Dimension two holomorphic distributions on four-dimensional projective space

We study two-dimensional holomorphic distributions on P4. 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 i...

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Bibliographic Details
Published in:	Bulletin des sciences mathématiques Vol. 197; p. 103517
Main Authors:	Calvo–Andrade, O., Corrêa, M., Fonseca–Quispe, J.
Format:	Journal Article
Language:	English
Published:	Elsevier Masson SAS 01-12-2024
Subjects:	Conformal structures Engel structures Foliations Holomorphic distributions Horrocks-Mumford bundle Moduli spaces secondary Moduli spaces Holomorphic distributions Horrocks-Mumford bundle Foliations Engel structures Conformal structures primary
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Summary:	We study two-dimensional holomorphic distributions on P4. 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 P4 whose 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 H5⋊SL(2,Z5)⊂Sp(4,Q), where H5 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≥6, on P4 has a family of degenerate flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by H5⋊SL(2,Z5) and singular along a degenerate abelian surface with (1,5)-polarization and level-5-structure.
ISSN:	0007-4497
DOI:	10.1016/j.bulsci.2024.103517