Minimal Fano threefolds: Exceptional sets and vanishing cycles

The (skew)symmetrized bilinear form of Euler characteristic of a Fano manifold calculated for classes of exceptional objects coincides with the pairing of vanishing cycles of the Dubrovin second structure connection. An algebraic variety and the bonded derivative category of coherent sheaves are considered, while the derivative categories are triangulated. Results show that the minimal Fano three-fold of anticanonical degree 22 satisfies the exceptional sets and vanishing cycles conjecture. As a result the monodromy of the regularized quantum D-module for the minimal Fano three-fold of anticanonical degree 22 is described. The minimal Fano threefold of anticanonical degree 40 satisfies the exceptional sets and vanishing cycles conjecture. The results of this proof shows describe the monodromies around certain points are reflections and the monodromy operators are orthogonal.