Chern numbers of algebraic varieties

June 10, 2009

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. As reported in the online edition of the journal Proceedings of the National Academy of Sciences of the United States of America (PNAS), Kotschick has solved Hirzebruch's problem.

Topology studies flexible properties of geometric objects that are unchanged by continuous deformations. In algebraic geometry some of these objects are endowed with additional structure derived from an explicit description by polynomial equations. Hirzebruch's problem concerns the relation between flexible and rigid properties of geometric objects.

Viewed topologically, the surface of a ball is always a sphere, even when the ball is very deformed: precise geometric shapes are not important in topology. This is different in algebraic geometry, where objects like the sphere are described by polynomial equations. Professor Dieter Kotschick has recently achieved a breakthrough at the interface of topology and algebraic geometry.

"I was able to solve a problem that was formulated more than 50 years ago by the influential German mathematician Friedrich Hirzebruch", says Kotschick. "Hirzebruch's problem concerns the relation between different mathematical structures. These are so-called algebraic varieties, which are the zero-sets of polynomials, and certain geometric objects called manifolds." Manifolds are smooth topological spaces that can be considered in arbitrary dimensions. The spherical surface of a ball is just a two-dimensional manifold.

In mathematical terminology Hirzebruch's problem was to determine which Chern numbers are topological invariants of complex-algebraic varieties. "I have proved that - except for the obvious ones - no Chern numbers are topologically invariant", says Kotschick. "Thus, these numbers do indeed depend on the algebraic structure of a variety, and are not determined by coarser, so-called topological properties. Put differently: The underlying manifold of an algebraic variety does not determine these invariants."

The solution to Hirzebruch's problem is announced in the current issue of PNAS Early Edition, the online version of PNAS.

Source: Ludwig-Maximilians-Universität München

Explore further: Mathematic innovator Raoul Bott dies

Related Stories

Mathematic innovator Raoul Bott dies

January 9, 2006

Raoul Bott, a mathematician who made innovative contributions to differential geometry and topology, has died at the age of 82.

Recommended for you

From a very old skeleton, new insights on ancient migrations

October 9, 2015

Three years ago, a group of researchers found a cave in Ethiopia with a secret: it held the 4,500-year-old remains of a man, with his head resting on a rock pillow, his hands folded under his face, and stone flake tools surrounding ...

Mexican site yields new details of sacrifice of Spaniards

October 9, 2015

Excavations at the site of one of the Spanish conquistadors' worst defeats in Mexico are yielding new evidence about what happened when the two cultures clashed—and a native people, at least temporarily, was in control.

Ancient genome from Africa sequenced for the first time

October 8, 2015

The first ancient human genome from Africa to be sequenced has revealed that a wave of migration back into Africa from Western Eurasia around 3,000 years ago was up to twice as significant as previously thought, and affected ...


Adjust slider to filter visible comments by rank

Display comments: newest first

not rated yet Jun 10, 2009
not rated yet Jun 10, 2009
I don't think I understood everything but from what did understand I thought it was interesting.

Please sign in to add a comment. Registration is free, and takes less than a minute. Read more

Click here to reset your password.
Sign in to get notified via email when new comments are made.