# Hodge numbers of a hypothetical complex structure on $S^6$

## TL;DR

This paper discusses the theoretical properties and Hodge numbers of a hypothetical complex structure on the six-dimensional sphere, synthesizing prior research without presenting new original results.

## Contribution

It compiles and analyzes existing results on Hodge numbers and cohomology of a hypothetical complex structure on $S^6$, highlighting what is known and what remains conjectural.

## Key findings

- Hodge numbers for the hypothetical complex structure are constrained by existing theories.
- The Frölicher spectral sequence's dimensions are investigated in this context.
- Bott-Chern cohomology provides insights into the structure of hypothetical complex $S^6$.

## Abstract

These are the notes for the talk "Hodge numbers of a hypothetical complex structure on $S^6$" given by the author at the MAM1 "(Non)-existence of complex structures on $S^6$" held in Marburg in March 2017. They are based on [A. Gray, A property of a hypothetical complex structure on the six sphere, Boll. Un. Mat. Ital. B (7)} $\textbf{11}$ (1997), Suppl. fasc. 2, 251--255.] and [L. Ugarte, Hodge numbers of a hypothetical complex structure on the six sphere, Geom. Dedicata $\textbf{81}$ (2000), no. 1-3, 173--179.], where Hodge numbers and the dimensions of the succesive pages of the Fr\"olicher spectral sequence for $S^6$ endowed with a hypothetical complex structure are investigated. We also add results from [Andrew McHugh, Narrowing cohomologies on complex $S^6$, Eur. J. Pure Appl. Math. $\textbf{10}$ (2017), no. 3, 440--454.], where the Bott-Chern cohomology of hypothetical complex structures on $S^6$ is studied. The material is not intended to be original.