Symplectic Geometry Seminar

Samuel Sottille (Stanford University)

Title: Almost Toric Fibrations in Higher Dimensions

Abstract:

The SYZ conjecture posits that special Lagrangian fibrations provide a geometric explanation for mirror symmetry. On the purely symplectic side, non-special Lagrangian torus fibrations provide a useful framework for understanding the geometry of symplectic manifolds. However, there is no general theory of these fibrations in dimensions higher than four. In this talk, I will present my results on constructing Lagrangian torus fibrations in higher dimensions. I will show that a wide class of manifolds with toric degenerations admit almost toric fibrations. I will show that Gross and Siebert's combinatorial duality for toric degenerations can be realized as a duality of these almost toric fibrations. I will construct almost toric fibrations on all three-dimensional Fano varieties. Time permitting, I will present applications to computing disk potentials of monotone torus fibers.