I would like to know an intuition behind perverse coherent sheaves. I am aware that it is induced by a heart of another t-structure on the derived category. Are there any better, probably more geometric, way to understand perverse coherent sheaves?

Just in case, let us recall the definition of perverse coherent sheaves. Let $X$ be a projective threefold with at worst Gorenstein terminal singularities and $f:Y\rightarrow X$ be a crepant resolution. Define a full subcategory $\mathrm{Per}(Y/X) \subset \mathrm{D}(Y)$ consisting of objects $E \in \mathrm{D}(Y)$ satisfying the following three conditions;

- $H^i(E)=0$ unless $i=0,-1$,
- $R^1f_*H^0(E)=0$ and $R^0f_∗H^{−1}(E)=0$,
- $Hom_Y(H^0(E),C)=0$ for any sheaf $C$ on $Y$ satisfying $Rf_∗(C)=0$.

We call the objects of $\mathrm{Per}(Y/X)$ perverse coherent sheaves.

coherentsheaves, not usual perverse sheaves. Incdidentally, I think thereisgood geometric intuition for usual perverse sheaves (though I am not the right one to convey it). Regards, Matthew $\endgroup$ – Emerton Nov 27 '12 at 16:55