Mirror symmetry for Fano manifolds. Part I.

We describe mirror symmetry for Fano manifolds in terms of the Strominger--Yau--Zaslow conjecture and the homological mirror symmetry conjecture of Kontsevich. The mirror of a Fano n-fold is a holomorphic fibration of Calabi--Yau (n-1)-folds over the affine line with maximally unipotent monodromy at infinity. According to the homological mirror symmetry conjecture, the derived category of coherent sheaves on the Fano is equivalent to the Fukaya-Seidel category of the mirror. The Fukaya--Seidel category admits a semiorthogonal decomposition with factors associated to the singular fibers, which gives a mirror interpretation of the Kuznetsov decomposition of the derived category of the Fano. This conjecture was proved in dimension n=2 by Auroux-Katzarkov-Orlov https://arxiv.org/abs/math/0506166. The survey https://arxiv.org/abs/0902.1595 of Auroux explains the heuristic picture in arbitrary dimension; see especially Section 5. We describe what is known and expected in dimension n=3 and the relation to the work of Kuznetsov on Fano 3-folds, based on work of UMass PhD Cristian Rodriguez.