Tomasz Pelka, Institute of Math of the Polish Academy of Sciences: Lagrangian tori at radius zero

Tomasz Pelka, Institute of Mathematics of the Polish Academy of Sciences: Lagrangian tori at radius zero
We will present a new technique to construct Lagrangian tori in degenerations of Kahler manifolds. For maximally degenerate families of Calabi-Yaus, these tori have some asymptotic properties expected from the SYZ picture: they fill almost all volume, collapse in the Gromov-Haudsdorff metric to the essential skeleton minus codimension 2 faces, etc. They naturally occur at the boundary of the A'Campo space, which extends a given degeneration from a punctured disk to the annulus. We will explain the construction of the A'Campo space and its hybrid coordinates. Using these coordinates, the proof of the above properties reduces to elementary computations at the boundary. The key part is the definition of a fiberwise Kahler form, which is asymptotically Ricci flat in the generic region, and allows us to move the tori from the boundary to the nearby fibers along its symplectic connection.