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.
Regards,
Peter
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. Regards, Peter 


See Braden, Tom and MacPherson, Robert, From moment graphs to intersection cohomology, Math. Ann. 321 (2001), no. 3, 533551. 


There is a simple and beautiful description in terms of commutative algebra (repeatedly calculating global sections and taking a projective cover). The work of BradenMacPherson cited by Alexander is relevant, but only for certain toric varieties (those admitting affine pavings). Also, the BradenMacPherson 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: SpringerVerlag. This was then abstracted to arbitrary (perhaps nonrational) polytopes here: Bressler, Paul and Lunts, Valery, Intersection Cohomology on Nonrational Polytopes,
Compositio Mathematica, Volume 135, Issue 3, pp 245278. There is parallel work by BBFK: Gottfried Barthel, JeanPaul Brasselet, KarlHeinz Fieseler, and Ludger Kaup Combinatorial intersection cohomology for fans, Tohoku Math. J. (2) Volume 54, Number 1 (2002), 141. All of this is summarized quite nicely in Kirwan and Wolf, An introduction to Intersection Cohomology Theory, Second Edition, Chapman and Hall, 2006. 

