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is there any algorithm known for computing (middle perversity)intersection homology of complex toric varieties based on their combinatorial data? I'm not looking for a computer program.



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I just found out that there is a chapter in Kirwan/Woolf-Introd. to IH". I will go thru it and write again if questions arise! – Peter Miller Jun 15 '10 at 11:42
up vote 5 down vote accepted


Braden, Tom and MacPherson, Robert, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), no. 3, 533--551.

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There is a simple and beautiful description in terms of commutative algebra (repeatedly calculating global sections and taking a projective cover). The work of Braden-MacPherson cited by Alexander is relevant, but only for certain toric varieties (those admitting affine pavings). Also, the Braden-MacPherson paper is really aimed at handling the case of flag varieties etc., which is more complicated than toric varieties.

I think the first combinatorial description was given by Bernstein and Lunts at the end of their book on equivariant sheaves:

Bernstein, Joseph; Lunts, Valery Equivariant sheaves and functors. LNM 1578. Berlin: Springer-Verlag.

This was then abstracted to arbitrary (perhaps non-rational) polytopes here:

Bressler, Paul and Lunts, Valery, Intersection Cohomology on Nonrational Polytopes, Compositio Mathematica, Volume 135, Issue 3, pp 245-278.

There is parallel work by BBFK:

Gottfried Barthel, Jean-Paul Brasselet, Karl-Heinz Fieseler, and Ludger Kaup Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) Volume 54, Number 1 (2002), 1-41.

All of this is summarized quite nicely in Kirwan and Wolf, An introduction to Intersection Cohomology Theory, Second Edition, Chapman and Hall, 2006.

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