János Kollár, external member of the Hungarian Academy of Sciences, awarded the János Bolyai International Mathematics Award

The award committee, comprising five full members of the Department of Mathematical Sciences of the Hungarian Academy of Sciences and five prominent foreign mathematicians, determined that while almost all areas of pure mathematics involve classification problems, a special feature of algebraic geometry is that it does not consider classification problems as completed once the objects in question are classified as elements of a set. Rather, it is interested in giving these classifying sets a geometric structure; this idea leads to what is called modular theory, where objects are classified by constructing moduli spaces, which can be used to parameterise the objects.

A classic example is the construction of a moduli space of smooth curves of at least two genus. Its construction was the crowning achievement of a hundred years of work that began with Riemann and ended with the work of Deligne, Grothendieck and Mumford. Mumford introduced geometric invariant theory (GIT) as a way to construct moduli spaces which can be expressed as the quotient of a variety by a reductive algebraic group. The end product is the moduli space of stable curves, which is a geometrically meaningful compactification of the moduli space of smooth curves. Gieseker made the first progress in the analogous higher dimensional case with a moduli space of surfaces of general type in 1977.

János Kollár

Unfortunately, no one has been able to apply the techniques of GIT beyond the case of curves. The 1988 publication of Kollár and N. Shepherd Barron, Threefolds and deformations of surface singularities, marked a breakthrough in the theory of moduli. Instead of GIT, this paper proposed to use ideas from birational geometry. The paper was the beginning of a monumental effort, including more than thirty major papers by Kollár, culminating in his book Families of varieties of general type.

The part of the programme dealing with surfaces was completed in the early 1990s. The construction of the moduli space in all dimensions required considerably more work. Kollár has always been the main driving force behind this programme, and he has contributed the lion’s share of understanding the overall picture in this regard.

It is impossible to understate how influential this general research programme has been in modern complex algebraic geometry. The solution to this problem not only originated with Kollár (and Barron), but substantially more than anyone else, has been developed by Kollár over his entire career. Almost every aspect of its construction has been deeply influenced by, or solved by, Kollár.

One important aspect of Kollár’s theory is its wide-ranging applicability. What Kollár achieves through this book is fundamental, across all moduli theories of varieties. As is true of all his books, Families of varieties of general type, although presenting the reader with some very serious technical challenges and being full of profound insights, has a clear structure, clear explanations and provides detailed exposition of the motivation of the concepts, making it an enjoyable read and an attractive way to study the moduli of varieties of general type; it is required reading for anyone working in the moduli theory of varieties. This book is written by the master of the field – it is a crowning achievement in algebraic geometry.