Please, Rapaire!! At least report with respect for verisimilitude.

What Kotschick actually SAID was the they had prove that 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 three, they further proved that only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. These results solve a long-standing problem of Hirzebruch's. He also claimed to determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers. This remains to be demonstrated.

(A diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.)

(Chern classes are characteristic classes. They are topological invariants associated to vector bundles on a smooth manifold. If you describe the same vector bundle on a manifold in two different ways, the Chern classes will be the same. When are two ostensibly different vector bundles the same? When are they different? These questions can be quite hard to answer. But the Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. (The converse is not true, though.))

(The Betti number of a topological space is, in intuitive terms, a way of counting the maximum number of cuts that can be made without dividing the space into two pieces. This defines, in fact, what is called the first Betti number. the Euler characteristic (or Euler-Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. The Pontryagin class lies in cohomology groups with index a multiple of four. It applies to real vector bundles.)


See, I thoughtfully provided informal definitions to make the whole matter clearer...


A