I have a question about fibrations on Irreducible log holomorphic symplectic manifolds. Lets give some introduction
Motivation; A holomorphic symplectic manifold (HSM) is a $2n$-dimensional compact K\"ahler manifold $X$ which admits a non-degenerate holomorphic $2$-form $\omega\in H^0(X,\Omega^2)$. One of the examples is $K3$ surfaces. The non-degeneracy here means that $\omega^{\wedge n}$ trivialises $\Omega^{2n}=K_X$ , i.e. $c_1=0$. A holomorphic symplectic manifold $(X,\omega)$ is irreducible (IHSM) if $H^0(X,\Omega^2)=\mathbb C\omega$. Matsushita showed that for $X$ be an IHSM, and $\pi:X\to B$ be a proper surjective morphism with connected fibres, and such that $B$ is smooth with $0<\dim B<\dim X=2n$, then the generic fibre is a complex torus (Abelian variety which is Calabi-Yau variety) and $\dim B=n$ which is the same as the dimension of the fibres. Moreover the restriction of $\omega$ on each fiber is zero, i.e. every fibre of $\pi$ is Lagrangian
Now if we consider the pair $(X,D)$ where $D$ is a snc divisor on $X$. Then is there Matsushita like theorem on pair $(X,D)$ where fibers are log Abelian varieties?