Special Lagrangian Fibrations II: Geometry

TL;DR

This paper advances the understanding of the Strominger-Yau-Zaslow mirror symmetry conjecture by analyzing special Lagrangian fibrations, their duals, and providing explicit evidence for K3 surfaces through differential geometric methods.

Contribution

It applies integrable systems theory to special Lagrangian fibrations, explores duality structures, and constructs mirror symmetry for K3 surfaces without Torelli theorems.

Findings

Established relations between cohomologies of dual fibrations.

Proposed a refined version of the SYZ conjecture.

Constructed mirror symmetry for K3 surfaces using differential geometry.

Abstract

We continue the study of the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, this states that if two Calabi-Yau manifolds X and Y are mirror partners, then X and Y have special Lagrangian torus fibrations which are dual to each other. Much work on this conjecture is necessarily of a speculative nature, as in dimension 3 it is still a very difficult problem of how to construct such fibrations. Nevertheless, assuming the existence of such fibrations there are many things one can prove. This paper covers a number of issues. First it applies results from the theory of completely integrable hamiltonian systems to understand some aspects of the geometry of such fibrations. From this, using reasonable regularity assumptions on the fibrations, one can understand how the cohomology of dual fibrations are related. We then study the question of how, given one such fibration, one would put a symplectic and complex structure on the dual fibrations, generalising work of Hitchin. While this question cannot be answered at this stage, these results should give insight into the nature of the problem. We sum up these ideas in a refined version of the Strominger-Yau-Zaslow conjecture. Finally, to give evidence for this conjecture, we prove it explicitly for K3 surfaces. One finds a construction of mirror symmetry for K3 surfaces which does not require the use of Torelli theorems, and is much more differential geometric in nature than previous constructions.