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There is a lot of work about compact complex surfaces of general type $X$ having ample cotangent bundle $\Omega_X$: for instance, one can read the recent works of Damian Brotbeck and collaborators in order to have an idea of the state of the art on the topic.

Instead, I am interested in compact complex surfaces of general type having globally generated cotangent bundle, namely, such that the evaluation map on global sections $$ H^0(X, \, \Omega_X) \otimes \mathcal{O}_X \to \Omega_X$$ is surjective. Note that ampleness of the cotangent bundle does not imply in general global generation: for instance, fake projective planes have ample cotangent bundle (since they are compact ball quotients) but $h^0(X, \Omega_X)=h^1(X, \, \mathcal{O}_X)=0$, namely, $\Omega_X$ has no global sections at all and so it cannot be globally generated.

The only general method I know in order to construct surfaces with globally generated cotangent bundle is taking subvarieties of varieties with the same property: for instance, subvarieties of abelian varieties or subvarieties of a product of curves with non-negative genus. In fact, if $\iota \colon X \hookrightarrow Y$, then there is a bundle epimorphism $\iota^*\Omega_Y \to \Omega_X \to 0$ and therefore, if $\Omega_Y$ is globally generated, the same holds for $\Omega_X$. But controlling such subvarieties is a difficult problem in general, unless one takes simple situations like complete intersections.

So, let me ask the following questions:

Question 1: Are there other general methods to construct surfaces of general type with globally generated $\Omega_X$? If so, are there any references ?

Question 2: Are there any interesting "sporadic" examples?

Added Note: I am particularly interested in the case where $\Omega_X$ is globally generated but not ample.

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    $\begingroup$ Quick comment: Isn't the map $H^0(X, \Omega_X)\otimes\mathcal{O}_X\to \Omega_X$ the differential of the Albanese map? If so then it seems that your condition is equivalent to the Albanese being unramified, which is quite close to being a subvariety of an abelian variety. $\endgroup$
    – Piotr Achinger
    Commented Jan 16, 2022 at 19:55
  • $\begingroup$ @PiotrAchinger: yes, it is the dual of the differential of the Albanese map $a \colon X \to \mathrm{Alb}(X)$. Unfortunately, knowing that something is a subvariety of an abelian variety does not provide in general concrete ways to construct it (unless one takes simple cases like complete intersections). It is like saying that every surface embeds in $\mathbb{P}^5$: it is true, but essentially useless if you want interesting examples. $\endgroup$
    – Francesco Polizzi
    Commented Jan 16, 2022 at 20:08
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    $\begingroup$ @FrancescoPolizzi You probably already have seen these: arxiv.org/abs/alg-geom/9410023 and arxiv.org/abs/2001.10475 In any case, these might contain useful suggestions. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jan 16, 2022 at 20:32
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    $\begingroup$ You might find some more inspiration in Sakai's survey MR0555717, but there seems to be a dearth of good examples in general as far as I know. $\endgroup$
    – Frank
    Commented Jan 17, 2022 at 9:46

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