Speaker
Helge Ruddat, University of Stavanger
Abstract
We prove the existence of Lagrangian torus fibrations on compact Calabi-Yau hypersurfaces in toric Fano manifolds associated with reflexive polytopes. The base is diffeomorphic to a sphere, the discriminant in the base has real codimension two, all singular fibers are half-dimensional Lagrangian skeleta, the critical set has real codimension four in the hypersurface, and the fibration is a smooth torus bundle away from the discriminant. By the Arnold-Liouville theorem, the regular part is identified with the cotangent bundle of the base equipped with its natural integral affine structure.
This work realizes a key prediction of the Strominger-Yau-Zaslow conjecture, making mirror symmetry act geometrically as T-duality near large complex structure limits. The proof uses local models of mixed type built inductively via Liouville flows, following an idea of Evans and Mauri. As a byproduct we obtain two Lagrangian sections whose difference generates the symplectic monodromy of certain one-parameter families determined by a divisor on the mirror dual whose holomorphic Euler number matches the intersection number of the Lagrangians. This is joint work with Cheuk Yu Mak, Diego Matessi, and Ilia Zharkov.