Holomorphic automorphisms of Markov-type surfaces

Rafael B. Andrist

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arXiv · 2502.15101

Holomorphic automorphisms of Markov-type surfaces

Rafael B. Andrist

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Abstract

Every complex surface of Markov type, i.e.\ the variety given by $x^2 + y^2 + z^2 + Exyz - Ax - By - Cz - D = 0$, has the symplectic density property and the Hamiltonian density property. We prove a singular symplectic version of the Anders{\'e}n--Lempert theorem for normal reduced affine complex varieties and apply it to describe the holomorphic symplectic automorphisms of a complex surface of Markov type. To this end, we also investigate the germs of vector fields in isolated singularities of type $A_k$ and $D_k$. Moreover, we show that any injective self-map of the set of ordered Markov triples can be realized by a holomorphic symplectic automorphism.

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BibTeX RIS

Rafael B. Andrist. 2025-02-20. Holomorphic automorphisms of Markov-type surfaces. https://arxiv.org/abs/2502.15101

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