Chern numbers of algebraic varieties

Jun 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: Researchers help Boston Marathon organizers plan for 2014 race

add to favorites email to friend print save as pdf

Related Stories

Mathematician's insight helps unravel knotty problem

Dec 05, 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 ...

Glimpses of a new (mathematical) world

Mar 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 ...

Mathematicians find new solutions to an ancient puzzle

Mar 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

Poll: Big Bang a big question for most Americans

3 minutes ago

Few Americans question that smoking causes cancer. But they have more skepticism than confidence in global warming, the age of the Earth and evolution and have the most trouble believing a Big Bang created the universe 13.8 ...

Egypt archaeologists find ancient writer's tomb

Apr 19, 2014

Egypt's minister of antiquities says a team of Spanish archaeologists has discovered two tombs in the southern part of the country, one of them belonging to a writer and containing a trove of artifacts including reed pens ...

Study finds law dramatically curbing need for speed

Apr 18, 2014

Almost seven years have passed since Ontario's street-racing legislation hit the books and, according to one Western researcher, it has succeeded in putting the brakes on the number of convictions and, more importantly, injuries ...

User comments : 2

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.

More news stories

Clippers and coiners in 16th-century England

In 2017 a new £1 coin will appear in our pockets with a design extremely difficult to forge. In the mid-16th century, Elizabeth I's government came up with a series of measures to deter "divers evil persons" ...

Making graphene in your kitchen

Graphene has been touted as a wonder material—the world's thinnest substance, but super-strong. Now scientists say it is so easy to make you could produce some in your kitchen.