Decomposition of Lagrangian classes on \(K3\) surfaces
The aim of the paper under review is to study a \(K3\) surface \(X\) together with a (Ricci-flat) Kähler form on \(X\). The main question is under what condition any Lagrangian class is special. The first part is devoted to giving criteria for it to happen when a Kähler form is given on \(X\). More precisely, if there exists a Lagrangian class of positive self-intersection, then, any Lagrangian is special, and otherwise, it is not the case, but any Lagrangian is special if and only if there exists a negative-definite lattice \(N\) such that both the Lagrangian and special Lagrangian lattices contain \(N\). In the second part, the authors turn to a study in deformation of Kähler forms on \(X\), and show that there exists a dense subset \(\mathcal{S}_X\) of the Kähler cone of \(X\) such that for any class in \(\mathcal{S}_X\), any associated Lagrangian is special. Finally, the authors conclude by describing possible special Lagrangian fibrations on \(X\).
Lagrangian classes on a Kähler \(K3\) surface
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