2021-10-28 Hot Topic

We are very happy to hear that Shigefumi Mori, Director-General and Distinguished Professor, KUIAS and Senior Advisor of RIKEN iTHEMS, will be awarded the Order of Culture in recognition of his “outstanding achievements in mathematics, especially in algebraic geometry, in creating Mori Theory, a theory of minimal models of algebraic varieties, which has had a significant impact on a wide range of fields in the mathematical sciences, and for his outstanding contributions to the development of this field”. Mori-san’s field of expertise is mathematics, especially algebraic geometry. He was awarded the Fields Prize at the International Congress of Mathematicians in 1990 for his great work on the classification problem of algebraic varieties, a central theme in algebraic geometry, and has continued to vigorously lead research in related areas.

An algebraic variety is, roughly speaking, the set of solutions of algebraic equations. For example, the (real) solution of the equation x2 + y2 = 1 is a unit circle, which is an algebraic variety. The notion of algebraic varieties enables us to transform algebraic problems into geometric problems.

It is an important task in algebraic geometry to study the properties of algebraic varieties. However, it is not efficient to study each algebraic varieties one by one. So, mathematicians identify similar varieties as the same “species”, and try to make a “biological dictionary” of algebraic varieties. Here, we say that two algebraic varieties are “similar” if they map to each other under an operation called birational transformation. Birational transformation is an operation to modify low-dimensional part of an algebraic variety. Since birational transformation preserves main part of algebraic varieties, we can naturally regard two varieties that are transformed to each other by this operation as the same species.

In such a biological species, there are many (non-isomorphic) individual algebraic varieties. Which individual should we choose to show in the dictionary as a “standard example”? In this kind of situation, mathematicians usually want to bring the simplest possible representative example, which algebraic-geometers call “minimal model”.

For example, let’s think of 1-dimensional algebraic varieties, i.e., curves. A curve has singularities in general, but we can birationally transform the curve into a non-singular curve. In fact, inside a “species”, there is only one non-singular curve. So it is obvious that we should choose this non-singular curve as a minimal model. In 2-dimensional case (surface case), we can still make any surface into non-singular surface. However, this time, there are many non-singular surfaces in a species, so not all of them are the simplest as possible. If a surface is not minimal, then we can alway find a special types of curve, called “(-1)-curve” on a surface and contract them into one points to get a smaller surface. By iterating this process, we can obtain a non-singular surface with no (-1)-curves, which is the minimal model.

What happens in higher dimensions? In any dimension, fortunately, we can always transform an algebraic variety into a non-singular one by birational transformation (This is an important result by Heisuke Hironaka, another Fields Medalist). But if the dimension is greater than or equal to 3, in the process of creating smaller algebraic varieties, it is sometimes inevitable to allow some singularities to re-appear. This makes the classification problem much more difficult in higher dimension. However, Mori-san proved that we can always find the minimal model in three dimension if we allow the existence of mild singularities called “terminal singularities”. It is very hard to construct smaller algebraic varieties without causing bad singularities, but Mori-san constructed a theory to overcome this essential difficulty. After Mori-san’s breakthrough, there has been a big development in classification problems in dimensions higher than 3. Moreover, the theory has had (and will have) vast applications in various areas of mathematics.

I have introduced only a small part of the great achievements of Shigefumi Mori. We congratulate Mori-san on his award and on the progress of higher dimensional minimal model program.

iTHEMS wishes him further progress in Mori Theory and in the promotion of mathematical sciences.

Hiroyasu Miyazaki
(on behalf of iTHEMS)

Related Link