# Measuring of geometric shapes and families of shapes

Algebraic varieties are geometric shapes defined by polynomial equations. For example, an ordinary sphere (the surface of a football) is an algebraic variety because the spatial coordinates of each of its points satisfy a polynomial equation of degree 2 (x^2+ y^2+z^2= r^2). Such shapes appear in all areas where scientists study phenomena described by polynomial equations: conic sections appear in geometry, cubic curves in cryptography and non-uniform rational basis splines in computer-aided graphic design. Birational geometry reduces the study of any geometric shape to understanding some basic building blocks.

These are Fano varieties (positively curved like a sphere), Calabi-Yau varieties (flat like a plane) and varieties of general type (negatively curved like a saddle). The meaning of curvature is intuitive, but to define it properly, we need a tool to measure the distance between points on the variety. Such a tool is a metric, and there is more than one choice for a metric so that it is important to pick a special metric with good properties, which would be chosen in a "canonical way". Geometers looked for a suitable condition defining a canonical metric for the first half of the 20th century. In 1957, Eugenio Calabi proposed that this canonical metric would satisfy both a certain algebraic property (being Kähler) and the Einstein (partial differential) equation.

These two conditions guarantee that the Kähler-Einstein metric is unique when it exists. Yet- it was unclear why such a metric should exist, and this became the subject of the Calabi conjecture. The solution of the conjecture for varieties with negative or zero curvature earned Shing-Tung Yau the Fields medal in 1978 - he had proved that the canonical metric always exists on such varieties. It soon became clear that some Fano varieties would not admit canonical metrics. In 2012, Xiuxiong Chen (Stony Brook), Donaldson and Song Sun (then at Imperial College) proved that a Fano variety admits a Kähler-Einstein metric if and only if it satisfies a (sophisticated) algebraic condition called K-polystability. This is an equivalence between two deep properties that are hard to verify on explicit examples! In dimension 3, an explicit description of Fano varieties is known, but in spite of the theoretical advances proved by Chen, Donaldson and Sun, we still don't know which Fano threefolds admit a canonical Kähler-Einstein metric.

Our project aims to answer this question completely.

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