The Calabi problem for smooth Fano threefolds

代数簇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计算机辅助图形设计中的非均匀有理基样条曲线。为了测量代数族的点之间的距离，我们可以给它配备一个称为度量的复杂的点积。测量距离引出了曲率的概念，这样人们就可以检查给定的代数族有多弯曲。这将代数族分为三种基本的(普适的)类型：负曲型、平坦型和正曲型。正曲线簇可以被认为是球面的高维推广。它们是以意大利数学家吉诺·法诺的名字命名的法诺变种。Fano变数经常出现在应用程序中，因为它们经常被有理函数参数化。与负曲变种不同，Fano变种受Caucher Birkar(Cambridge)的一个定理的约束，后者在2018年因证明了这一点而获得菲尔兹奖。对于一个代数变种，度量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量，该度量将以“规范的方式”选择。几何学家们寻找一个合适的条件来定义20世纪上半叶的标准度量衡。1957年，Eugenio Calabi提出，这个正则度规将同时满足某种代数性质(为Kähler)和爱因斯坦(偏微分方程式)。这两个条件保证了卡勒-爱因斯坦度规在存在时是唯一的。不清楚的是为什么会存在这样的度量，所以Calabi提出了一个问题。1978年，游成栋解决了负曲率或零曲率的变种的Calabi问题。Yau证实了卡拉比的预测，并表明这些变种永远是卡勒-爱因斯坦；他因此而获得了菲尔兹奖。另一方面，松岛洋三观察到，卡拉比问题可能对一些法诺品种有负面的解决方案。也就是说，他证明了Kähler-Einstein Fano簇的对称性必须满足一种称为约化的代数性质。这阻碍了卡勒-爱因斯坦度规的存在。在过去的30年里，关于Fano簇的Calabi问题引起了许多几何学家的注意，其中包括菲尔兹奖牌获得者西蒙·唐纳森爵士(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Yau-Tian-Donaldson猜想，该猜想指出一个Fano簇允许Kähler-Einstein度量当且仅当它满足一个称为K-多稳定性的(复杂的)代数条件。2012年，陈秀雄(石溪)、唐纳森和孙松(当时在帝国理工学院)解决了这个猜想。由于这一结果，Chen、Donaldson和Sun获得了著名的Oswald Veblen几何奖，Donaldson还获得了突破奖和Wolf奖。在解决Yau-Tian-Donaldson猜想的理论方面取得了快速而令人印象深刻的进展，然而，在大多数显性情况下，它们不允许我们解决原始的Calabi问题。例如，如果一个Fano变种是由一个多项式方程给出的，我们并不总是知道它是不是卡勒-爱因斯坦(但我们希望它是，这是一个长期悬而未决的问题)。在二维Kähler-Einstein Fano变元中，Tian在1990年显式地解决了Calabi问题。不幸的是，在维度3中，Fano变种被归类为105个家族可以追溯到20世纪80年代初，我们不知道哪些三维Fano变种(Fano三重)符合Kähler-Einstein度量。这个项目的目标是在三维空间中做田对Fano曲面所做的事情：也就是找到所有的Kahler-Einstein Fano三重。

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.