Tensor-generating line bundles on toric varieties

Abstract

A line bundle is $\otimes$-generating if its powers generate the derived category of perfect complexes. Refining the derived Nakai--Moishezon criterion of Ito--Olander, we characterize $\otimes$-generating line bundles on proper toric varieties by a combinatorial criterion on the fan, leading to the notion of $\otimes$-generating fans characterizing $\otimes$-generating canonical bundles. Furthermore, we prove an analogue of Kleiman's criterion of ampleness for $\otimes$-generation. As applications, we provide several examples including: (i) smooth proper fourfolds with no $\otimes$-generating line bundle, by showing that products of two or more positive-dimensional proper toric varieties admit a $\otimes$-generating line bundle only when they are projective, (ii) families of smooth non-projective proper threefolds with $\otimes$-generating canonical bundles. Finally, we study subvarieties obstructing $\otimes$-generation of canonical bundles and implications to autoequivalences of derived categories.