In this post we’ll follow Macphearson’s paper Chern Classes for singular algebraic varieties
In Fulton’s intersection theory, one of the basic constructions is a natural transformation
which maps a cycle to it’s rational equivalence class. A different way of looking at this is as a map from constructible functions to . Because by definition, is the group of functions generated by the characteristic functions for subvarieties $V\subset $.
The essential properties of the functor are captured by the following proposition:
There is a unique covariant functor from compact complex algebraic varieties to abelian groups whose value on a variety is the group of constructible functions from the variety to the integers and whose value on a map satisfies:
Where denotes the topological Euler characteristic.
Now, we can associate to each smooth variety a natural homology class by taking the Poincare dual of the total chern class of it’s tangent bundle. Our primary goal is to generalize this construction to arbitrary varieties. Macphearson accomplishes this by proving the following theorem.
There exists a natural transformation from to homology which, on a smooth variety , takes the constant function to the Poincare dual of the total chern class of .
Given the theorem, we can define the chern class of a variety to be .
One useful trick that Macphearson uses is the following application of resolution of singularities. Let be a constructible function on . By a successive resolution of singularities of the support of , we can find integers and maps from smooth varieties to such that
By combining this trick with the projection formula for homology, we can reduce many statements about to the case where the ambiant variety is smooth and the function is the identity.
The two main tools in the construction of are the Mather class and the Euler obstruction
The Mather class
The Mather class attempts to solve the same problem as the Macphearson class. On the one hand, it manages to define a class which reduces to the chern class on smooth varieties. However, it fails to be functorial under pushforward. In some sense, the Macphearson class can be seen as an attempt to fix this failure. It does this by carefully breaking up a variety into pieces and applying the Mather class to each piece, together with integer weights.
The idea of the Mather class is fairly straightforward. Given a singular variety , we embed it in a smooth variety and use the tangent bundle to construct a natural bundle on that agrees with the tangent bundle on the smooth part of . We then take the chern class of this new bundle.
We do this as follows. Let be the dimension of and the dimension of . Over the smooth part of there is a natural section of , the grassmanian of -hyperplanes in , given by the tangent space of . The Nash blowup of is defined to be the closure of the image of this section. It comes equipped with a map and a natural bundle given by the restriction of the tautological bundle on the grasmanian. We can then define the Mather class as:
It is important to note that the Mather class can be calculated in this way using any proper resolution with a subbundle which, over the smooth part of , agrees with the pullback of the tangent bundle of . One way to see this is to note that any two such resolutions are dominated by a third. So when comparing the resolutions and , we can assume that we’re in the situation such that the bundle on is the pullback of the bundle on . We then apply the projection formula for proper maps.
The Euler obstruction
The Euler obstruction will give us a new way to associate a constructible function to a variety . The naive approach is to take the function and this is equal to the Euler obstruction when is smooth. However, when is singular, the Euler obstruction gives better information about the singularities.
As before, let be the Nash blowup. Let be coordinates around a point in . Let be the real valued function . Then is a section of which pulls back to give us a section .
Lemma: For small , is nonzero over for .
The proof of the theorem uses Whitney stratification.
Now, let be the -ball around and the -sphere. We thus have a non-zero map which we can think of as a map to the corresponding sphere bundle. By chapter 4.3 in Hatcher, the obstruction to lifting this map to a map from , which we'll denote , lies in the cohomology group . By the theory in Hatcher, the group of coefficients in this case is of the sphere bundle of the rank bundle which is why we get .
Let denote the orientation class of the pair. Then we can the define the value of the Euler obstruction at to be:
Using this function, we can define a map from algebraic cycles to constructible functions on by defining:
The Main Result
We now have enough machinery to define the Macphearson class. Symbolically, we define . In other words,
As we remarked earlier, the analogous mapping of the Mather class is which is equivalent to when is smooth since in that case and so . However, when is singular, carries additional information about the singular strata and this is what grants it’s power and functoriality.
So we now want to prove that given a proper map , . By the reduction we mentioned above, it will be enough to prove this in the case that is smooth and the constructible function is the identity. Since is in this case , we need to show that
By using the definitions of and , it will suffice to find an algebraic cycle on such that
Interestingly, we’ll first obtain the cycle by a completely separate construction, and then show that it satisfies 1 and 2.
Before we do this, note that 1 tells us that we can write the pushforward of the chern class of using canonically defined classes on which a priori have nothing to do with $X$. This fact is interesting in it’s own right and does not require any of the Euler obstruction machinery.