For a finite subgroup $G$ of SL($3,\mathbb{C}$), the $G$-Hilbert scheme is a moduli space of certain $G$-invariant subschemes of ${\mathbb{C}}^3$. It is a crepant resolution of the quotient singularity ${\mathbb{C}}^3/G$, and the universal family induces an equivalence between the derived category of $G$-equivariant coherent sheaves on ${\mathbb{C}}^3$ and that of coherent sheaves on $G$-Hilb. It is conjectured by Cautis and Logvinenko that this derived equivalence sends the simple $G$-sheaves associated to nontrivial irreducible representations of $G$ to pure sheaves on $G$-Hilb. In the talk, I will show that this conjecture holds under a fairly mild assumption and explain its relationship with the wall-crossing phenomena. This  is a joint work with Alastair Craw.

 

Organizer

 

Hsueh-Yung Lin (NTU)