Splitting of the tangent sheaf for codimension one foliations without invariant hypersurfaces or rational first integrals

Determine whether the tangent sheaf TF of a codimension one holomorphic foliation F on complex projective space P^n with locally free tangent sheaf necessarily splits when F does not admit any invariant hypersurface; more generally, determine whether TF necessarily splits when F admits no rational first integral.

Background

The paper investigates conditions under which tangent sheaves of holomorphic distributions and foliations split, with a focus on two-dimensional distributions and codimension one foliations on projective spaces. Among its main results, Theorem C shows that for codimension one foliations on P^3 tangent to a nontrivial holomorphic vector field, the tangent sheaf splits. Additional criteria and division properties are established to guarantee splitting in various scenarios.

The authors note that there exist codimension one foliations on projective space whose tangent sheaf is locally free but does not split, with recent explicit constructions given in [13]. Importantly, all known non-splitting examples have rational first integrals. Building on Corollary 6.5, which ensures splitting for certain low-degree cases or under the presence of specific geometric structures, the authors pose an open problem asking whether the absence of invariant hypersurfaces or rational first integrals suffices to force splitting of TF.