The strong hyperkahler SYZ conjecture

Theorem 1.1: if $c_1(L)\ne0$, $c_1(L)$ is in the closure of the Kahler cone and $q_X(c_1(L))=0$, then $L$ is semiample: some $L^m\simeq f^*A$ for a surjective connected-fibre map $f:X\to B$ onto a normal projective $n$-dimensional $B$ with $A$ ample, every fibre component Lagrangian of dimension $n$. Corollary 1.2: in each dimension there are finitely many deformation types with $b_2\ge5$. Corollary 1.3 (projective $X$, via the companion): $B\simeq\mathbb P^n$. Not shown: existence of a nef isotropic class on a given $X$, or anything for $b_2\le4$ in the finiteness corollary.

Theorem 1.1: if $c_1(L)\ne0$, $c_1(L)$ is in the closure of the Kahler cone and $q_X(c_1(L))=0$, then $L$ is semiample: some $L^m\simeq f^*A$ for a surjective connected-fibre map $f:X\to B$ onto a normal projective $n$-dimensional $B$ with $A$ ample, every fibre component Lagrangian of dimension $n$. Corollary 1.2: in each dimension there are finitely many deformation types with $b_2\ge5$. Corollary 1.3 (projective $X$, via the companion): $B\simeq\mathbb P^n$. Not shown: existence of a nef isotropic class on a given $X$, or anything for $b_2\le4$ in the finiteness corollary.