resolved the case was later provided by Datar–Székelyhidi, and several other proofs are now known. See the Yau–Tian–Donaldson conjecture for more details. Kähler–Ricci flow and the minimal model program A central program in birational geometry is the minimal model program, which seeks to generate models of algebraic varieties inside every birationality class, which are in some sense minimal, usually in that they minimize certain measures of complexity (such as the arithmetic genus in the case of curves). In higher dimensions, one seeks a minimal model which has nef canonical bundle. One way to construct minimal models is to contract certain curves inside an algebraic variety which have negative self-intersection. These curves should be thought of geometrically as subvarieties on which has a concentration of negative curvature. In this sense, the minimal model program can be viewed as an analogy of the Ricci flow in differential geometry, where regions where curvature concentrate are expanded or contracted in order to reduce the original Riemannian manifold to one with uniform curvature (precisely, to a new Riemannian manifold which has uniform Ricci curvature, which is to say an Einstein manifold). In the case of 3-manifolds, this was famously used by Grigori Perelman to prove the Poincaré conjecture. In the setting of Kähler manifolds, the Kähler–Ricci flow was first written down by Cao. Here one fixes a Kähler metric with Ricci form , and studies the geometric flow for a family of Kähler metrics parametrised by : When a projective variety is of general type, the minimal model admits a further simplification to a canonical model , with ample canonical bundle. In settings where there are only mild (orbifold) singularities to this canonical model, it is possible to ask whether the Kähler–Ricci flow of converges to a (possibly mildly singular) Kähler–Einstein metric on , which should exist by Yau and Aubin's existence result for . A precise result along these lines was proven by Cascini and La Nave, and around the same time by Tian–Zhang. Theorem: The Kähler–Ricci flow on a projective variety of general type exists for all time, and after at most a finite number of singularity formations, if the canonical model of has at worst orbifold singularities, then the Kähler–Ricci flow on converges to the Kähler–Einstein metric on , up to a bounded function which is smooth away from an analytic subvariety of . In the case where the variety is of