Macphearson’s Chern Class

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

Z_{*}X\rightarrow A_{*}X

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 A_{*}X. Because by definition, CF(X) is the group of functions generated by the characteristic functions 1_{W} for subvarieties $V\subset $.

The essential properties of the functor CF are captured by the following proposition:

Proposition:
There is a unique covariant functor CF 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 f satisfies:

f_{*}(1_{W})(p) = \chi(f^{-1}(p)\cap W)

Where \chi denotes the topological Euler characteristic.

Now, we can associate to each smooth variety X 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.

Theorem:
There exists a natural transformation c_* from CF to homology which, on a smooth variety X, takes the constant function 1 to the Poincare dual of the total chern class of X.

Given the theorem, we can define the chern class of a variety V to be c_{*}(1_{V}).

One useful trick that Macphearson uses is the following application of resolution of singularities. Let \alpha be a constructible function on V. By a successive resolution of singularities of the support of \alpha, we can find integers k_i and maps g_i from smooth varieties X_i to V such that

\alpha =  \sum k_i g_{i*}(1)

By combining this trick with the projection formula for homology, we can reduce many statements about c_* to the case where the ambiant variety is smooth and the function is the identity.

The two main tools in the construction of c_* 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 V, we embed it in a smooth variety N and use the tangent bundle TN to construct a natural bundle on V that agrees with the tangent bundle on the smooth part of V. We then take the chern class of this new bundle.

We do this as follows. Let v be the dimension of V and n the dimension of N. Over the smooth part of V there is a natural section of G_v(TN), the grassmanian of v-hyperplanes in TN, given by the tangent space of V. The Nash blowup \hat{V} of V is defined to be the closure of the image of this section. It comes equipped with a map \nu:\hat{V}\rightarrow V and a natural bundle TV\rightarrow\hat{V} given by the restriction of the tautological bundle on the grasmanian. We can then define the Mather class as:

c_M(V) = \nu_{*}(c(TV)\cap[\hat{V}])

It is important to note that the Mather class can be calculated in this way using any proper resolution \nu:\hat{V}\rightarrow V with a subbundle TV \subset \nu^{*}TN which, over the smooth part of V, agrees with the pullback of the tangent bundle of V. One way to see this is to note that any two such resolutions are dominated by a third. So when comparing the resolutions X\rightarrow V and Y\rightarrow V, we can assume that we’re in the situation X\rightarrow Y\rightarrow V such that the bundle on X is the pullback of the bundle on Y. 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 V. The naive approach is to take the function V and this is equal to the Euler obstruction when V is smooth. However, when V is singular, the Euler obstruction gives better information about the singularities.

As before, let \nu:\hat{V}\rightarrow V be the Nash blowup. Let (z_1,\dots,z_n) be coordinates around a point p in N. Let f be the real valued function z_1\bar{z_1}+\dots +z_n\bar{z_n}. Then df is a section of T^{*}N which pulls back to give us a section r:\hat{V}\rightarrow T^{*}V.

Lemma: For small \epsilon, r is nonzero over \nu^{-1}(z) for 0<f(z)\leq\epsilon.

The proof of the theorem uses Whitney stratification.

Now, let B_{\epsilon} be the \epsilon-ball around p and S_\epsilon the \epsilon-sphere. We thus have a non-zero map r:\nu^{-1}S_{\epsilon}\rightarrow T^{*}V 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 \nu^{-1}B_{\epsilon}, which we'll denote Eu(T^{*}V,r), lies in the cohomology group H^{v}(\nu^{-1}B_{\epsilon}, \nu^{-1}S_{\epsilon}; \mathbb{Z}). By the theory in Hatcher, the group of coefficients in this case is \pi_v of the sphere bundle of the rank v bundle T^{*}V which is why we get \mathbb{Z}.

Let \mathcal{O}_{(\nu^{-1}B_{\epsilon},\nu^{-1}S_{\epsilon})} denote the orientation class of the pair. Then we can the define the value of the Euler obstruction at p to be:

Eu_{p}(V) = \langle Eu(T^{*}V,r), \mathcal{O}_{(\nu^{-1}B_{\epsilon},\nu^{-1}S_{\epsilon})}\rangle

Using this function, we can define a map T from algebraic cycles to constructible functions on V by defining:

T(\sum n_i V_i)(p) = \sum n_i Eu_{p}(V_i)

The Main Result

We now have enough machinery to define the Macphearson class. Symbolically, we define c_{*} = c_{M}\circ T^{-1}. In other words,

c_{*}(Eu(V))=c_{M}(V)

As we remarked earlier, the analogous mapping of the Mather class is c_{M}(1_V)=c_{M}(V) which is equivalent to c_{*} when V is smooth since in that case V=\hat{V} and so Eu(V)=1_V. However, when V is singular, Eu(V) carries additional information about the singular strata and this is what grants c_* it’s power and functoriality.

So we now want to prove that given a proper map X\rightarrow Y, c_M T^{-1}f_{*} = f_{*}c_{M}T^{-1}. By the reduction we mentioned above, it will be enough to prove this in the case that X is smooth and the constructible function is the identity. Since is in this case c_{M}T^{-1}(1)=c(X)\cap[X], we need to show that

f_{*}(c(X)\cap[X]) = c_{M}(T^{-1}(f_{*}(1)))

By using the definitions of T and f_{*}(1), it will suffice to find an algebraic cycle \sum n_{i}V_{i} on Y such that

  1. f_{*}(c(X)\cap[X]) = \sum n_{i}c_{M}(V_i)
  2. \chi f^{-1}(p) = \sum n_i Eu_p(V_i)

Interestingly, we’ll first obtain the cycle \sum n_{i}V_{i} 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 X using canonically defined classes on Y 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.

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