On algebraically coisotropic submanifolds of holomorphic symplectic manifolds (Q6543022)

The authors study cosisotropic submanifolds of holomorphic symplectic manifolds. For the case of a hypersurface, the situation was already well understood. Indeed, it can be easily shown that if the submanifold is a smooth uniruled hypersurface, it is algebraically coisotropic. On the other hand, let \(D\) be a smooth algebraically coisotropic hypersurface in a holomorphic symplectic manifold \(M\). In earlier work [J. Lond. Math. Soc., II. Ser. 95, No. 1, 115--127 (2017; Zbl 1402.14010)], the authors showed that \(D\) is uniruled or up to finite étale cover\N\[\ND = C \times Y \subset M = S\times Y\N\]\Nwhere \(Y\) is a holomorphic symplectic manifold and \(C\) is a curve in the holomorphic symplectic surface \(S\). \textit{J.-M. Hwang} and \textit{E. Viehweg} [Compos. Math. 146, No. 2, 497--506 (2010; Zbl 1208.37031)] had previously worked out that a smooth algebraically coisotropic hypersurface of general type is a curve in a holomorphic symplectic surface.\N\NRecall that the higher codimension analogue of a curve in a holomorphic symplectic surface is a Lagrangian subvariety. Here, the authors study the higher codimension question:\N\NLet \(X\) be a non-uniruled algebraically coisotropic submanifold of a projective holomorphic symplectic manifold \(M\). Up to finite étale cover, is it true that\N\[\NX = L\times Y \subset M = N\times Y\N\]\Nwhere \(Y,N\) are holomorphic symplectic and \(L\) is Lagrangian in \(N\)?\N\NFor the case where \(M\) is an abelian variety, they show that this is true. They also show the following.\N\N{Theorem. } Let \(X\) be an algebraically coisotropic submanifold of a holomorphic symplectic manifold \(M\). Let \(f\colon X \to B\) be the corresponding characteristic fibration. Suppose \(K_X\) is semiample. Then \(f\) is isotrivial and \(\kappa(X) = \kappa(F)\), where \(F\) is a smooth fibre of \(f\).\N\NUsing results of \textit{B. Taji} [Eur. J. Math. 9, No. 4, Paper No. 88, 44 p. (2023; Zbl 1530.14026)], they extend this result to the case where \(K_X\) is not necessarily semiample but we assume only that the smooth fibres of \(f\) have good minimal models. As a corollary, they prove a higher dimensional analogue of Hwang and Viehweg's result.