Hartshorne Complete-Intersection Conjecture

Hartshorne conjectured in 1974 that smooth projective varieties of sufficiently small codimension must be complete intersections: if $X\subset\mathbb P^n_\mathbb C$ is smooth with $\dim X>2n/3$, its homogeneous ideal should be generated by exactly $\operatorname{codim}X$ forms [Hartshorne1974SmallCodimension]. The conjecture reflects the fact that low-codimension subvarieties of projective space obey strong topological and cohomological constraints that make them resemble complete intersections.

The first genuinely open situation is codimension two, where by the Serre correspondence the conjecture is closely tied to the splitting of rank-two vector bundles on projective space; the decomposability criterion of Barth and Van de Ven for $2$-bundles belongs to this circle of ideas [BarthVanDeVen1974]. Positive results are known in various small-codimension and positivity ranges, including the work of Ionescu and Russo on conic-connected manifolds [IonescuRusso2012Hartshorne], but the general statement near the boundary $\dim X>2n/3$ resists current methods.

The conjecture remains open; a resolution must either produce a complete-intersection structure for every such $X$ or exhibit a smooth low-codimension subvariety of projective space that is not one.