D-Branes on Noncompact Calabi-Yau Manifolds: K-Theory and Monodromy

We study D-branes on smooth noncompact toric Calabi-Yau manifolds that are resolutions of abelian orbifold singularities. Such a space has a distinguished basis {S_i} for the compactly supported K-theory. Using local mirror symmetry we demonstrate that the S_i have simple transformation properties under monodromy; in particular, they are the objects that generate monodromy around the principal component of the discriminant locus. One of our examples, the toric resolution of C^3/(Z_2 x Z_2), is a three parameter model for which we are able to give an explicit solution of the GKZ system.Comment: 40 pp, substantial revision

A derived approach to geometric McKay correspondence in dimension three

We propose a three dimensional generalization of the geometric McKay correspondence described by Gonzales-Sprinberg and Verdier in dimension two. We work it out in detail in abelian case. More precisely, we show that the Bridgeland-King-Reid derived category equivalence induces a natural geometric correspondence between irreducible representations of G and subschemes of the exceptional set of G-Hilb. This correspondence appears to be related to Reid's recipe

Reid's recipe and derived categories

We prove two conjectures from Cautis and Logvinenko (2009) [CL09] which describe the geometrical McKay correspondence for a finite, abelian subgroup of SL3(C). We do it by studying the relation between the derived category mechanics of computing a certain Fourier–Mukai transform and a piece of toric combinatorics known as ‘Reid's recipe’, effectively providing a categorification of the latter

Derived McKay correspondence via pure-sheaf transforms

In most cases where it has been shown to exist the derived McKay correspondence D(Y)−→∼DG(Cn) can be written as a Fourier–Mukai transform which sends point sheaves of the crepant resolution Y to pure sheaves in DG(Cn) . We give a sufficient condition for E∈DG(Y×Cn) to be the defining object of such a transform. We use it to construct the first example of the derived McKay correspondence for a non-projective crepant resolution of C3/G . Along the way we extract more geometrical meaning out of the Intersection Theorem and learn to compute θ-stable families of G-constellations and their direct transforms