351 research outputs found
Affine hom-complexes
For two general polytopal complexes the set of face-wise affine maps between
them is shown to be a polytopal complex in an algorithmic way. The resulting
algorithm for the affine hom-complex is analyzed in detail. There is also a
natural tensor product of polytopal complexes, which is the left adjoint
functor for Hom. This extends the corresponding facts from single polytopes,
systematic study of which was initiated in [6,12]. Explicit examples of
computations of the resulting structures are included. In the special case of
simplicial complexes, the affine hom-complex is a functorial subcomplex of
Kozlov's combinatorial hom-complex [14], which generalizes Lovasz' well-known
construction [15] for graphs.Comment: final version, to appear in Portugaliae Mathematic
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 be a proper toric
variety of dimension 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 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 . This equivalence is monoidal---it intertwines the
tensor product of coherent sheaves on 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
- …