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

These are the notes for the talk "Hodge numbers of a hypothetical complex structure on S6" given by the author at the MAM1 "(Non)-existence of complex structures on S6" held in Marburg in March 2017. They are based on [A. Gray, A property of a hypothetical complex structure on the six sphere, {\em 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, {\em 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 S6 endowed with a hypothetical complex structure are investigated. We also add results from [Andrew McHugh, Narrowing cohomologies on complex S6, to appear in {\em Eur. J. Pure Appl. Math.}.], where the Bott-Chern cohomology of hypothetical complex structures on S6 is studied. The material is not intended to be original.