WEEK 1: THE SYZ CONJECTURE AND COLLAPSING CALABI-YAU MANIFOLDS
The Strominger-Yau-Zaslow (SYZ) conjecture dates from 1996. It was proposed as a geometric mechanism underlying mirror symmetry for Calabi-Yau manifolds. The proposal is that, at least near the “large complex structure” limit in moduli space, a Calabi-Yau manifold has a fibration whose generic fibers are Special Lagrangian tori. The mirror manifold should then be obtained by taking the duals of the torus fibers.
The existence of such Special Lagrangian fibrations, and particularly the understanding of the singular fibers is a difficult problem, and many of the prominent developments around the SYZ conjecture over the past quarter century involve related algebro-geometric and symplectic-topological constructions. However there have also been great advances and activity in complex differential geometry and in the study of Special Lagrangian submanifolds, and calibrated submanifolds more generally. There are also new connections with non-Archimedean geometry.
One substantial recent advance towards the SYZ conjecture came in the work of Yang Li, who established (under reasonable hypotheses) the existence of a Special Lagrangian fibration outside a set of very small volume. In metric terms, working near the large complex structure limit means that fibers of the conjectured fibration are very small compared with the size of the base. This fits into a wider discussion of “collapsing” in Calabi-Yau geometry (and Riemannian geometry more generally). There have been many other recent developments in this area, for example the classification by Sun and Zhang of collapsed Gromov-Hausdorff limits of Calabi-Yau metrics on K3 surfaces and descriptions of metrics as the complex structure degenerates, involving multiscale collapsing. These are very relevant to questions of moduli space compactification.
The aim of this week of the SCGP summer programme will be to involve both experts in these differential geometry and geometric analysis developments and experts in relevant aspects of neighboring fields such as algebraic geometry and symplectic topology. This week will include lecture courses by Yang Li and Song Sun, and talks by Simon Donaldson and Ruobing Zhang.