Question

The generalized homological mirror symmetry conjecture states that for mirror dual models $(X_E, w)$ and $(X_E', w')$ , if $L$, a lattice polytope which is a Newton polytope of a nonsingular projective toric variety, is in $M_{R}$ , then isomorphisms exist between: $D^b(X_E,w)$ and $DFS(X_E',w')$, $D^b(X_E',w')$ and $DFS(X_E,w)$, where $DFS$ is the derived Fukaya category and $D^b$ is derived category of coherent sheaves. What cases of this conjecture are open? Any references are greatly appreciated.

Answer 1

The conjecture has been solved for elliptic curves, abelian varieties, non-singular torus bundles over affine manifolds, and quartic surfaces. It remains to find a unification from algebraic geometry. A few references for this subject are the following:

Kontsevich, Maxim (1994), Homological algebra of mirror symmetry, arXiv:alg-geom/9411018. Kontsevich, Maxim; Soibelman, Yan (2000), Homological Mirror Symmetry and torus fibrations, arXiv:math.SG/0011041. Seidel, Paul (2003), Homological mirror symmetry for the quartic surface, arXiv:math.SG/0310414. Hausel, Tamas; Thaddeus, Michael (2002), Mirror symmetry, Langlands duality, and the Hitchin system, arXiv:math.DG/0205236