The Calabi problem for smooth Fano threefolds

代数变量是由多项式方程给出的几何形状。它们自然地出现在纯数学和应用数学中，例如几何中的二次曲线，密码学中的三次曲线，或者计算机辅助图形设计中的非均匀有理基样条。为了测量一个代数变量点之间的距离，我们可以用一个叫做度规的复杂点积来装备它。测量距离引出了曲率的概念，这样人们就可以检查给定代数变量的弯曲程度。这将代数变种分成三种基本（普遍）类型：负弯曲、平弯曲和正弯曲变种。正弯曲的变体可以被认为是球的高维推广。它们被称为法诺变种，以意大利数学家吉诺·法诺的名字命名。范诺变量经常出现在应用中，因为它们经常被有理函数参数化。与负曲线品种不同，Fano品种受Caucher Birkar（剑桥大学）的定理约束，他在2018年因证明这一事实而获得了菲尔兹奖。对于一个代数变量，度规的选择从来都不是唯一的，所以人们可以尝试找到一个具有良好性质的特殊度规，它将以“规范的方式”被选择。几何学者在20世纪上半叶寻找一个合适的条件来定义标准度量。1957年，Eugenio Calabi提出，这个标准度规将同时满足某些代数性质（Kähler）和爱因斯坦（偏微分）方程。这两个条件保证Kähler-Einstein度量在存在时是唯一的。不清楚的是为什么这样的度量应该存在，所以卡拉比提出了一个问题。Calabi问题是由邱成东在1978年解决的。丘证实了卡拉比的预测，并表明这些变化总是卡勒-爱因斯坦；他因此获得了菲尔兹奖。另一方面，Yozo Matsushima观察到，对于一些范诺品种来说，卡拉比问题可能有一个负面的解决方案。也就是说，他证明了Kähler-Einstein Fano变体的对称性必须满足称为约化性的代数性质。这就阻碍了Kähler-Einstein指标的存在。然而，也有范诺变种与约化群对称，不是Kähler-Einstein。在过去的30年里，法诺变量的Calabi问题引起了许多几何学家的注意，包括菲尔兹奖获得者西蒙·唐纳森爵士（英国帝国理工学院）和中国数学家田刚（北京大学）。这导致了著名的yu - tian - donaldson猜想，该猜想指出，当且仅当Fano变体满足称为k -多稳定性的（复杂）代数条件时，它承认Kähler-Einstein度量。2012年，陈秀雄（石溪分校）、唐纳森（Donaldson）和孙嵩（当时在帝国理工学院）解决了这个猜想。由于这一结果，陈、唐纳森和孙获得了著名的奥斯瓦尔德·凡勃伦几何奖，唐纳森还获得了突破奖和沃尔夫奖。在解决you - tian - donaldson猜想方面的理论进展是迅速而令人印象深刻的，然而，在大多数显式情况下，它们不允许我们解决最初的Calabi问题。例如，如果Fano变量由单个多项式方程给出，我们并不总是知道它是Kähler-Einstein（但我们期望它是，这是一个长期存在的开放问题）。在维度2中，田在1990年通过找到所有二维Kähler-Einstein Fano变量明确地解决了Calabi问题。不幸的是，在维度3中，范诺品种分为105个科，可以追溯到20世纪80年代初，我们不确切地知道哪些三维范诺品种（范诺三倍）允许Kähler-Einstein度量。这个项目的目标是在三维空间中做田对法诺曲面所做的事情：也就是说，找到所有的卡勒-爱因斯坦法诺三倍。

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.