Topologically invariant Chern numbers of projective varieties

TL;DR

This paper characterizes which Chern numbers of smooth complex projective varieties are invariant under diffeomorphisms, showing only Euler and Pontryagin numbers are invariant, and identifies those bounded by Betti numbers.

Contribution

It proves that only Euler and Pontryagin numbers are diffeomorphism invariants among Chern numbers, solving a longstanding problem of Hirzebruch.

Findings

Only Euler and Pontryagin numbers are oriented diffeomorphism invariants.

In dimension at least three, only multiples of the top Chern number are invariant.

Certain Chern number combinations can be bounded by Betti numbers.

Abstract

We 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 we prove 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. We also determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers.