All complete intersection varieties are Fano visitors

Abstract

We prove that the derived category of a smooth complete intersection variety is equivalent to a full subcategory of the derived category of a smooth projective Fano variety. This enables us to define some new invariants of smooth projective varieties and raise many interesting questions.

Fano varieties of K3 type and IHS manifolds

“…More precisely the result is the following Theorem 2.3 (Thm. 2.10 in [Orl06], Thm 2.4 in [KKLL15]). Let q : E → U be a vector bundle of rank r ≥ 2 over a smooth projective variety U and let S = s −1 (0) ⊂ U denote the zero locus of a regular section s ∈ H…”

Fano 3-folds from homogeneous vector bundles over Grassmannians

“…Lemma 2.8 admits a classical generalisation in higher codimension, known as the Cayley trick (see [23,Thm 2.4], or [18, 3.7]), which in turn can be considered as a generalisation of Orlov's formula for the derived category of blow ups. The setup is the following: assume that we have Y = Z (A) ⊂ S, where A is ample of rank r ≥ 2.…”

Derived categories of curves as components of Fano manifolds

“…In fact, this conjecture would give a positive answer in dimension 1 to a general question asked by Alexei Bondal some time ago: whether for any algebraic variety X there is Fano variety Y and a fully faithful embedding D(X) → D(Y ) (see [1,6,7] for other results in this direction).…”