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Introduction to Intersection Homology Theory, Second Edition

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Introduction to Intersection Homology Theory, Second Edition INTERSECTION HOMOLOGY SIDDHARTH VENKATESH Abstract. These are notes for a talk given in the MIT Graduate Seminar on D-modules and Perverse Sheaves in Fall 2015. In this talk, I give an introduction to the intersection homology of topologically stratified spaces using the construction of singular and simplicial Borel-Moore intersection chains. I list several formal properties satisfied by intersection homology and then use these properties to compute some examples. Subsequently, I develop the sheaf-theoretic framework for intersection homology and give Deligne's construction of the minimal extension of a local system. Contents 1. Introduction 1 2. Simplicial and Singular Intersection Homology 2 3. Some Computations 4 4. Homology with Local Coefficients 6 5. Some Useful Properties of Intersection Homology 7 6. Sheaf-Theoretic Intersection Homology 8 References 10 1. Introduction Intersection homology was first introduced by Goresky and Macpherson to fix one big problem with the homology of singular spaces: the failure of Poincare duality. To fix this issue, they constructed a new homology theory for triangulated stratified spaces with mild singularities by only considering chains that intersected the singular strata in low enough dimension. They called this homology theory intersection homology and showed that it was equipped with a Poincare duality structure. As a fortunate accident, it turns out that intersection homology possesses much richer structure than just Poincare duality. For open embeddings, it is possible to define a relative intersection homology for which excision and a Mayer-Vietoris long exact sequence held. Additionally, while intersection homology is not homotopy invariant, it is is independent of stratification. These two results combined to make intersection homology a relatively easy to compute measure of the singularities of a space. Further structure can be found in the intersection homology of complex quasi-projective and projective varieties. In this setting, intersection homology, like regular homology, is equipped with the Lefschetz hyperplane theorem and the hard Lefschetz theorem and that the Hodge decomposition and Hodge signature theorem also holds. While the above properties show that intersection homology is an extremely interesting geometric invariant of complex algebraic varieties, the goal of this seminar is to study perverse sheaves as they appear in representation theory. The connection of this goal with intersection homology comes with the construction of a sheaf-theoretic version of intersection homology. Following a suggestion by Deligne, Goresky and Macpherson sheafified their definition of intersection homology to get a complex of sheaves, called the IC sheaf of the space, whose hypercohomology gave the intersection homology of the space. The cohomology sheaves of this complex satisfiy the support conditions which define the perverse t-structure (after applying a shift). In fact, these IC sheaves furnish all possible examples of 1 simple perverse sheaves: every simple perverse sheaf on a stratified complex variety is the IC sheaf of a closed stratum (with coefficients in a potentially nontrivial local system defined on the smooth part of the stratum.) In this talk, we will go through the formal properties that help in computing intersection homology and the IC sheaves that appear in representation theory. Emphasis will be on computational tools and examples and most of the technical details will be left to the references. In section 2, we give definitions and examples of simplicial and singular intersection homology. In section 3, we give an explicit description of the properties of intersection homology mentioned above and show how these properties can be used to compute examples. In section 4, we construct the IC sheaf via Goresky and Macpherson's sheafification process and via Deligne's construction. In section 5, we define the constructible derived category and the perverse t-structure and show that the IC sheaf is perverse. 2. Simplicial and Singular Intersection Homology The main reference for this section is [KW, Chapter 4]. All the definitions and computations can be found there. We begin with an inductive definition of a topologically stratified space. Definition 2.1. We say that paracompact X is a topologically stratified space of dimension m if there exists a filtration X = Xm ⊇ Xm−1 ⊇ · · · ⊇ X0 ⊇ X−1 = ; of X by closed subspaces Xi such that for all x 2 Xk − Xk−1 , there exists a neighborhood Nx of x in X, a compact topologically stratified space L = Lm−k−1 ⊇ · · · ⊇ L0 of dimension m − k − 1 called the link of x, and a homeomorphism k φ : Nx ! R × C(L) with C(L) the open cone on L such that k 1. φ takes Nx \ Xk+i+1 homoeomorphically onto R × C(Li) for 0 ≤ i ≤ m − k − 1 k 2. φ takes Nx \ Xk homeomorphically onto R × fvertex of the coneg: We call the connected components of the smooth, locally closed subspaces XinXi−1 the strata of X. Up to homeomorphism the link L of x depends only on the stratum that x is in. To define intersection homology, we demand an additional condition from our spaces. Definition 2.2. We say that X is a topological pseudomanifold of dimension m if it has a topological stratification with Xm−1 = Xm−2. Note that if X is a topological pseudomanifold and L is the link of x 2 XknXk−1, then the k homeomorphism φ takes R × C(Lm−k−2) onto Nx \ Xm−1 = Nx \ Xm−2, which is the image of k R × C(Lm−k−2). This shows for every point in a pseudomanifold X, the associated link is also a pseudomanifold. Hence, we can henceforth forget about stratified spaces that aren't pseudomanifold. Here are some examples of topological pseudomanifolds: Example 2.3. 1. The most obvious example is the trivial case of a manifold M. Here the stratification is M = Xm ⊇ ;: 2 2. The first nontrivial example would be a cone C(M) on a manifold M of dimension > 1. This is topologically stratified by C(M) ⊇ fvg where v is the vertex of the cone. 3. In similar vein, we can consider the wedge sum M W M of a manifold with itself (as long as M has dimension > 1.) 4. Eventually the main example we will be working with are complex quasi-projective varieties. These will be stratified by closed subvarieties and hence the stratification will be purely even. An important example is that of the affine cone over a (smooth) projective variety. Intersection homology is defined for topological pseudomanifolds using a function called the perversity that controls the chains under consideration. Definition 2.4. A perversity p is a function Z≥2 ! Z that satisfies p(2) = 0 and p(i + 1) = p(i) or p(i) + 1: There are four examples of perversities that are particularly important. 1. p(i) = 0 for all i. This is the bottom perversity. 2. p(i) = i − 2 for all i. This is the top perversity. i 3. p(i) = b 2 c − 1. This is the lower middle perversity. i 4. p(i) = d 2 e − 1. This is the upper middle perversity. We now define singular intersection homology as the homology of the complex whose chains are a suitable subset of the standard singular homology chains. Fix a perversity p. Definition 2.5. A singular i-simplex σ : ∆i ! X is said to be p-allowable, if −1 σ (Xm−k) ⊆ (i − k + p(k)) − skeleton of∆i: Similarly, a Borel-Moore singular i-chain is p-allowable if each simplex that appears with nonzero coefficient is allowable. p;BM Define I Si(X) to be the vector space over Q consisting of p-allowable Borel-Moore singular chains p with allowable boundary. Similarly, define I Si(X) to be the vector space over Q consisting of allowable singular chains with compact support with allowable boundary. Note that this does not mean that each simplex in the chain has allowable boundary since cancellation may occur. We define the intersection homology of X with respect to the perversity p (a priori dependent on the p;BM given stratification) as the homology of the complex I Si(X) and define the intersection homology p with compact supports to be the homology of I Si(X). Remark. The perversity function p should be seen as the distance the p-allowability condition is from transversality. More precisely, if p(k) = 0 for all k, then a singular simplex is p-allowable if it is transverse to each stratum. As in the case with ordinary homology, singular intersection homology is useful theoretically but difficult to compute with. So, we now develop a theory of simplicial intersection homology. For this, we need some extra structure on the stratification. Namely, we need a triangulation T on X such that each Xi is a union of simplices. We define p-allowability in exactly the same manner as in the singular case and then define the simplicial intersection homology and compactly supported homology as the limit of the homology of the analagous intersection complexes under refinement of the triangulation. 3 Remark. Unlike the case of ordinary homology, the homology of the simplicial intersection complex associated to a particular triangulation is not isomorphic to the simplicial intersection homology of X. But if T is flag-like, i.e., if the intersection of each simplex in T with Xi is a single face, then simplicial intersection homology can be directly computed using T . The simplicial intersection homology and singular intersection homology (and those with compact supports) of X are the same (for a fixed perversity). From now on, we only consider the middle perversity function p(i) = b and denote the corresponding homology as just intersection homology. We end this section by defining relative intersection homology.
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