Reality of Beauville Surfaces

Definitions and mathematical statement

Let $S$ be the complex Beauville surface intended in the source, and let $\overline S$ be its complex conjugate. Beauville surfaces are rigid surfaces isogenous to a product: they have a presentation as a free finite-group quotient of a product of smooth projective curves of genera at least two, with additional Beauville conditions. In the unmixed setting one commonly writes $S=(C_1\times C_2)/G$, with the group preserving the two factors and each quotient map $C_i\to C_i/G\cong{\mathbb{P}}^1$ branched over three points. The extracted record gives no such presentation, group, branch data, or indication whether mixed actions are included.

The two requested determinations are distinct: \[ \text{Does there exist a biholomorphism }f:S\longrightarrow\overline S? \] \[ \text{Does there exist an antiholomorphic involution }\tau:S\longrightarrow S, \qquad\tau^2=\operatorname{id}_S? \] In the second question, $S$ is called real if it has a model over ${\mathbb{R}}$, equivalently a real structure given by such an involution. A biholomorphism to $\overline S$ corresponds to an antiholomorphic automorphism of $S$, which is not required to have order two. Descent to ${\mathbb{R}}$ requires the existence of a choice of $f$ obeying the cocycle condition \[ \overline f\circ f=\operatorname{id}_S. \] Consequently the first existence condition must not be identified with the second.

The source asks these questions about "the Beauville surface $S$" without specifying it. It may seek criteria in terms of construction data, or a classification over a stated family, rather than assert a universal yes answer. The supplied assessment checked 2026-08-21 reports examples and group-theoretic criteria giving both conjugation-invariant and non-conjugation-invariant Beauville surfaces, including non-real examples, and says that a complete classification is broader than those results. No particular example or complete criterion is reproduced in the record.

Short English statement

Determine whether the intended Beauville surface is isomorphic to its conjugate and whether it admits a real structure. The latter requires an antiholomorphic involution, a stronger condition than merely having an antiholomorphic automorphism.

Sources

• Local UnsolvedMath registry, permanent ID 30000119, catalogue entry, Reality of Beauville Surfaces.
• Komplexe Analysis (2005), Oberwolfach Reports, pp. 2171--2216, DOI https://doi.org/10.4171/owr/2004/42.
• Bauer--Catanese--Grunewald, supplied preprint link https://arxiv.org/abs/math/0408025; no full article title or theorem number is reproduced.
• Supplied literature triage checked 2026-08-21, reporting examples and group-theoretic criteria but not a complete classification.