Special Lagrangian Fibrations II: Geometry
Abstract
We continue the study of the StromingerYauZaslow mirror symmetry conjecture. Roughly put, this states that if two CalabiYau 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 StromingerYauZaslow 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 1998
 DOI:
 10.48550/arXiv.math/9809072
 arXiv:
 arXiv:math/9809072
 Bibcode:
 1998math......9072G
 Keywords:

 Algebraic Geometry;
 Differential Geometry;
 14J32
 EPrint:
 72 Pages