Hyperbolic geometry and moduli of real curves of genus three (Q1709771)

The moduli space of smooth plane quartic curves admits a period map that maps the moduli space to a ball quotient [\textit{S. Kondō}, J. Reine Angew. Math. 525, 219--232 (2000; Zbl 0990.14007)]. In this article the authors study the question of reality for this period map. More precisely, the space of smooth real plane quartic curves has six connected components. The authors prove that each of the components is isomorphic to a divisor complement in an arithmetic real ball quotient, which corresponds to six real forms of certain hyperbolic Gaussian lattice. Moreover, the authors construct a Coxeter diagram for the connected component of maximal real quartic curves and describe its geometry more explicitly.