Moduli spaces of spacefilling branes in symplectic 4-manifolds

On a symplectic manifold $(M,\omega )$, a spacefilling brane structure is a closed 2-form $F$ which determines a complex structure, with respect to which $F+i\omega$ is holomorphic symplectic. For holomorphic symplectic compact Kähler 4-manifolds, we show that the moduli space of spacefilling branes is smooth, and determine its dimension. The proof relies on the local Torelli theorem for K3 surfaces and tori.