Algebraic structures with unbounded Chern numbers

TL;DR

This paper characterizes which Chern numbers of high-dimensional smooth complex projective varieties are topologically determined, providing bounds and optimality results for their linear combinations.

Contribution

It identifies all Chern numbers determined by the underlying smooth manifold in dimensions four and higher, and establishes bounds on their linear combinations.

Findings

All such Chern numbers are classified in dimensions ≥4.

An upper bound on the dimension of the space of these Chern numbers is proven.

Optimality of the bound is demonstrated in dimension four.

Abstract

We determine all Chern numbers of smooth complex projective varieties of dimension at least four which are determined up to finite ambiguity by the underlying smooth manifold. We also give an upper bound on the dimension of the space of linear combinations of Chern numbers with that property and prove its optimality in dimension four.