SYZ conjecture

The SYZ conjecture provides a geometric framework for understanding homological mirror symmetry by establishing a correspondence between Calabi-Yau manifolds through toroidal fibrations. Each point on the base space B corresponds to a torus in the manifold X, and its mirror X̂ is constructed using line bundles over these tori. Singularities on B play a crucial role in this relationship, affecting the geometry of both manifolds.

Homological mirror symmetry connects the Fukaya category of Lagrangian submanifolds in X with the derived category of coherent sheaves on X̂. Simple geometric objects like sections become algebraic structures such as line bundles and skyscraper sheaves. Complex objects like multisections correspond to higher-rank sheaves, requiring attention to details like instanton corrections counted via Gromov-Witten theory.

The conjecture bridges geometry and algebra, showing that dual manifolds have concrete geometric correspondences beyond abstract duality. It's akin to a Fourier transform, decomposing geometric data into tori and reinterpreting them algebraically, revealing deep symmetries. Understanding the singularities in B is key to constructing these equivalences, highlighting the interconnectedness of different mathematical domains.

In summary, SYZ conjecture offers a tangible geometric bridge between geometry and algebra, illustrating how seemingly unrelated areas can be deeply connected through advanced mathematical structures.