Resolutions of toric subvarieties by line bundles and applications

Abstract

Given any toric subvariety $Y$ of a smooth toric variety $X$ of codimension $k$, we construct a length $k$ resolution of ${\mathcal{O}}_Y$ by line bundles on $X$. Furthermore, these line bundles can all be chosen to be direct summands of the pushforward of ${\mathcal{O}}_X$ under the map of toric Frobenius. The resolutions are built from a stratification of a real torus that was introduced by Bondal and plays a role in homological mirror symmetry. As a corollary, we obtain a virtual analogue of Hilbert's syzygy theorem for smooth projective toric varieties conjectured by Berkesch, Erman, and Smith. Additionally, we prove that the Rouquier dimension of the bounded derived category of coherent sheaves on a toric variety is equal to the dimension of the variety, settling a conjecture of Orlov for these examples. We also prove Bondal's claim that the pushforward of the structure sheaf under toric Frobenius generates the derived category of a smooth toric variety and formulate a refinement of Uehara's conjecture that this remains true for arbitrary line bundles.

arXiv comment64 pages, 15 figures. Some minor changes following referee report. Version accepted to Forum of Mathematics, Pi

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Cited by in corpus (2)

• A Resolution of the Diagonal for Toric Deligne-Mumford Stacks
• Explicit constructions of short virtual resolutions of truncations