Characteristic numbers of algebraic varieties

TL;DR

This paper characterizes which linear combinations of Chern numbers are invariants under diffeomorphisms of smooth complex projective varieties, revealing that only Euler and Pontryagin numbers are invariant in general, with specific distinctions in higher dimensions.

Contribution

It provides a complete characterization of diffeomorphism invariants among Chern numbers and clarifies the structure of their subspaces in the context of algebraic varieties.

Findings

Only Euler and Pontryagin numbers are diffeomorphism invariants in all dimensions.

In higher dimensions, only multiples of the top Chern number (Euler characteristic) are invariant.

The space of Chern numbers has two distinguished subspaces with a well-understood intersection.

Abstract

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 only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. In the space of Chern numbers there are two distinguished subspaces, one spanned by the Euler and Pontryagin numbers, the other spanned by the Hirzebruch--Todd numbers. Their intersection is the span of the Euler number and the signature.