Characteristic foliations -- a survey (Q6570059)

Let \(X\) be a compact hyperkähler manifold, that is a simply connected compact Kähler manifold of dimension \(2n\) such that \(H^0(X, \Omega_X^2)\) is generated by a symplectic form \(\sigma\). While \(\sigma\) is by definition non-degenerate on \(X\) its restriction to any smooth prime divisor \(D \subset X\) has a one-dimensional kernel defining a regular foliation \(\mathcal F \subset T_D\). It is expected that the geometry of this foliation is determined by the Beauville-Bogomolov square \(q_X(D)\) of the divisor. More precisely, let \(L \subset D\) be a general leaf of the foliation \(\mathcal F\), and denote by \(\bar L \subset D\) its closure in the Zariski topology. Then one expects\N\begin{itemize}\N\item \(\dim \bar L=1\) (i.e. \(L\) is algebraic) if and only if \(q_X(D)<0\);\N\item \(\dim \bar L=n\) if and only if \(q_X(D)=0\);\N\item \(\dim \bar L=2n-1\) (i.e. \(L\) is dense) if and only if \(q_X(D)>0\).\N\end{itemize}\NMoreover, these properties should be determined by the Kodaira dimension of the divisor. \newline In this paper the authors give a very well written overview of the various techniques used for attacking these conjectures, from the initial work of \textit{J.-M. Hwang} and \textit{E. Viehweg} [Compos. Math. 146, No. 2, 497--506 (2010; Zbl 1208.37031)] to the more recent contributions of \textit{E. Amerik} and \textit{F. Campana} [J. Lond. Math. Soc., II. Ser. 95, No. 1, 115--127 (2017; Zbl 1402.14010)] and \textit{R. Abugaliev} [``Characteristic foliation on hypersurfaces with positive Beauville-Bogomolov-Fujiki square'', Preprint, \url{arXiv:2102.02799}]. They also give essentially all the proofs of their results, the classical examples and comments about a possible extension to singular divisors.