Rigid Fano Targets of Holomorphic Maps

Definitions and mathematical statement

A Fano manifold is a smooth connected complex projective variety $X$ with ample anticanonical line bundle $K_X^{-1}$. Let $\pi:T_X\to X$ be the total space of its holomorphic tangent bundle, rather than its projectivization. A section is a holomorphic map $s:X\to T_X$ with $\pi\circ s=\operatorname{id}_X$.

Use the source's following rigid-target condition: \[ \begin{gathered} \text{For every irreducible projective subvariety }Z\subset T_X \text{ with }\overline{\pi(Z)}=X,\\ \text{there is a section }s:X\to T_X\text{ such that }Z=s(X). \end{gathered} \] Equivalently in the last line, $\pi|_Z:Z\to X$ is an isomorphism. The classification problem is to determine which Fano manifolds satisfy this condition. Here projective subvariety means a complete projective subvariety inside the total space, not a subvariety of ${\mathbb{P}}(T_X)$.

The source also describes rigidity as the property that every deformation of a surjective holomorphic map $f:Y\to X$ arises from an automorphism of $X$. In the fixed-domain, local-family interpretation, a holomorphic deformation $f_t$ with $f_0=f$ should have the form \[ f_t=a_t\circ f,\qquad a_t\in\operatorname{Aut}(X),\quad a_0=\operatorname{id}_X, \] locally in the deformation parameter. The extract does not specify the allowed source spaces $Y$ or whether a local, infinitesimal, or global deformation statement is intended. Its claimed equivalence is retained as source content, while the explicit tangent-bundle condition supplies the main classification target.

Short English statement

Classify Fano manifolds for which every projective subvariety of the tangent-bundle total space that dominates the manifold is a section. The source relates this to rigidity of surjective maps up to target automorphisms, but does not specify the source spaces or deformation conventions.

Sources

• Supplied report: Komplexe Analysis (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/40, pp. 2399--2446; proposed year 2006. https://doi.org/10.4171/owr/2006/40.
• Supplied reference labelled "Fano manifolds and rigid targets": https://intlpress.com/site/pub/files/_fulltext/journals/pamq/2011/0007/0004/PAMQ-2011-0007-a015.pdf.
• Local UnsolvedMath registry, permanent ID 30000582, catalogue entry; created 25 August 2026, updated 16 September 2026. Its corrected_verified note describes editorial cleanup. The supplied literature triage of 21 August 2026 labels the classification partially solved, reporting a complete Fano-surface classification and broad higher-dimensional sufficient criteria, but no general classification. This dated assessment is retained without independent verification or PDF access.