Projective Subspaces with Trivial Normal Bundle in Fano Manifolds

Definitions and mathematical statement

Fano immersion conjecture. Let $X$ be a Fano manifold of complex dimension $n$, with $b_2(X)=1$, and assume $X\not\cong{\mathbb{P}}^n$. For every integer $k$ with $1<k<n$, is there no holomorphic immersion $f:{\mathbb{P}}^k\to X$ whose normal bundle is trivial? In the usual complex projective convention, Fano means smooth projective with ample anticanonical line bundle, and $b_2(X)=\dim_{\mathbb{Q}} H^2(X;{\mathbb{Q}})$.

An immersion requires $df:T_{{\mathbb{P}}^k}\to f^*T_X$ to be injective at every point. Its holomorphic normal bundle is defined by\[0\longrightarrow T_{{\mathbb{P}}^k}\xrightarrow{df}f^*T_X\longrightarrow N_f\longrightarrow0.\] The prohibited condition is $N_f\cong\mathcal O_{{\mathbb{P}}^k}^{\oplus(n-k)}$ as a holomorphic vector bundle. This is stronger than vanishing first Chern class or topological triviality. An immersion is not assumed globally injective, so replacing it by an embedding would weaken the proposed prohibition. Nor does projective space as the domain mean a linear subspace in a chosen projective embedding of $X$.

The quantifiers exclude an immersion for each individual permitted $k$; they do not merely deny a collection of immersions existing in all dimensions at once. The strict range excludes projective lines and the equal-dimension case; for $n\le2$ it is empty. The hypothesis $X\not\cong{\mathbb{P}}^n$ is retained even if a separate argument might make it unnecessary in another formulation.

The supplied triage checked 2026-08-21 labels the conjecture open and reports no located authoritative resolution. The report is its sole reference and no precise later theorem is supplied. This dated status assertion is preserved without independent verification of present-day status.

Short English statement

The conjecture prohibits every holomorphically immersed ${\mathbb{P}}^k$, $2\le k<n$, with a holomorphically trivial normal bundle in the stated Picard-type-one Fano setting. Immersions are not restricted to embeddings or linear subspaces, and triviality is not replaced by a Chern-class condition.

Sources

• Local UnsolvedMath registry, permanent ID 30000583, code catalogue entry; Projective Subspaces with Trivial Normal Bundle in Fano Manifolds, Oberwolfach Reports Open Problems.
• Komplexe Analysis (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/40, pp. 2399--2446. https://doi.org/10.4171/owr/2006/40. Proposed year and year tag: 2006; no proposer named.
• The same full report is the sole supplied literature source. No separate authoritative immersion theorem or conjecture attribution is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field claims editorial and notation cleanup only. Verify dated claims independently.