Sheaves in combinatorics and discrete geometry

Question

I am looking for examples for the application of sheaves, sheaf-like constructions or the (co)homology of sheaves to problems in combinatorics and discrete geometry.

For example given a poset $(P,\leq)$ one can look at the topology given by declaring that the open sets are order filters $U \subseteq P$, i.e. if $x \in U$ and $x \leq y$ then $y \in U$. Now, any functor $\mathcal{F}$ from $P$ to some category, e.g. $\mathcal{F}:P \to \mathbf{Ab}$ to the category of abelian groups, gives a sheaf (e.g. of abelian groups) on the topological space described before. This is called a sheaf on $P$.

I am aware of the following applications of sheaves on posets:

• K. Baclawski used sheaves on posets in Whitney numbers of geometric lattices, in particular sheaf cohomology on posets to answer a question of G.-C. Rota: the Whitney numbers of the first kind of a geometric lattice are the Betti numbers of some homology theory on posets, namely the cohomology of suited sheaves on posets.

• S. Yuzvinsky used sheaf cohomology in Cohomology of local sheaves on arrangement lattices to give an interesting characterization of freeness of hyperplane arrangements. In the subsequent paper The first two obstructions to the freeness of arrangements he used this characterization to prove a conjecture of Falk and Randell that free arrangements are formal.

I suppose there should be plenty more examples and I am looking forward to your answers. Thanks!

Answer 1

One from mathematical physics.

The enummeration of plane partitions in the image of the moment map of a toric variety compute the Donaldson-Thomas theory of the toric variety $\mathcal{X}$ by identifying plane partitions with monomial ideals of the structural sheaf of $\mathcal{X}$, namely ideal sheaves of $\mathcal{X}$.

The procedure can be generalized by giving a planar bipartite graph and constructing with it the image of the moment map of a toric variety (see Quantum Calabi-Yau and Classical Crystals). In that context the enummeration of perfect matchings on the graph is the same problem as the ennumeration of plane partitions in the image of the moment map of the aforementioned toric variety.