Sunday, July 26, 2009

Chern numbers of algebraic varieties

Important news of developments in mathematics doesn't come out all that often, but here's some:

Chern numbers of algebraic varieties (6/10/09)
A problem at the interface of two mathematical areas, topology and algebraic geometry, that was formulated by Friedrich Hirzebruch, had resisted all attempts at a solution for more than 50 years. The problem concerns the relationship between different mathematical structures. Professor Dieter Kotschick, a mathematician at the Ludwig-Maximilians-Universität (LMU) in Munich, has now achieved a breakthrough. ... Kotschick has solved Hirzebruch's problem.

It would be somewhat hopeless to try to explain in a few paragraphs what this is all about. But since the result is a good example of the kinds of things mathematicians work on, a little discussion seems worthwhile.

A couple of important mathematicians of the 20th century are directly involved in the explanation. Friedrich Hirzebruch, who is still alive, worked in a number of active related fields, including topology, algebraic geometry, and complex manifolds. All of these are a legacy of the now well-known 19th century mathematician Bernhard Riemann.

Shiing-Shen Chern, who died in 2004 at the age of 93, worked primarily in another field closely related to those mentioned above, namely differential geometry.

The subject matter of all these fields comprises various types of abstract geometric objects. Perhaps the most familiar examples of such objects are plane curves defined by algebraic equations, such as circles, defined as the set of points (x,y) in the plane that satisfy x2+y2=R2, where R is a constant (the radius of the circle).

Such curves can be generalized to any number of dimensions, as subsets in the space of n-tuples of numbers (x1, ..., xn) where all the coordinates xi simultaneously satisfy some specific set of polynomial equations. The coordinates may be complex numbers (the set of which is denoted by ℂ). Such generalized curves in the space ℂn are called algebraic varieties. The branch of mathematics that studies such things is called algebraic geometry.

Mathematicians like to classify things. In plane geometry, for example, polygons are classified as triangles, rectangles, pentagons, etc. according to the number of sides they have. All members of one of these classes of polygons are defined in terms of having the same number of distinct straight sides, say n.

A more general way in which geometric objects (or even more generally, topological objects) can be classified is in terms of objects which are all related by some sort of 1-to-1 mapping between their points. Many different types of mappings may be considered, such as mappings that preserve only the most basic topological properties (in the sense that donuts and coffee cups have the same "shape" in 3 dimensional space). Other types of mappings might be more narrowly defined, such as being "differentiable" (in the sense of calculus). A class of objects is then specified in terms of all objects that are related by some 1-to-1 mapping of a given type.

Normally, it is quite difficult to determine whether two objects are related by some mapping, and hence belong to the same class, because it's usually necessary to specify the mapping explicitly in order to do this.

But sometimes there are shortcuts that make the task of determining relatedness between topological objects easier. One of these is to find some number or set of numbers that is easily calculated and has to be the same for all members of the class of interest. For example, the class of n-gons in the plane is defined simply as all closed curves with the same number (n) of straight sides. Such numbers are called "invariants", because they are the same no matter what transformation (of the specified sort) is applied to any object of the class.

The best sort of invariant (or set of invariants) to have is one that is not only necessary for membership in the class, but also sufficient for membership. In that case one doesn't need to explicitly construct appropriate 1-to-1 mappings – one only has to calculate a few numbers (which may or may not actually be much easier).

Invariants need not be limited to ordinary numbers. They can also be more complicated sorts of mathematical objects, such as polynomials, groups, rings, or even topological objects like manifolds. All that one asks is that there be some way of calculating or determining the invariant associated with an object specified in some other way.

The "Chern numbers" mentioned above are important examples of such invariants, and the result which has now been proven is a statement about the properties of Chern numbers in a broad range of cases.

Why are mathematicians interested in such seemingly abstract constructs? Actually, there are many "real world" applications. Solutions of sets of polynomial equations – i. e., algebraic varieties – are often important in physics, especially mechanics. And more exotic sorts of topological objects crop up in problems involving differential equations.

Here's the research abstract:

Characteristic numbers of algebraic varieties
A rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least 3, only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation preserving. In the space of Chern numbers, there are 2 distinguished subspaces, one spanned by the Euler and Pontryagin numbers, and the other spanned by the Hirzebruch–Todd numbers. Their intersection is the span of the Euler number and the signature.

The complete research paper can be found here.

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