Intersection Homology - Properties of The Complex IC(X)

Properties of The Complex IC(X)

The complex IC_p(X) has the following properties

• On the complement of some closed set of codimension 2, we have

is 0 for i+m≠ 0, and for i=−m the groups form the constant local system C

• is 0 for i + m < 0
• If i > 0 then is zero except on a set of codimension at least a for the smallest a with p(a) ≥ m − i
• If i>0 then is zero except on a set of codimension at least a for the smallest a with q(a) ≥ (i)

As usual, q is the complementary perversity to p. Moreover the complex is uniquely characterized by these conditions, up to isomorphism in the derived category. The conditions do not depend on the choice of stratification, so this shows that intersection cohomology does not depend on the choice of stratification either.

Verdier duality takes IC_p to IC_q shifted by n = dim(X) in the derived category.