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: Mathematician's insight helps unravel knotty problem

Related Stories

Mathematician's insight helps unravel knotty problem

December 5, 2005

The latest insight from Rice University assistant professor Shelly Harvey is the kind of idea that comes along rarely for a theorist in any discipline: It's an idea that is both simple and capable of explaining much.

Russian mathematician wins science award

December 22, 2006

A Russian mathematician's solution to a 100-year-old math puzzle was voted Breakthrough of the Year by Science, a leading scientific journal.

Glimpses of a new (mathematical) world

March 13, 2008

A new mathematical object was revealed yesterday during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental ...

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.

Mathematicians find new solutions to an ancient puzzle

March 14, 2008

Many people find complex math puzzling, including some mathematicians. Recently, mathematician Daniel J. Madden and retired physicist, Lee W. Jacobi, found solutions to a puzzle that has been around for centuries.

Recommended for you

Study casts new light on fishing throughout history

November 12, 2018

A new study from The Australian National University (ANU) has revealed new insights into ancient fishing throughout history, including what type of fish people were regularly eating as part of their diet.

A toast to the proteins in dinosaur bones

November 9, 2018

Burnt toast and dinosaur bones have a common trait, according to a new, Yale-led study. They both contain chemicals that, under the right conditions, transform original proteins into something new. It's a process that may ...


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.