On linear combinations of cohomological invariants of compact complex manifolds

TL;DR

This paper investigates the relationships and invariance properties of cohomological invariants like Hodge, Betti, and Chern numbers in compact complex manifolds, extending known results and developing a framework for broader invariants.

Contribution

It proves the absence of unexpected universal relations among these invariants and extends the analysis to all cohomological invariants, providing solutions in low dimensions.

Findings

No unexpected universal integral linear relations among Hodge, Betti, and Chern numbers.

Identified linear combinations that are bimeromorphic or topological invariants.

Solved specific construction problems for general cohomological invariants in many cases.

Abstract

We prove that there are no unexpected universal integral linear relations and congruences between Hodge, Betti and Chern numbers of compact complex manifolds and determine the linear combinations of such numbers which are bimeromorphic or topological invariants. This extends results in the K\"ahler case by Kotschick and Schreieder. We then develop a framework to tackle the more general questions taking into account `all' cohomological invariants (e.g. the dimensions of the higher pages of the Fr\"olicher spectral sequence, Bott-Chern and Aeppli cohomology). This allows us to reduce the general questions to specific construction problems. We solve these problems in many cases. In particular, we obtain full answers to the general questions concerning universal relations and bimeromorphic invariants in low dimensions.