Carrell Minimality Conjecture

Definitions and mathematical statement

Carrell's conjecture, as supplied. If a smooth projective variety $X$ of general type admits a nowhere-vanishing holomorphic one-form, then $X$ is minimal. In the usual complex setting, let $X$ be connected of positive complex dimension $d$, with canonical line bundle $K_X=\bigwedge^d\Omega_X^1$. General type means $\kappa(X)=d$, equivalently that $K_X$ is big. The hypothesis is the existence of $\alpha\in H^0(X,\Omega_X^1)$ such that $\alpha_x\ne0$ in $T_x^*X$ for every $x\in X$. Under the usual smooth minimal-model convention, the conclusion is that $K_X$ is nef, meaning $K_X\cdot C\ge0$ for every irreducible curve $C\subset X$. The excerpt does not define minimality; this convention is explicit rather than a claim that all alternative birational meanings agree.

The supplied literature triage checked 2026-08-21 labels the implication open. It reports results about zeros of holomorphic one-forms for minimal varieties, simple Albanese varieties, and other cases, but describes the general minimality implication as unresolved. It cites Hacon--Kovacs and Popa--Schnell without giving their exact theorem statements. There is a material logical issue to verify: a theorem saying that every holomorphic one-form on every smooth projective variety of general type has a zero would exclude the antecedent altogether, and would imply the displayed conditional vacuously, regardless of minimality. Such a theorem must be distinguished from a zero theorem restricted to already minimal varieties or other subclasses. The supplied assessment does not settle that scope comparison. Accordingly, the original implication and attribution are preserved, but its open label is not adopted as an independently verified present-day status.

Short English statement

The supplied implication says that a general-type variety with a nowhere-zero holomorphic one-form must be minimal. The definition of minimality is made explicit conditionally. Its literature-status note needs review: a general zero theorem for such one-forms would rule out the hypothesis and hence settle the implication vacuously.

Sources

• Local UnsolvedMath registry, permanent ID 30000186, code catalogue entry; Carrell Minimality Conjecture, attributed to Carrell in the original statement; Oberwolfach Reports Open Problems.
• Komplexe Algebraische Geometrie (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/07, pp. 407--463. https://doi.org/10.4171/owr/2005/07. Proposed year 2005; no separate proposer field supplied.
• Supplied literature label Hacon--Kovacs, holomorphic one-forms on varieties of general type: https://doi.org/10.1016/j.ansens.2004.12.002.
• Supplied literature label Popa--Schnell, zeros of holomorphic one-forms: https://doi.org/10.4007/annals.2014.179.3.6.
• Supplied literature label original OWR report: https://doi.org/10.4171/owr/2005/07.
• Literature triage checked 2026-08-21, registry status open; record updated 2026-09-16. The corrected_verified label concerns source-label removal and notation; the dated theorem-scope assessment was not independently checked here.