Results on intersection cohomology

Lemma 6.5 ([10, Proposition 1.4],[31, Proposition 3]). Let G he a finite group acting on a nonsingular irreducible variety X. Then IC X/G) = Qx/G[diniX]. 

This lemma is equivalent to saying that X/G is a rational homology manifold. It holds because we have the Poincare duality isomorphism for the cohomology groups with rational coefficients on X/G. 

Lemma 6.6 ([31, Lemma 1]). Let k X Y be a finite birational morphism between irredueible varieties. Then k IG X) = IG Y). 

Let 7T Z y be a proper projective morphism between algebraic varieties. Suppose that Y decomposes into a finite number of irreducible nonsingular subvarities, called strata Y = JyOy. Here y denotes a distinguished point in Oy. We further assume the following condition For each stratum Oy, the restriction of vr to K iOy) is a topological fiber bundle with base Oy and fiber 

Consider the top degree cohomology group i/ dmiTr of the fiber ir y), 

Consider the top degree cohomology group H2dimn 1 (,x L(y), Q) of the fiber 7r-1(y), and denote it by Vy. It has a basis dual to the irreducible components of 7r-1(y). The fundamental group iri(Oy) of the stratum Oy acts on Vy by monodromy. The action is simply the permutation of irreducible components. Let 

Theorem 6.8 (Decomposition theorem for a semismall morphism [10, Theorem 1.7]). 

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These lectures were intended for graduate students who have basic knowledge on algebraic geometry and ordinary topology. The only results which will be used but not proved in this note are Grothendieck s construction of the Hilbert scheme (Theorem 1.1) and results on intersection cohomology ( 6.1). I recommend to the reader to accept these results when he/she is not familiar with them. 

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