The Calabi problem for smooth Fano threefolds

平滑法诺三重的卡拉比问题

• 批准号: EP/V054597/1
• 负责人: Ivan Cheltsov
• 金额: $ 9.94万
• 依托单位: University of Edinburgh
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV054597%2F1
• 关键词: Calabi; problem; smooth; Fano; threefolds

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.

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，例如几何学中的圆锥曲线，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。为了测量代数簇中点之间的距离，我们可以给它配备一个复杂的称为度量的点积。测量距离引出了曲率的概念，这样人们就可以检查一个给定的代数簇有多弯曲。这将代数簇分为三种基本的（普遍的）类型：负曲簇、平坦簇和正曲簇。正弯曲的变体可以被认为是球体的高维泛化。他们被称为法诺品种后，意大利数学家吉诺法诺。Fano簇经常出现在应用程序中，因为它们通常由有理函数参数化。与负曲簇不同，Fano簇受Caucher Birkar（剑桥）的一个定理约束，他在2018年因证明这一事实而获得菲尔兹奖。对于代数簇，度量的选择从来都不是唯一的，因此人们可以尝试找到一个具有良好性质的特殊度量，这将以“规范方式”选择。几何学家在寻找一个合适的条件来定义世纪上半叶的规范度量。1957年，Eugenio Calabi提出这个正则度规同时满足一个代数性质（即Kähler）和爱因斯坦方程（偏微分方程）。这两个条件保证了凯勒-爱因斯坦度规在存在时是唯一的。但不清楚的是为什么这样的度量应该存在，所以卡拉比提出了它作为一个问题。卡拉比问题是解决了品种的负或零曲率的成东丘在1978年。丘证实了卡拉比的预测，并表明这些品种总是卡勒-爱因斯坦，他收到了菲尔兹奖的这一证明。另一方面，Yozo Matsushima观察到，Calabi问题可能对某些Fano品种有负解。也就是说，他证明了凯勒-爱因斯坦法诺簇的对称性必须满足一个称为约化性的代数性质。这给凯勒-爱因斯坦度规的存在带来了障碍。在过去的30年里，Fano簇的Calabi问题引起了包括菲尔兹奖获得者Sir Simon唐纳森（帝国理工学院）和中国数学家田刚（北京大学）在内的众多几何学家的关注。这导致了著名的姚田-唐纳森猜想，其中指出，法诺品种承认凯勒-爱因斯坦度量当且仅当它满足一个（复杂的）代数条件称为K-多稳定性。2012年，这个猜想被陈秀雄（斯托尼布鲁克）、唐纳森和孙松（当时在帝国理工学院）解决。由于这一结果，Chen，唐纳森和Sun被授予了著名的Oswald Veblen几何奖，唐纳森还被授予了突破奖和Wolf奖。Yau-Tian-唐纳森猜想的理论进展是快速而令人印象深刻的，然而，它们并不允许我们在大多数显式情况下解决原始的Calabi问题。例如，如果一个法诺簇由一个多项式方程给出，我们并不总是知道它是凯勒-爱因斯坦（但我们期望它是，这是一个长期存在的开放问题）。在2维中，田在1990年通过找到所有二维Kähler-Einstein Fano簇明确解决了卡拉比问题。不幸的是，在第3维中，Fano变种的分类可以追溯到1980年代初，我们不知道哪些三维Fano变种（Fano threefolds）承认Kähler-Einstein度量。这个项目的目标是在三维中做田为Fano曲面做的事情：也就是说，找到所有的Kahler-Einstein Fano三重。

项目成果

One-dimensional components in the K-moduli of smooth Fano 3-folds

光滑 Fano 3 重的 K 模中的一维分量

• DOI: 10.48550/arxiv.2309.12518
• 发表时间: 2023
• 作者: Abban H

K-STABLE DIVISORS IN OF DEGREE

K 阶除数

• DOI: 10.1017/nmj.2023.5
• 发表时间: 2023
• 期刊: Nagoya Mathematical Journal
• 影响因子: 0.8
• 作者: CHELTSOV I

K-stable Fano threefolds of rank 2 and degree 30

K 稳定 Fano 三倍的 2 级和 30 度

• DOI: 10.1007/s40879-022-00569-x
• 发表时间: 2022
• 期刊: European Journal of Mathematics
• 影响因子: 0.6
• 作者: Cheltsov I

Equivariant pliability of the projective space

射影空间的等变柔度

• DOI: 10.1007/s00029-023-00869-4
• 发表时间: 2023
• 期刊: Selecta Mathematica
• 作者: Cheltsov I

The Calabi problem for Fano threefolds

法诺的卡拉比问题有三重

• 发表时间: 2022
• 作者: Hiramatsu Naoya;Kento Fujita
• 通讯作者: Kento Fujita