Derived categories of toric varieties.

*(English)*Zbl 1159.14026In this paper, the author investigates the structure of the derived category of a toric variety. If \(X\) is a projective toric variety with at most quotient singularities, and \(B\) is an invariant \(\mathbb Q\)-divisor with coefficients of the form \((r-1)/r\) with \(r\) a nonnegative integer, then one can consider the smooth Deligne-Mumford stack \(\mathcal X\) associated to the pair \((X,B)\), as in Y. Kawamata [J. Math. Sci. Univ. Tokyo 12, 211–231 (2005; Zbl 1095.14014)]. The author shows that the derived category \(D^b({\mathcal X})\) has a complete exceptional collection consisting of sheaves.

In order to prove the result, the author considers a projective space, which is known to have a complete exceptional collection. Using the toric minimal program, he then constructs a complete exceptional collection on any toric variety with at most quotient singularities. Indeed, by a covering trick he proceeds from projective spaces to log Fano varieties. It is now enough to work out Mori fiber spaces: the presence of multiple fibres shows that boundary cases have to be taken into account, and this introduces the branch divisor \(B\) into the picture. Even if a Mori fiber space can have singular fibers, the associated morphism of stacks is smooth. Then a careful study of “stacky” sheaves gives the required collection (which consists indeed of sheaves) on any toric variety with at most quotient singularities.

In order to prove the result, the author considers a projective space, which is known to have a complete exceptional collection. Using the toric minimal program, he then constructs a complete exceptional collection on any toric variety with at most quotient singularities. Indeed, by a covering trick he proceeds from projective spaces to log Fano varieties. It is now enough to work out Mori fiber spaces: the presence of multiple fibres shows that boundary cases have to be taken into account, and this introduces the branch divisor \(B\) into the picture. Even if a Mori fiber space can have singular fibers, the associated morphism of stacks is smooth. Then a careful study of “stacky” sheaves gives the required collection (which consists indeed of sheaves) on any toric variety with at most quotient singularities.

Reviewer: Marcello Bernardara (Bonn)

##### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |

18E30 | Derived categories, triangulated categories (MSC2010) |

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