The projective-space base conjecture for Lagrangian fibrations of compact hyperkahler manifolds

Theorem 1.1: if $f:X\to B$ is a projective surjective connected-fibre morphism from a compact IHS Kahler manifold of dimension $2n$ onto a normal projective $n$-dimensional $B$, with every fibre component $n$-dimensional and isotropic for $\sigma$, then $B\cong\mathbb P^n$, with no smoothness assumption on $B$, no bound on $b_2$ and no restriction on deformation type. With Kim-Oguiso-Shinder it excludes codimension-one multiple fibres. Not shown: the case of a nonprojective $X$ (Kahler $X$ with a non-projective morphism).

Theorem 1.1: if $f:X\to B$ is a projective surjective connected-fibre morphism from a compact IHS Kahler manifold of dimension $2n$ onto a normal projective $n$-dimensional $B$, with every fibre component $n$-dimensional and isotropic for $\sigma$, then $B\cong\mathbb P^n$, with no smoothness assumption on $B$, no bound on $b_2$ and no restriction on deformation type. With Kim-Oguiso-Shinder it excludes codimension-one multiple fibres. Not shown: the case of a nonprojective $X$ (Kahler $X$ with a non-projective morphism).