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I am trying to read "Tilting exercises" and have trouble to see any geometric pictures behind the formulas.

So my questions are, how to think about tilting perverse sheaves?

Are they just formal gadgets or have they a "geometric meaning"? Can one draw a pictiure of a tilting sheaf? etc.

A closely related question is the connection between tilting sheaves and glueing. Let $U$ be an open subset of an algebraic variety $X$ and $i:Z\rightarrow X$ its complement. Let $M$ be a perverse sheaf on $U$ with tilting extension $M^{tilt}$ on $X$. Define

$\Psi:=i^! M^{tilt}$ and $\Psi':=i^* M^{tilt}$ and $\tau: \Psi\rightarrow \Psi' $ the map induced from $i_* i^! \rightarrow id \rightarrow i_* i^*$

Then BBM show (Prop 1.2) that the category of extensions of of $M$ to $X$ is equivalent to the category of diagrams $\Psi \rightarrow \Phi \rightarrow \Psi'$ such that the composition equals $\mathcal \tau$.

How should one think of $\Psi$ and $\Psi'$? What would be a good name for $\Psi$ and $\Psi'$? What is the geometric picture behind this glueing construction?

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I'm not sure about a geometric picture. It might be helpful to play with tiliting objects in the category O and the maximal extension from Beilinson's gluing paper. –  Moosbrugger Nov 30 '11 at 12:56
    
Tilting modules (in the derived categories of quiver representations) can sometimes be displayed using the Auslander-Reiten quiver. This is the case in particular for coeherent sheaves on P¹. See for examples images at the end of math.jussieu.fr/~keller/publ/dct.pdf –  F. C. Nov 30 '11 at 20:41
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