A line bundle is ⊗-generating if its powers generate the derived category of perfect complexes. Refining the derived Nakai–Moishezon criterion of Ito–Olander, we characterize ⊗-generating line bundles on proper toric varieties by a combinatorial criterion on the fan, leading to the notion of ⊗-generating fans characterizing ⊗-generating canonical bundles. Furthermore, we prove an analogue of Kleiman's criterion of ampleness for ⊗-generation. As applications, we provide several examples including: (i) smooth proper fourfolds with no ⊗-generating line bundle, by showing that products of two or more positive-dimensional proper toric varieties admit a ⊗-generating line bundle only when they are projective, (ii) families of smooth non-projective proper threefolds with ⊗-generating canonical bundles. Finally, we study subvarieties obstructing ⊗-generation of canonical bundles and implications to autoequivalences of derived categories. Status: preprint. Tags: combinatorics, algebraic geometry, toric varieties, derived categories.
2026 · preprint
A line bundle is ⊗-generating if its powers generate the derived category of perfect complexes. Refining the derived Nakai–Moishezon criterion of Ito–Olander, we characterize ⊗-generating line bundles on proper toric varieties by a combinatorial criterion on the fan, leading to the notion of ⊗-generating fans characterizing ⊗-generating canonical bundles. Furthermore, we prove an analogue of Kleiman's criterion of ampleness for ⊗-generation. As applications, we provide several examples including: (i) smooth proper fourfolds with no ⊗-generating line bundle, by showing that products of two or more positive-dimensional proper toric varieties admit a ⊗-generating line bundle only when they are projective, (ii) families of smooth non-projective proper threefolds with ⊗-generating canonical bundles. Finally, we study subvarieties obstructing ⊗-generation of canonical bundles and implications to autoequivalences of derived categories.