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Codimension-one holomorphic Anosov diffeomorphisms
Jiesong Zhang proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This settles Ghys's conjecture in that case, and so for all compact complex threefolds, where Xu and Zhang (2024) recorded it as open. The proof removes the topological transitivity assumption from Ghys's Theorem B to get a topological conjugacy to a hyperbolic toral automorphism, proves that both invariant distributions are holomorphic (the stable one by an estimate), and shows a torus with holomorphic invariant distributions carries the standard complex structure. Cantat (2004) had the codimension-one case for projective manifolds. The paper also gives a nilmanifold example whose invariant distributions are smooth but not holomorphic, so the method stops at codimension one. The general conjecture remains open.
Exact FrontierDelta
Prior state unknown→proved
Scope and record
Occurred: Sep 29, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: lean-checked. Publication: preprint. AI contribution: ai-assisted. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: Codimension-one holomorphic Anosov diffeomorphisms · Ghys conjecture (codim. 1)
Confidence: Not scored
Registry verification: lean checked · preprint · partial
Attribution
VibeMathed
registry · event recorded by
Intern-S2 (internal version)
model · ai model contributor · Shanghai AI Lab
Jiesong Zhang
human · human collaborator
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Codimension-one holomorphic Anosov diffeomorphisms evidence for this event
Xu and Zhang, On holomorphic partially hyperbolic systems (records the conjecture as open in dimension 3) evidence for this event
VibeMathed record: Codimension-one holomorphic Anosov diffeomorphisms evidence for this event
Jiesong Zhang proves that every codimension-one holomorphic Anosov diffeomorphism of a compact connected complex manifold is biholomorphically conjugate to a hyperbolic automorphism of a complex torus. This settles Ghys's conjecture in that case, and so for all compact complex threefolds, where Xu and Zhang (2024) recorded it as open. The proof removes the topological transitivity assumption from Ghys's Theorem B to get a topological conjugacy to a hyperbolic toral automorphism, proves that both invariant distributions are holomorphic (the stable one by an estimate), and shows a torus with holomorphic invariant distributions carries the standard complex structure. Cantat (2004) had the codimension-one case for projective manifolds. The paper also gives a nilmanifold example whose invariant distributions are smooth but not holomorphic, so the method stops at codimension one. The general conjecture remains open. parent of this event
Lean formalisation (frenzymath/HolomorphicAnosov) evidence for this event
Codimension-one holomorphic Anosov diffeomorphisms parent of this event
This event attributed to Jiesong Zhang
This event attributed to Intern-S2 (internal version)
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