The Fano–VB finiteness conjecture for Geigle–Lenzing projective spaces

Let $\mathbb{X}$ be a Geigle–Lenzing projective space and let $d$ be its dimension parameter. A $d$-tilting bundle on $\mathbb{X}$ is a tilting bundle whose endomorphism algebra has global dimension at most $d$; $\mathbb{X}$ is $d$-VB finite when its category of vector bundles has a $d$-cluster tilting subcategory with finitely many indecomposable objects up to the relevant shifts. The Fano–VB finiteness conjecture. The following conditions are equivalent:

• $\mathbb{X}$ is Fano;
• $\mathbb{X}$ is $d$-VB finite;
• $\mathbb{X}$ has a $d$-tilting bundle;
• $\mathbb{X}$ is derived equivalent to a $d$-representation-infinite algebra.

Several implications are established in the paper, while the equivalence of all four conditions is posed as a conjecture. In particular, the unresolved issue is whether the geometric Fano condition, finiteness, existence of a suitable tilting bundle, and the representation-theoretic derived description coincide in general.