Characteristic foliations

The restriction of the holomorphic symplectic form on a hyperkähler manifold X to a smooth hypersurface D Ă X leads to a regular foliation F Ă TD of rank one, the characteristic foliation. We survey, with essentially complete proofs, a series of recent results concerning the geometry of the characteristic foliation, starting with work by Hwang–Viehweg [HV10], but also covering articles by Amerik–Campana [AC14] and Abugaliev [Ab19, Ab21]. The picture is complete in dimension four and shows that the behavior of the leaves of F on D is determined by the Beauville–Bogomolov square qpDq of D. In higher dimensions, some of the results depend on the abundance conjecture for D. 1. Main theorem and motivation Throughout, D Ă X denotes a smooth connected hypersurface in a compact hyperkähler manifold X of complex dimension 2n, i.e. X is a simply connected, compact Kähler manifold such that H0pX,Ω2Xq is spanned by a holomorphic symplectic form σ. Usually X will be in addition assumed to be projective, although one expects all results to hold in general. The symplectic form σ induces a regular foliation of rank one on D, i.e. a line sub-bundle F Ă TD. We shall denote a generic leaf of the foliation by L and its Zariski closure by L̄. The space of leaves will be denoted D{F . These notions will all be recalled in Sections 2 and 3. Our goal to discuss the following table and establish equivalence of all assertions in each row. We will throughout assume n ą 1, but see Remark 1.1.