Chern Classes - Part 1 of Several

14 Apr 2012

Chern classes are a collection of invariants in differential and algebraic geometry. In the analytic setting, Chern classes are characteristic invariant of vector bundles on complex smooth manifolds; in algebraic geometry, Chern classes are invariants of locally free coherent sheaves over projective schemes. For the latter, if such schemes are \mathbb{C}-schemes, in which case such coherent sheaves are the algebraic analogues of vector bundles, via duality theories in the respective domains, the two Chern classes agree.

In general when it comes to manifolds, there are a lot of parallels between analytic and algebraic techniques. Whether or not it is the aim of these two camps to cooperate, certainly understanding how to transition from one to another greatly increases the level of understanding regarding manifold theory. In view of this, I aim to answer the following: what is an intuitive way to understand why the two approaches to Chern classes are the same?

The exposition is broken roughly into the following sections:

1. Chern classes of locally free coherent sheaves over projective schemes
2. Chern classes of complex vector bundles over smooth manifolds
3. Intersection theory
4. Dictionary: bridging the gap

These sections will not correspond to entries: I intend to split this exposition into manageable chunks to write. The references will be composed of bits and pieces from [BoTu], [F], [GrHa], and [V]. If other references arise, I will make these explicit.

I would like to give thanks to AM and VD for innumerable conversations that attempt to break down the language barriers between geometers and a stubborn algebraist such as myself. I would also like to thank CW for the conversations that motivated this side project.

Section 1 – Locally Free Coherent Sheaves of Projective Schemes

The story of Chern classes began for me with the Intersection Theory appendix of [H]: Let X be a projective \mathbb{k}-scheme. For simplicity, assume \mathbb{k} = \overline{\mathbb{k}}. There is a unique way to associate to each locally free coherent sheaf \mathcal{E} on X a formal polynomial c_t(E) = \sum_{i=0} c_i(E)t^i with c_i(E) \in A_i(X) such that

1. Functoriality. For f: Y \rightarrow X, for locally free coherent sheaf \mathcal{E} on X, then c_t(f^*\mathcal{E}) = f^*c_t(E).
2. Normality. For any any line bundle \mathcal{E} = \mathcal{L}(D), we have that c_t(\mathcal{E}) = 1 + Dt.
3. Additivity. For any short exact sequence
   \[0 \longrightarrow \mathcal{E}\,' \longrightarrow \mathcal{E} \longrightarrow\mathcal{E}\,'' \longrightarrow 0,\] we have c_t(\mathcal{E}) = c_t(\mathcal{E}\,’)c_t(\mathcal{E}\,”).

The statement is proven in [Gr1]. Here, Chern classes c_i(\mathcal{E}) is defined by
\[\sum_{i = 0}^r (-1)^i\pi^*c_i(\mathcal{E}).\xi^{r - i} = 0\] where \xi is the generator of A^*(\mathbb{P}\mathcal{E}) as a free module over A^*(X). It is unique of all characteristic classes satisfying functoriality, normality, and additivity. In fact, [Gr1] offers an explanation, albeit unintuitive, of how the geometric Chern class is the same as its algebraic counterpart.

In the following, we briefly introduce the notions necessary to define and understand Chern classes in the algebraic setting.

1.1 Projectivization of Locally Free Coherent Sheaves

For the remainder, let \mathcal{E} be a locally free coherent sheaf on a scheme X. That is, there exist affine open subsets \{U_i \subseteq X\} of X, for which \mathcal{E}|_{U_i} = \tilde{M_i} where M_i is a free \mathcal{O}_X(U_i) module, and \tilde{M_i} is the sheaf of \mathcal{O}_{U_i}-module associated to M_i (see [H] 2.5).

Let S(\mathcal{E}) be the symmetric algebra on \mathcal{E}: \[S(\mathcal{E}) = \bigoplus_{i \geq 0} S^i(\mathcal{E}).\] Locally, it sends an open subset U of X to S(\mathcal{E}(U)) = \bigoplus_{i \geq 0} S^i(\mathcal{E}(U)). One can check via ([H] Ex 2.1.22) that S(\mathcal{E}) is a sheaf on X. In particular, locally at least, the module is graded by tensor power (and you may wish to verify that the grading is preserved along transition maps). It is easy to see that for U = \mathrm{spec} R of X, the generators of \Gamma(U, \mathcal{E}) generate the symmetric algebra S(\mathcal{E}(U)) over R.

For example, let \mathcal{O}(1) be the hyperplane bundle over \mathbb{P}^n = \mathbf{Proj}\mathbb{k}[x_0,...,x_n], then locally over U_i = D_+(x_i), \mathcal{O}(1) is generated by x_i over R_i = \mathbb{k}[x_0,\dots,x_n]_{x_i}, and S(\mathcal{O}(1)(U_i)) is the graded ring R_i[x_it]. The transition maps of S(\mathcal{E}) from U_i to U_j is obtained from the (degree 0) ring map which sends x_it \in R_i[x_it] to x_jt\cdot x_i/x_j \in R_j[x_jt].

At this point, we construct the projective fibre bundle from the sheaf S(\mathcal{E}), which is a scheme \mathbb{P}(\mathcal{E}) together with a map \pi: \mathbb{P}(\mathcal{E}) \rightarrow X such that the fibre over each p \in X is \mathbf{Proj}\,S(\mathcal{E}_p). The construction is exactly that of ([H], p 160), applied to S(\mathcal{E}):

Construction: to each affine open subset U of X on which \mathcal{E} is trivial associate the projective space \mathbf{Proj}\,S(\mathcal{E}(U)) — \mathbf{Proj}\,S(\mathcal{E}(U)) is well-defined since S(\mathcal{E}(U)) is graded (see [H] p 76-77). This gives a family of projective schemes \{P_U = \mathbf{Proj} S(\mathcal{E}(U))\,|\,U \subset X\textrm{ is affine open}\}. The end goal is to use ([H], Ex 2.2.12). To do so, we must obtain a subcollection \{U \subset X \| U \textrm{ open in }X\} of the open sets covering X such that for every U, V \subset X we have that P_{UV} \subset P_U, P_{VU} \subset P_V and that P_{UV} \simeq P_{VU}. In our case, notice that over U on which \mathcal{E} is trivial, P_U = \mathbb{P}^{n – 1} \times U, where n is the rank of \mathcal{E}. With this in mind, let i_U (resp. i_V) be the inclusion of U \cap V into U (resp. V). Then, set P_{UV} = \mathbb{P}^n \times i_U(U \cap V) and P_{VU} = \mathbb{P}^n \times i_V(U \cap V). If we let the subcollection of open subsets of X be those on which \mathcal{E} is trivial, it is easy to see that the conditions of ([H], Ex 2.2.12) is satisfied. We obtain from the gluing: a scheme \mathbb{P}(\mathcal{E}) together with a map \pi: \mathbb{P}(\mathcal{E}) \rightarrow X such that at the fibre of each point p of X is a projective scheme.

Example. If \mathcal{E} is trivial, then \mathbb{P}(\mathcal{E}) = \mathbb{P}^{n – 1} \times X, where n (assuming X connected) is the (global) rank of the bundle.

What about \mathbb{P}(\mathcal{O}(1)) on \mathbb{P}^n?? On each U_i = D_+(x_i), we have trivial line-bundles, hence \pi^{-1}(U_i) = \mathbb{P}^0 \times U_i. One might think that \mathbb{P}(\mathcal{O}(1)) = \mathbb{P}(\mathcal{O}) = \mathbb{P}^n. In fact, this is not true: no trivialization exists for \mathbb{P}(\mathcal{O}(1)). To see this, it is enough to show that there are no constant nonzero sections of \mathbb{P}(\mathcal{O}(1)). To see this, notice that

(to be continued)