reference quest: surface whose $1$-forms have a common zero

Question

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

Answer 1

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, 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. See also Enumeration of the 50 fake projective planes.

Answer 2

If $S$ is any surface with $h^{1,0}(S) = 1$, then the Albanese morphism is a surjection $a \colon S \to E$ to an elliptic curve, and the pullback $H^0(E,\Omega_E^1) \to H^0(S,\Omega_S^1)$ is an isomorphism (see for instance this post). So the only question is whether the pullback $a^* \omega$ of a nontrivial $1$-form $\omega$ on $E$ has a zero somewhere.

If $a$ is not smooth, then the pullback $a^* \omega$ vanishes at any singular point $s \in S$ for $a$. Indeed, note that $a$ is singular at $s$ if and only if $\dim_{\kappa(s)}\bigl(\Omega^1_{S/E} \otimes \kappa(s)\bigr) > 1$, which means that the pullback $a^* \colon \Omega^1_E\otimes \kappa(a(s)) \to \Omega^1_S \otimes \kappa(s)$ is the zero map (since $\dim_{\kappa(s)} \Omega_S^1 \otimes \kappa(s) = 2$).

So any surface whose Albanese morphism is a surjection $a \colon S \to E$ with at least one singular fibre gives a counterexample. For instance, if $S$ is a surface of general type with $h^{1,0}(S) = 1$ (in this case, the vanishing of its global $1$-form also follows from [PS14]).

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References.

[PS14] M. Popa, C. Schnell, Kodaira dimension and zeros of holomorphic one-forms. Ann. Math. (2) 179.3 p. 1109-1120 (2014). ZBL1297.14011.

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P.S. If you want a more precise grip on numerical invariants of surfaces, there is a great tool by Pieter Belmans and Johan Commelin with loads and loads of data. See Le superficie algebriche at https://superficie.info.