive. It was conjectured by Yau in 1993, in analogy with the similar problem of existence of Hermite–Einstein metrics on holomorphic vector bundles, that the obstruction to existence of a Kähler–Einstein metric should be equivalent to a certain algebro-geometric stability condition similar to slope stability of vector bundles. In 1997 Tian Gang proposed a possible stability condition, which came to be known as K-stability. The conjecture of Yau was resolved in 2012 by Chen–Donaldson–Sun using techniques largely different from the classical continuity method of the case , and at the same time by Tian. Chen–Donaldson–Sun have disputed Tian's proof, claiming that it contains mathematical inaccuracies and material which should be attributed to them. Tian has disputed these claims. The 2019 Veblen prize was awarded to Chen–Donaldson–Sun for their proof. Donaldson was awarded the 2015 Breakthrough Prize in Mathematics in part for his contribution to the proof, and the 2021 New Horizons Breakthrough Prize was awarded to Sun in part for his contribution. Theorem: A compact Fano manifold admits a Kähler–Einstein metric if and only if the pair is K-polystable. A proof based along the lines of the continuity method which