A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli's description of the K-theory of a smooth projective toric variety. Specifically, let $X$ be a proper toric variety of dimension $n$ and let M_\bR = \mathrm{Lie}(T_\bR^\vee)\cong \bR^n be the Lie algebra of the compact dual (real) torus T_\bR^\vee\cong U(1)^n. Then there is a corresponding conical Lagrangian \Lambda \subset T^*M_\bR and an equivalence of triangulated dg categories \Perf_T(X) \cong \Sh_{cc}(M_\bR;\Lambda), where \Perf_T(X) is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on $X$ and \Sh_{cc}(M_\bR;\Lambda) is the triangulated dg category of complex of sheaves on M_\bR with compactly supported, constructible cohomology whose singular support lies in $\Lambda$. This equivalence is monoidal---it intertwines the tensor product of coherent sheaves on $X$ with the convolution product of constructible sheaves on M_\bR.Comment: 20 pages. This is a strengthened version of the first half of arXiv:0811.1228v3, with new results; the second half becomes arXiv:0811.1228v

A Modified View on Octocorals: Heteroxenia fuscescens Nematocysts Are Diverse, Featuring Both an Ancestral and a Novel Type

Cnidarians are characterized by the presence of stinging cells containing nematocysts, a sophisticated injection system targeted mainly at prey-capture and defense. In the anthozoan subclass Octocorallia nematocytes have been considered to exist only in low numbers, to be small, and all of the ancestral atrichous-isorhiza type. This study, in contrast, revealed numerous nematocytes in the octocoral Heteroxenia fuscescens. The study demonstrates the applicability of cresyl-violet dye for differential staining and stimulating discharge of the nematocysts. In addition to the atrichous isorhiza-type of nematocysts, a novel type of macrobasic-mastigophore nematocysts was found, featuring a shaft, uniquely comprised of three loops and densely packed arrow-like spines. In contrast to the view that octocorals possess a single type of nematocyst, Heteroxenia fuscescens features two distinct types, indicating for the first time the diversification and complexity of nematocysts for Octocorallia