Maximal lengths of exceptional collections of line bundles

Abstract

In this paper, we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant c>34 there exist infinitely many toric Fano varieties Y with Picard number three, such that the maximal length of exceptional collection of line bundles on Y is strictly less than crkK0(Y). To obtain varieties without full exceptional collections of line bundles, it suffices to put c=1. On the other hand, we prove that for any toric nef‐Fano DM stack Y with Picard number three, there exists a strong exceptional collection of line bundles on Y of length at least 34rkK0(Y). The constant 34 is thus maximal with this property.

Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties

“…Although there is always a full strong exceptional collection of thimbles, Conjecture 4.12 would not guarantee that D b Coh(X) admits a full strong exceptional collection of line bundles. In fact, it is known by work of Efimov [17] that even Fano toric varieties do not admit such an exceptional collection in general. A necessary condition for Conjecture 4.12 to produce an exceptional collection of line bundles is that the monodromy action must be transitive on critical points of W Σ .…”

Tilting bundles on toric Fano fourfolds

“…It is important to note that the existence of full strong exceptional collections of line bundles is rare; Hille-Perling [HP06] constructed smooth toric surfaces that do not have such collections. Even when only considering smooth toric Fano varieties, there exist examples in dimensions ≥ 419 that do not have full strong exceptional collections of line bundles, as demonstrated by Efimov [Efi10].…”

On Landau–Ginzburg systems, quivers and monodromy

“…The abundance of examples led experts to ask whether, in fact, any toric Fano manifold admits a full strongly exceptional collection of line bundles in P ic(X), see [12,17]. However, in a recent surprising work [20] Efimov discovered examples of toric Fano manifolds which do not admit any full strongly exceptional collections of line bundles. In particular, the question of which toric Fano manifolds admit full strongly exceptional collections in P ic(X) is currently still open.…”