Examples of calculating perverse sheaves on algebraic varieties with easy stratification. This question is also asked in mathoverflow https://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book 
http://www.amazon.com/Introduction-Intersection-Homology-Theory-Edition/dp/1584881844.
Then, I read chapter $8$ of the book http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf which introduced the theory of perverse sheaves using the language of $t-$ structure. After reading the abstract construction of the category of perverse sheaves for an algebraic variety or analytic space, I hope to see examples of calculating the perverse sheaves for spaces with easy stratification, such as $\mathbb{CP}^n$. Could anyone please provide some interesting examples? Thanks!
 A: I remember reading this example somewhere but I cannot find the source now:
Consider $\mathbb{P}^1_{\mathbb{C}}$ the projective line and the stratification $\mathbb{A}^1, \{\infty\}$. Now what are the irreducible perverse sheaves here?
Well to get an irreducible perverse sheaf you must choose one of your strata and then a local system there. In this case, local systems both on the point and on the affine line are trivial (since the fundamental group is trivial), so you only have the two irreducible perverse sheaves $IC_{\mathbb{A}^1}, \delta_{\{\infty\}}$ where the latter is the scyscraper sheaf.
Now if we were in a category like the representations of finite groups, where objects decompose in irreducibles, you would have basically described the whole category. But there are indecomposable non-irreducible perverse sheaves, so a total description should also find them:
For starters, the obvious thing to do now is to play around with pushfowards. Indeed, consider the distinguished triangle, for $i: \{\infty \}\rightarrow \mathbb{P}^1$ and $j:\mathbb{A}^1\rightarrow  \mathbb{P}^1$ the obvious inclusions, and apply it to $j_* IC_{\mathbb{A}^1}.$
$$ 0\rightarrow j_{!}j^* j_* IC_{\mathbb{A}^1}\rightarrow j_* IC_{\mathbb{A}^1}\rightarrow i_*i^*j_* IC_{\mathbb{A}^1}\rightarrow 0$$
Notice that for the last term, one may naively think that $i^*j_* IC_{\mathbb{A}^1}=0$. This is not true if you remember more carefully the definition of the inverse image: You have to look at the neighborhoods around the image, which will be nonempty for every neighborhood, and indeed $i^*j_* IC_{\mathbb{A}^1}$ is the trivial sheaf. Then the pushfoward will give you $\delta_{\infty}.$
For the first term,$j^*j_*$ basically acts trivially, and then notice that $j_!$ will give you exactly the middle extension, so the first term is $IC_{\mathbb{P}^1}.$
So we get
$$ 0\rightarrow IC_{\mathbb{P}^1}\rightarrow j_* IC_{\mathbb{A}^1}\rightarrow \delta_{\infty}\rightarrow 0.$$
The middle sheaf is indecomposable (why?) so you found one of your indecomposable sheaves.
In a similar fashion, you can show $j_! IC_{\mathbb{A}^1}$ is also an indecomposable sheaf, and it satisfies the dual triangle.
Now notice that perverse sheaves have enough projectives. What is the projective cover of $\delta_{\infty}?$ If you try to identify it with a direct sum of the ones we have already found you will see that it is impossible. Indeed, this projective cover, let it be $P_{\infty},$ is also indecomposable, and these are all your indecomposable sheaves. 
By the way, you may want to try playing around with $P_{\infty},$ it is a nice exercise.
Also it is nice to work out the Riemann-Hilbert correspondance in this case.
