Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on $\mathbb{P}^4$. We classify dimension two distributions, of degree at most $2$, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non-integrable to integrable distributions. In the maximally non-integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in $\mathbb{P}^4$ whose the conormal sheaves are the Horrocks-Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non-integrable, uniquely determined by their singular schemes and invariant by a group $H_5 \rtimes SL(2,\mathbb{Z}_5) \subset Sp(4, \mathbb{Q})$, where $H_5$ is the Heisenberg group of level $5$. We prove that the moduli spaces of Horrocks-Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree $d\geq 6$, on $\mathbb{P}^4$ have a family of degenerated flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by $H_5 \rtimes SL(2,\mathbb{Z}_5)$ and singular along a degenerate abelian surface with $(1,5)$-polarization and level-$5$-structure.