What are classical/simple examples of smooth projective surfaces $S$ (over $\mathbb C$) for which all the $1$-forms have a common zero (and $h^{1,0}(S)>0$)?



reference quest: surface whose $1$-forms have a common zero
If $S$ admits a global holomorphic 1-form without zeros, $S$ would admit a $C^\infty$ real closed 1-form which has no zeros, equivalently, $S$ would admit a $C^\infty$ fibre bundle structure over the circle.


In this way, the Euler number $\chi(S)$ and the signature $\sigma (S)$ vanish: see [Holomorphic one-forms, fibrations over the circle, and characteristic numbers of Kähler manifolds](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/holomorphic-oneforms-fibrations-over-the-circle-and-characteristic-numbers-of-kahler-manifolds/DAC4529082DBF6F70474970ACFA70426), Lemma 2.


So a simple choice may be a surface with either $\chi(S)$ or $\sigma (S)$ non-vanishing, and $h^{1,0}(S)=1$. A concrete example may be the [Cartwright-Steger surface](https://www.maths.usyd.edu.au/u/donaldc/cs-surface/cs-surface.pdf). See also [Enumeration of the 50 fake projective planes](https://www.sciencedirect.com/science/article/pii/S1631073X09003987).


