Fibrations by Lagrangian Tori for Maximal Calabi–Yau Degenerations and Beyond

ABSTRACT We introduce a general technique to construct Lagrangian torus fibrations in degenerations of Kähler manifolds. We show that such torus fibrations naturally occur at the boundary of the A'Campo space . This space extends a degeneration over a punctured disc to a hybrid space involving its real oriented blowup and the dual complex of the central fiber; equipped with an exact modification of a given Kähler form which becomes fiberwise symplectic at the boundary. Therefore, it allows to move Lagrangian tori from the “radius zero” fibers at the boundary to the nearby fibers of the original degeneration using the symplectic connection. As a consequence, for any maximal Calabi–Yau degeneration we prove that a generic region of each nearby fiber admits a Lagrangian torus fibration with asymptotic properties expected from the (conjectural) Strominger–Yau–Zaslow fibration.