## Geometric intuition behind perverse coherent sheaves?

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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 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 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 at 20:50

## 1 Answer

This is the definition appear in Bridgeland's paper which shows that flops of smooth 3-folds induces equivalence of derived category of coherent sheaves. From your question I think you know the word "perverse" is kind of related to t-structures.

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.)

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Dear 36min, As you probably know, the adjective "perverse" is inherited from its usage in the usual theory of perverse sheaves, which served as a partial motivation for Bridgeland's work.  Regards, Matthew – Emerton Nov 27 at 16:56
I said in the first paragraph the word "perverse" is related to t-structures. – 36min Nov 28 at 2:00
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