Holomorphic automorphisms of Markov-type surfaces
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915345055350784 |
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| author | Andrist, Rafael B. |
| author_facet | Andrist, Rafael B. |
| contents | 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{é}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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15101 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holomorphic automorphisms of Markov-type surfaces Andrist, Rafael B. Complex Variables Algebraic Geometry Number Theory 32M17, 32M25, 14J50, 32Q56, 14J17 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{é}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. |
| title | Holomorphic automorphisms of Markov-type surfaces |
| topic | Complex Variables Algebraic Geometry Number Theory 32M17, 32M25, 14J50, 32Q56, 14J17 |
| url | https://arxiv.org/abs/2502.15101 |