Intersection Homology and Perverse Sheaves
Intersection Homology and Perverse Sheaves
In the fall semester of 2025, I am reading intersection homology and perverse sheaves following [1][2] under the guidance of Professor Maxim. Aiming to get a better understanding in the Riemann-Hilbert correspondence on a more topological view. With me there is also my friends Yanfei and Xingzhi.
The reason why I am reading this is because perverse sheaves are equivalent to the regular holonomic D-modules. The perverse t-structure, which don't make much sense to me. Has a clear topological picture from intersection homology.
Also, I have a small goal, which is to understand the description of simple objects, in D-modules, all simple objects comes from minimal extensions of simple connections, which for me enhances the information given by representation of fundamental group and pretty interesting.
We met Max weekly, asking about questions from reading. We plan to cover the intersection homology, Degline's sheaf and theory of IC sheaves. My personal goal is to 1. proof the finite length and simple objects in Perverse category(done) and 2. see the idea of BBGD decomposition theorem and calculate a good example of it.
Laurenţiu G. Maxim. Intersection homology & perverse sheaves-with applications to singularities, volume 281 of Graduate Texts in Mathematics. Springer, Cham, [2019] ©2019.
Frances Kirwan and Jonathan Woolf. An introduction to intersection homology theory. Chapman & Hall/CRC, Boca Raton, FL, second edition, 2006.
McCrory, C. (1975). Cone complexes and PL transversality. Transactions of the American Mathematical Society, 210, 183–200.
Goresky, M., & MacPherson, R. (1980). Intersection homology theory. Topology, 19(2), 135–162.
Goresky, M., & MacPherson, R. (1983). Intersection homology II. Inventiones Mathematicae, 71, 77–129.
Beilinson, Alexander A.; Bernstein, Joseph; Deligne, Pierre; Gabber, Ofer (1982). "Faisceaux pervers". Astérisque (in French). 100. Paris: Société Mathématique de France. MR 0751966.