A resolution of the diagonal for some toric D-M stacks

Abstract

Beilinson’s resolution of the diagonal for complex projective spaces gives a locally free resolution of the structure sheaf of the diagonal as a Koszul complex, which gives a Fourier-Mukai transform inducing the identity on objects in the derived category of coherent sheaves of P[superscript n]. Since P[superscript n] is a toric variety, we can ask for a resolution of the diagonal of an arbitrary toric variety. While resolutions of the diagonal are known for toric varieties which are unimodular in the sense of Bayer-Popescu-Sturmfels, in order to generalize a resolution of the diagonal for simplicial toric varieties, we must consider smooth Deligne-Mumford toric stacks associated to a given simplicial toric variety. Here, the diagonal can fail to be a closed substack because of non-zero stabilizers at the origin, so that a direct generalization of Beilinson’s resolution will not work. ... See PDF file for full abstract.