  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.
