# Examples of calculating perverse sheaves on algebraic varieties with easy stratification

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!

Geordie Williamson has a very nice set of notes on perverse sheaves: http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf it deals with some examples on curves (section 10).

You can also look at De Cataldo and Migliorini's paper (section 2.2): http://www.ams.org/journals/bull/2009-46-04/S0273-0979-09-01260-9/S0273-0979-09-01260-9.pdf

• Thank you so much for your answer! The examples included in the paper are precisely the ones I hope to see and calculate by myself. Thanks! – David Lucien Mar 4 '16 at 20:54

I would suggest you two papers

Gudiel Rodríguez, Félix; Narváez Macarro, Luis: Explicit models for perverse sheaves. Rev. Mat. Iberoamericana 19 (2003), no. 2, 425–454.

This deals with the two-strata case. And

Gudiel-Rodríguez, F.; Narváez-Macarro, L.: Explicit models for perverse sheaves. II. Algebr. Represent. Theory 11 (2008), no. 2, 149–178.

This treats the case of an arbitrary stratification. The techniques are more complicated but it extends the basic ideas of the previous paper.

Things are easiest when the automorphism group of $M$ (with its stratification) acts with finitely many orbits on $T^* M$: see Perverse sheaves on Grassmannians, by Tom Braden.

• Thanks for the answer. I will take a careful look at the paper. – David Lucien Sep 2 '16 at 6:32