Remarks on the nonequivariant coherent-constructible correspondence for toric varieties

We prove the following result of Bondal's: that there is a fully faithful embedding $\kappa$ of the perfect derived category of a proper toric variety into the derived category of constructible sheaves on a compact torus. We compare this result to a torus-equivariant version considered in joint work with Fang, Liu, and Zaslow. There we showed that in the torus-equivariant version the image of the embedding is cut out by microlocal conditions. To establish a similar characterization of the image of $\kappa$ is an open problem.