“Holomorphic symplectic automorphisms of Markov surfaces”
Giovedì 18 giugno 2026, ore 14:15 11:15 - Aula 2AB45 1AD100 - Rafael Andrist (University of Ljubljana)
Abstract
The Diophantine solutions of the equation $x^2 + y^2 + z^2 = 3xyz$ were originally considered by Markov, and are now called Markov triples. Later, this equation was studied over the complex numbers and considered as an algebraic surface. The group of algebraic automorphisms of the Markov surface is discrete and acts transitively on the Markov triples. The Markov surface admits a natural meromorphic symplectic form which is preserved by all algebraic automorphisms.
We describe the identity component of the group of holomorphic symplectic automorphisms of the Markov surface. In contrast to the algebraic case, this group is infinite-dimensional and interpolates any permutation of (ordered) Markov triples. The results can be extended to so-called Markov-type surfaces of the form $x^2 + y^2 + z^2 – 3xyz – Ax – By – Cz – D = 0$.
