A remark on minimal Fano threefolds

TL;DR

This paper proves Dubrovin's conjecture for certain minimal Fano threefolds, linking algebraic K-theory pairings with geometric intersection pairings via modularity results.

Contribution

It confirms Dubrovin's conjecture for specific minimal Fano threefolds, expanding the class of varieties where the conjecture is verified.

Findings

Conjecture holds for V_{22} and V_5 threefolds.

Confirmed cases include P^3 and three-dimensional quadric.

Utilized modularity results for rank 1 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 $K_0$ 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 $V_{22}$, a minimal Fano threefold of anticanonical degree~22, and for $V_5$, the minimal Fano threefold of anticanonical degree~40, by applying the modularity result for rank 1 Fano threefolds. The truth of the conjecture for $\P ^3$ and the three--dimensional quadric is known; we consider these cases for the sake of completeness.