The Calabi problem for smooth Fano threefolds

点击翻译按钮获取中文摘要

Algebraic varieties are geometric shapes given by polynomial equations. They appear naturally in pure and applied mathematics, e.g. conic sections in geometry, cubic curves in cryptography, or non-uniform rational basis splines in computer-aided graphic design. To measure distances between points of an algebraic variety, we can equip it with a sophisticated dot product called metric. Measuring distances leads to the notion of curvature, so that one can check how curved a given algebraic variety is. This splits algebraic varieties into three basic (universal) types: negatively curved, flat and positively curved varieties. Positively curved varieties can be thought of as higher dimensional generalisations of a sphere. They are called Fano varieties after the Italian mathematician Gino Fano. Fano varieties frequently appear in applications, because they are often parametrised by rational functions. Unlike negatively curved varieties, Fano varieties are bounded by a theorem by Caucher Birkar (Cambridge), who received a Fields medal in 2018 for proving this fact.For an algebraic variety, the choice of a metric is never unique, so that one can try to find 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. What was unclear is why such metric should exist, so Calabi posed it as a problem.The Calabi problem was solved for varieties with negative or zero curvature by Shing-Tung Yau in 1978. Yau confirmed Calabi's prediction and showed that these varieties are always Kahler-Einstein; he received the Fields medal for this proof. On the other hand, Yozo Matsushima observed that the Calabi problem may have a negative solution for some Fano varieties. Namely, he proved that symmetries of a Kähler-Einstein Fano variety must satisfy an algebraic property known as reductivity. This gives an obstruction to the existence of Kähler-Einstein metrics. Yet there are also Fano varieties with reductive group of symmetries that are not Kähler-Einstein.In the past 30 years, Calabi problem for Fano varieties attracted attention of many geometers including Fields Medalist Sir Simon Donaldson (Imperial College) and Chinese mathematician Gang Tian (Peking University). This resulted in the famous Yau-Tian-Donaldson conjecture which states that a Fano variety admits a Kähler-Einstein metric if and only if it satisfies a (sophisticated) algebraic condition called K-polystability. In 2012 this conjecture was solved by Xiuxiong Chen (Stony Brook), Donaldson and Song Sun (then at Imperial College). For this result, Chen, Donaldson and Sun were awarded the prestigious Oswald Veblen Prize in Geometry, and Donaldson was also awarded Breakthrough and Wolf prizes.The theoretical advances in the solution to the Yau-Tian-Donaldson conjecture have been fast and impressive, yet, they do not allow us to solve the original Calabi problem in most of the explicit cases. For example, if a Fano variety is given by a single polynomial equation, we do not always know that it is Kähler-Einstein (but we expect it to be, and this is a long standing open problem). In dimension 2, Tian explicitly solved the Calabi problem in 1990 by finding all two-dimensional Kähler-Einstein Fano varieties. Unfortunately, in dimension 3, where the classification of Fano varieties into 105 families dates back to the early 1980s, we do not know exactly which three-dimensional Fano varieties (Fano threefolds) admit a Kähler-Einstein metric. The goal of this project is to do in dimension three what Tian did for Fano surfaces: that is, to find all Kahler-Einstein Fano threefolds.