where the -point compactification. is covariant with respect to proper maps.

Alternatively,

where (dualising sheaf).

(Poincaré duality) If is an oriented manifold of dimension then

where the requires a choice of orientation.

If is a complex manifold, it comes with a canonical orientation, e.g., take or something as long as one is consistent.

For instance, taking

gives

There exist other definitions in Chriss-Ginzburg §2.6.

Take

One has a triangle

out of which one gets a long exact sequence

Let be a variety over . Then one has a fundamental class , where . Look at , the singular locus of , which has real codimension . Then is smooth, and one gets by poincare duality. Now the long exact sequence yields

where the terms on the sides vanish due to dimension reasons (supported only between and twice the dimension). Hence one has an isomorphism

Recall that one has

where each is affine. One has . From the long exact sequence, one can derive that is free of rank . Moreover, the graded rank is

where are the degrees of and . Therefore (via some Poincaré duality argument or something), is free of rank with basis (possibly but Peter didn’t check this).

Consider

where is smooth, is proper, and there exists a Zariski dense open such that . Then one can take

Pick and the sequence . Start with standard flag, arrows are containments.

Take the flag on the right-hand side. Choices of flags give . In general, one has

where .

If is a reduced expression for then is a resolution of singularities.

given by mapping a point to the flag .

The must be . Over the subset in where , the fibre is a point because there is only one choice of . But if (in ), then the fibre is .

Recall that we had the “divided difference operators”

We have a -fibration

where is given by forgetting the th step of the flag, so a partial flag.

Using , one gets the key stepkey step.

We have and for :

We have for the pairings on and

This tells you that must kill . .