Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient

Abstract

For a finite subgroup G ⊂ SL(3, C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C 3 /G. This paper considers the moduli spaces M θ , introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G -Hilb for a particular choice of GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C 3 /G is isomorphic to M θ for some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier-Mukai transform. We also uncover explicit equivalences between the derived categories of moduli M θ for parameters lying in adjacent GIT chambers.

Derived category automorphisms from mirror symmetry

“…We make no attempt to relate this work to these developments, although there seem to be many interesting connections. We only mention the papers [42], [43], [17], [5], which are intimately related to the results of the present work and provide more applications than those presented in the last section of this paper.…”

D-BRANES ON Calabi–Yau MANIFOLDS

“…This gives a degree of ambiguity to how we may associate the derived category of sheaves to the derived category of quivers. We refer to [164,179] for further details on how to compute the correspondence exactly. Let ∆ m 0 m 1 m 2 denote a quiver representation of the form (283) and consider the fractional branes F 0 = ∆ 100 , F 1 = ∆ 010 and F 2 = ∆ 001 .…”

On G/N-Hilb of N-Hilb

“…Equivalently, when X = C n a G-equivariant coherent sheaf F on X is a representation of the McKay quiver Q satisfying the relations R. This identification was first stated in [IN,§3] and rewritten in the language of G-constellations in [CI,§2.1]. For an explicit description of the relations R see [BSW].…”