Chern Numbers of Algebraic Varieties: The Evolution of a Classification Problem
When Friedrich Hirzebruch was a Member in the School of Mathematics in 1954, his paper, "Some problems on differentiable and complex manifolds", was published in the Annals of Mathematics. In it he asked whether Chern numbers in albegraic geometry could be understood topologically. Specifically, which Chern numbers are topological invariants of complex-algebraic varieties?
Classification problems exist in both topology, which studies qualitative properties of geometric objects (flexible properties unchanged by continuous deformation), and algebraic geometry, which studies more quantitative properties. "Classification problems are very interesting in both subjects because they tend to be solved slowly over long periods of time," says Robert MacPherson, Hermann Weyl Professor in the School of Mathematics. "You have questions where successive generations of people all contribute to the same problem."
An example of this type of progress led to the recent solution by Dieter Kotschick, former Member in the School of Mathematics (1989-90, 2008-09) and Professor at Ludwig-Maximilians-Universität München, of the classification problem that Hirzebruch proposed at the Institute more than fifty years ago. The solution was made possible by several mathematicians (and surprisingly physicists) at the Institute and elsewhere. It concerns Chern numbers, which emerged from work Shiing-Shen Chern conducted at the Institute as a Member between 1943 and 1945.
To solve a classification problem one needs to find "invariants" (ways to distinguish topologically different objects, or different objects in algebraic geometry) and also one needs to develop techniques to show that two apparently different objects are topologically the same. In algebraic geometry, Chern numbers are invariants that distinguish between algebraic varieties. Hirzebruch was interested in how they could be understood as a classification problem in topology.
The first progress on this problem was made at the Institute in 1958-59 when Armand Borel, then a Professor at the Institute, and Hirzebruch developed a technique that showed how certain pairs of different algebraic varieties were the same topologically. A subsequent breakthrough ccame in 1994 with the introduction of equations developed for theoretical physics by Nathan Seiberg, who would become a Professor in the School of Natural Sciences in 1997, and Edward Witten, a Professor in the School since 1987. The Seiberg-Witten equations provided powerful techniques for determining when two algebraic surfaces are smoothly different although they may be topologically the same. Such examples later turned out to be crucial for the solution of Hirzebruch's problem.
Kotschick had been thinking about Borel and Hirzebruch's joint work for years. Some time after arriving at the Institute last September, he began to ponder a seemingly unrelated question about the Pontryagin numbers of smooth manifolds admitting Riemannian metrics of nonnegative sectional curvature. He was able to prove that there is one such number, the signature, that is up to multiples the only Pontryagin number that is bounded on connected manifolds of nonnegative curvature.
It was this work that motivated his interest in the structure result for the rational complex cobordism ring by John Milnor, former Institute Professor (1970-90). Kotschick was able to construct a new sequence of generators for this ring using differences of diffeomorphic algebraic varieties, which are constructed as projective space bundles over certain algebraic surfaces.
This construction led to Kotschick's solution of Hirzebruch's problem in all dimensions. He concluded that, with the obvious exceptions, Chern numbers do depend on the algebraic structure of a variety, not just on topological properties. "It was the peace and quiet at the IAS that allowed me to zero in on the solution," says Kotschick. "This is one of those cases where the freedom to ponder things quitely and to concentrate without distractions led to unexpected payoffs. Sometimes the crucial idea appears through thinking about seemingly unrelated problems, or by transposing methods from one area of mathematics to another."