Homological mirror symmetry via tensor-triangulated geometry

Homological mirror symmetry (HMS), proposed by Kontsevich, conjectures a categorical equivalence between the Fukaya category of a symplectic manifold and the derived category of coherent sheaves on its mirror scheme. We recall the basic formulation of HMS together with several classical examples that illustrate this correspondence.

We then introduce a tensor-triangulated approach and present a necessary and sufficient condition for a Fukaya category to be realized as the perfect derived category of a Noetherian scheme. This provides a conceptual reconstruction of the mirror scheme from purely Fukaya-categorical data and leads to a natural construction of an A∞-functor. Finally, we explain how classical HMS examples can be recovered within this tensor-triangulated framework.