Let X be a compact connected complex manifold of complex dimension n, and let f:X→ X be a holomorphic Anosov diffeomorphism. Thus there is an invariant splitting T_X=E^s oplus E^u and constants C>0 and 0<lambda<1 such that lVert Df^k|_E^srVert≤ C lambda^k and lVert Df^-k|_E^urVert≤ C lambda^k for every k≥ 0. If f is codimension one, meaning dim_ CE^s=n-1 or dim_ CE^u=n-1, then there exist a complex torus T= C^n/Lambda, a hyperbolic complex-linear torus automorphism A:T→ T, and a biholomorphism h:X→ T such that h circ f circ h^-1=A. Here hyperbolic means that no eigenvalue of the linear map inducing A has absolute value 1.

Ghys’s codimension-one holomorphic Anosov conjecture

Let $X$ be a compact connected complex manifold of complex dimension $n$ , and let $f:X\to X$ be a holomorphic Anosov diffeomorphism. Thus there is an invariant splitting $T_X=E^s\oplus E^u$ and constants $C>0$ and $0<\lambda<1$ such that $\lVert Df^k|_{E^s}\rVert\le C\lambda^k$ and $\lVert Df^{-k}|_{E^u}\rVert\le C\lambda^k$ for every $k\ge 0$ . If $f$ is codimension one, meaning $\dim_{\mathbb C}E^s=n-1$ or $\dim_{\mathbb C}E^u=n-1$ , then there exist a complex torus $T=\mathbb C^n/\Lambda$ , a hyperbolic complex-linear torus automorphism $A:T\to T$ , and a biholomorphism $h:X\to T$ such that $h\circ f\circ h^{-1}=A$ . Here hyperbolic means that no eigenvalue of the linear map inducing $A$ has absolute value $1$ .

References

Primary source

arXiv

Additional references

Codimension-one holomorphic Anosov diffeomorphisms — arXiv — Jiesong Zhang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims to prove the classification, but the claim has not been independently checked.

Ghys’s conjecture classifies codimension-one holomorphic Anosov systems, with a corresponding consequence in complex dimension three. Ghys introduced the relevant classification results in 1995.

Known results

Ghys, 1995: holomorphic Anosov diffeomorphisms of compact complex surfaces are holomorphically conjugate to linear torus automorphisms.
Ghys, 1995: transitive systems with real $2$ -dimensional unstable foliation are topologically conjugate to linear toral automorphisms.
Ghys, 1995: holomorphic Anosov flows on compact complex $3$ -manifolds are classified up to finite covers, including suspensions, twisted examples, and roots of $\mathbb{C}^{*}$ -actions.

September 2026 claimed proof

Jiesong Zhang’s preprint claims the codimension-one classification and its complex-dimension-three consequence. No independent expert assessment, referee report, or verification was found; another 2026 preprint still presents the broader Ghys conjecture as open.

Current status (as of September 2026): The codimension-one classification is claimed in an unrefereed preprint, but remains unverified; the broader conjectural picture is not settled.

Sources

No solutions have been posted yet.