Speaker
Alastair Craw
(U. of Bath)
Description
For a finite subgroup $G$ of $\operatorname{SL}(3,\mathbb{C})$, the
$G$-Hilbert scheme is a crepant resolution of the quotient singularity
$\mathbb{C}^3/G$, and the universal family induces a derived equivalence
between coherent sheaves on $G$-Hilb and $G$-equivariant coherent sheaves
on $\mathbb{C}^3$. In 2009, Cautis and Logvinenko conjectured that this
equivalence sends the simple $G$-sheaves, one for each nontrivial irreducible
representation of $G$, to pure sheaves on $G$-Hilb. I will describe how the
recent proof of this conjecture with Lin and Yamagishi generalises naturally
to all 3-fold noncommutative crepant resolutions for which the vertex simple
modules exist.