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Suppose $X$ is an irreducible compact variety of dimension $n$(possibly singular) over $\mathbb{C}$, from Etale cohomology theories, there is a version Poincare duality. I do not know much Etale cohomology, could anyone explain it in this special case in a more geometric way? i.e. since the closed points of $X$ is a very nice topological space, could the Poincare duality be explained using its singular cohomology, $H^i(X,\mathbb{C}), 0 \leq i \leq 2n$? Could we at least deduce that the Betti number $b^i$ equals $b^{2n-i}$?

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    poincare duality only works for smooth varieties in etale cohomology. – 2017-02-11
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    There is a version of Poincaré duality for singular spaces in the complex or étale topologies, by the theory of intersection cohomology and perverse sheaves. Is this what you are asking about? – 2017-02-12
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    @TakumiMurayama Yes, could you give a references? Thank you! – 2017-02-12
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    @Wenzhe [This historical survey](https://arxiv.org/abs/math/0701462v3) by Kleiman is very nice. Kleiman also lists some survey articles in the introduction (also look at Endpoint 2), so look there for some references. – 2017-02-12
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    I also advice the book by Kirwan and Woolf about intersection homology which was really helpful for me. There are not so much details so you'll probably need to read "D-modules, perverse sheaves and representation theory". – 2017-03-20

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