# Geometric intuition behind perverse coherent sheaves?

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;

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

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

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Your definition is far too specific. I don't think there's a really good geometric intuition. They're the natural objects to look at on singular spaces which imitate good "cohomological properties" of smooth spaces (e.g. duality theorems, purity of etale cohomology, etc.). –  David Hansen Nov 26 '12 at 23:36
Dear David, The OP is asking about perverse coherent sheaves, not usual perverse sheaves. Incdidentally, I think there is good geometric intuition for usual perverse sheaves (though I am not the right one to convey it). Regards, Matthew –  Emerton Nov 27 '12 at 16:55
Dear Matthew: Oops, I didn't realize these were distinct concepts! Thanks for the correction. Best, Dave --- Pooya: Sorry for the tone of my comment. :) –  David Hansen Nov 28 '12 at 20:50

The main theorem of that paper, indicates that for a flop $Y\to X\leftarrow W$, $Per(Y/X)$ will be send to $Coh(W)$ under that isomorphism. In other words, these objects are sheaves on another scheme which you can construct from the data $Y\to X$! In my opinion that's pretty cool, not "perverse" at all. But as for the name, so be it.
BTW, you don't need $X$ and $Y$ to be three fold in the definition. If you check Bridgeland's paper, most of the time he work with birational morphism such that $Rf_*\mathcal{O}_Y=\mathcal{O}_X$ and fibers have dimension at most 1. (For 3-folds that's just a small resolution.)