Geometric intuition behind perverse coherent sheaves? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T05:48:29Z http://mathoverflow.net/feeds/question/114601 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114601/geometric-intuition-behind-perverse-coherent-sheaves Geometric intuition behind perverse coherent sheaves? Pooya 2012-11-26T22:56:17Z 2012-11-27T07:36:12Z <p>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? </p> <p>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;</p> <ol> <li>$H^i(E)=0$ unless $i=0,-1$,</li> <li>$R^1f_*H^0(E)=0$ and $R^0f_∗H^{−1}(E)=0$,</li> <li>$Hom_Y(H^0(E),C)=0$ for any sheaf $C$ on $Y$ satisfying $Rf_∗(C)=0$.</li> </ol> <p>We call the objects of $\mathrm{Per}(Y/X)$ perverse coherent sheaves.</p> http://mathoverflow.net/questions/114601/geometric-intuition-behind-perverse-coherent-sheaves/114637#114637 Answer by 36min for Geometric intuition behind perverse coherent sheaves? 36min 2012-11-27T07:36:12Z 2012-11-27T07:36:12Z <p>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. </p> <p>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.</p> <p>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.)</p>