Intersection Homology
This is a reading seminar organized by professors of Algebraic depertment. In this seminar we will learn about Intersection homology and Perverse Sheaves from the book of Kirwan and Wolf.
1. Overview Talk
12/1
Overview talk by Prof. Suresh Nayak
Notes taken by me (Latexed) - Overview Talk .
2. Review Talk
19/1
Prognadipto Majumdar and Eeshan Pandey
Notes by me (Latexed)- Prerequisites.
In this lecture we will cover some prerequisite needed for this seminar like, review of singular, simplicial, Borel-Moore (co)homology etc and then Sheaf theory, sheaf cohomology (both from resolution and Derived functor point of view), cech cohomology, their relations (in some case).3. Pseudomanifold and Intersection Homology
02/02
Soumya Dasgupta
Notes by Soumya - Lecture-3.
Introduction to stratified spaces, Psudomanifold, Perversity and `Intersection Homology' with examples. Heading towards resolving Poincare duality - Normalization of topological spaces.4. First Properties of I.H. and I.H for Quasi projective Varites
09/02
Trishan Mondal
Notes by me - Lecture 4.
In this talk, we will discuss the homological properties of intersection homology like pushforward maps, excision and Mayer-Vietoris. We will compute the intersection homology of cones. We then discuss Whitney stratifications for complex quasi-projective varieties and the associated pseudomanifold structure on their underlying topological space. We will conclude with a discussion of Poincaré duality, Lefschetz hyperplane and hard Lefschetz theorems in the context of intersection homology.5. \(L^2\)- Cohomology and Intersection Cohomology.
16/02
Aaratrick Basu
Notes by Aaratrick - Lecture 5.
We introduce \(L^2\)-cohomology of smooth manifolds with riemannian metric, which is closely related to de Rham cohomology. We will then discuss strong Hodge theorem and a conjecture of Cheeger about intersection cohomology and \(L^2\)-cohomology. In the case of complex projective varieties with simple singularities, we show that \(L^2\)-cohomology does coincide with its intersection cohomology. If time permits, we will discuss the relation between the \(L^2\)-cohomology of a locally symmetric space and the intersection cohomology of its Baily-Borel compactification.6. Sheaf theoretic Intersection homology
24/02
Jishnu Biswas
In this talk, we sheafify the construction of the intersection homology we have seen so far by showing that it can be computed as the homology group of a certain complex of sheaves. As a consequence, it is shown that intersection homology is a topological invariant of a pseudomanifold, i.e, it does not depend on the choice of a stratification. This is done using Deligne's construction which given any stratification associates to a complex of sheaves, a new complex of sheaves. This construction is then applied to the complex computing intersection homology for the canonical (coarsest) stratification to derive the independence of stratification.
7. Continuation
01/03
Jishnu Biswas,Trishan Mondal
In this talk we will first look into The Kähler package, then we will get back proving the fact Intersection homology sheaf is invariant of stratification.
8. Perverse sheaves
08/03
Animesh Renanse
Notes by Animesh - Lecture 5.
Let X be an n-pseudomanifold. In the last two talks, we constructed the simplicial intersection complex of sheaves over X and showed that its hypercohomology is the intersection homology. Furthermore, it is completely characterized in the derived category by a list of axioms. Continuing with this, we will see that these axioms ensure that this complex is a perverse sheaf with a twist. We will see that the category of perverse sheaves enjoys special properties in the derived category; it is abelian and is closed under Verdier duality. Thinking of perverse sheaves as an enlargement of intersection homology with local coefficients, we will see that every perverse sheaf can be “approximated” by finitely many of them; that is, the category of perverse sheaves is artinian. We will end with some more remarks and examples from complex varieties.8. Nearby and Vanishing cycle
15/03
Raushon toor nair
We define nearby and vanishing cycles functors, associated with a function \(f: \mathbb{C}^n \to \mathbb{C}\), from the bounded derived category of constructible sheaves on \(\mathbb{C}^n\) to those on the special fiber of 0 of the function \(f\). We will then discuss the Beilinson-Bernstein-Deligne-Gabber decomposition theorem for intersection homology.
9,10. Weil's conjecture for the singular case
05/04, 12/04
Kannappan Sampath
We will introduce Weil's zeta function of a smooth projective variety over finite fields and Weil's conjectures about them. We will then explain how one could deduce these conjectures from a reasonable cohomology theory; we will then explain that such a cohomology theory with coefficients in \(\mathbb{Q}_p\) that is functorial for morphisms between smooth projective varieties cannot exist. We will then mention that \(\ell\)-adic etale cohomology (with ell not equal to p) is a Weil-cohomology theory. We will end by computing the "naive" zeta function of some explicit examples of singular varieties.
Main Text : F. Kirwan, J. Woolf, An introduction to intersection homology theory [CRC Press, 2006] - Pdf
- Web page : of Charanya Ravi related to this seminar.