A Remark on Minimal Fano Threefolds

Abstract. We prove in the case of minimal Fano threefolds a conjecture stated by Dubrovin at the ICM 1998 in Berlin. The conjecture predicts that the symmetrized/alternated Euler characteristic pairing on K0 of a Fano variety with an exceptional collection expressed in the basis of the classes of the exceptional objects coincides with the intersection pairing of the vanishing cycles in Dubrovin’s second connection. We show that the conjecture holds for V22, a minimal Fano threefold of anticanonical degree 22, and for V5, the minimal Fano threefold of anticanonical degree 40, by applying the modularity result for the rank 1 Fano threefolds established in [Gol07]. The truth of the conjecture for P and the three–dimensional quadric is known; we consider these cases for the sake of completeness.