Kähler-Einstein metrics on Fano manifolds

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The fabric of modern geometry is designed around questions that lead to the existence of canonical metrics on manifolds. A classical example is the Riemannian metrics with constant Gauss curvature on Riemann surfaces. The higher dimensional analogue sparks the hope of finding an "Einstein metric" on a given manifold. When the manifold in question is Kähler, then the desired metric is called Kähler-Einstein. Manifolds can be simplified to have positive or negative curvature, or be flat. The existence of a Kähler-Einstein metric when the curvature is negative or flat is known, thanks to the celebrated work of Aubin and Yau. The existence of such metric is obstructed in the positive curvature case. Due to the pioneering work of Donaldson et al, the existence of a Kähler-Einstein metric in this case is determined by an algebraic stability condition on the underlying Fano variety. However, it is difficult to verify such stability condition for a given Fano variety.Based on some recent developments in the field, we aim to produce and fine-tune a new method to check whether a given Fano variety is K-(semi)stable or not. The plan is to apply this new method to various challenging examples.