In the previous post, we introduced the Fano scheme of a subscheme of projective space, as the Hilbert scheme of planes of a certain dimension on that subscheme. In this post, I’d like to work out an explicit example, of the 27 lines on a smooth cubic surface in $\mathbb{P}^3$; as we’ll see, the Fano scheme is 27 reduced points, and the count can be made with a little calculation on the Grassmannian. Although the calculation is elementary, I found it worthwhile to work carefully through it, not only for its intrinsic interest but also as motivation for the study of intersection theory on moduli spaces in general. Once again, most of this material is from Eisenbud-Harris’s draft book 3264 and All That.

1. The normal bundle as self-intersection

Suppose ${X = S}$ is a smooth surface, imbedded in some projective space, and consider the scheme ${F_1 S}$ of lines in ${S}$.

Fix a line ${L}$ in $S$. In this case, the normal sheaf ${N_{S/L}}$ is actually a vector bundle of normal vector fields, given by the adjunction formula

$\displaystyle N_{S/L} = \left(\mathcal{I}_L/\mathcal{I}_L^2\right)^{\vee} = \left(\mathcal{O}_S(-L)/\mathcal{O}_S(-2L)\right)^{\vee} = \mathcal{O}_L(L).$

In particular, ${N_{S/L}}$ is a line bundle on ${L \simeq \mathbb{P}^1}$ and has a well-defined degree. This degree is in fact the self-intersection ${L.L}$ of ${L}$, considered as a divisor on the smooth surface ${S}$. (more…)

Let ${E}$ be a multiplicative cohomology theory. We say that ${E}$ is complex-oriented if one is given the data of an element ${t \in \widetilde{E}^2(\mathbb{CP}^\infty)}$ which restricts to the canonical generator of ${\widetilde{E}^2(\mathbb{CP}^1) \simeq \widetilde{E}^0(S^0)}$. It turns out that one has a bit more: a complex orientation gives on a functorial, multiplicative choice of Thom classes for complex vector bundles. In fact, this is a perhaps more natural definition of such a theory.

What does this mean? Given a vector bundle ${\zeta \rightarrow X}$, one can form the Thom space ${T(\zeta) = B(\zeta)/S(\zeta)}$: in other words, the quotient of the unit ball bundle ${B(\zeta)}$ in ${\zeta}$ (with respect to a choice of metric) by the unit sphere bundle ${S(\zeta)}$. When ${X}$ is compact, this is just the one-point compactification of ${\zeta}$.

Definition 1 The vector bundle ${\zeta}$ is orientable for a multiplicative cohomology theory ${E}$ if there exists an element ${\theta \in \widetilde{E}^*( T(\zeta)) = E^*(B(\zeta), S(\zeta))}$ which restricts to a generator on each fiberwise ${E^*(B^n, S^{n-1})}$, where ${\dim \zeta = n}$. Such a ${\theta}$ is called a Thom class.

Observe that for each point ${x \in X}$, there is a restriction map ${\widetilde{E}^*(T(\zeta)) \rightarrow E^*(B_x^n, S_x^{n-1})}$ if the dimension of ${\zeta}$ is ${n}$.

The existence of a Thom class implies a Thom isomorphism, as for ordinary homology.

Theorem 2 (Thom isomorphism) A Thom class ${\theta \in \widetilde{E}^*(T(\zeta))}$ induces an isomorphism

$\displaystyle E^*(X) \simeq \widetilde{E}^*(T(\zeta))$

given by cup-product with ${\theta}$.

In the case of ordinary homology, a Thom class is unique (up to sign) if it exists; in general, though, a Thom class is highly non-unique, and an orientation is additional data than simple orientability.

Here are a few basic cases:

1. Any vector bundle is orientable for ${\mathbb{Z}/2}$-cohomology.
2. An oriented (in the usual sense: i.e., the top wedge power is trivial) vector bundle is one oriented for ${\mathbb{Z}}$-cohomology.
3. Complex vector bundles are oriented for ${K}$-theory. We will see this below.
4. Spin bundles are oriented for ${KO}$-theory. An explicit construction of Thom classes can be made, as virtual bundles arising from Clifford modules: this is in Atiyah-Bott-Shapiro’s paper.
5. A trivial bundle is orientable for any cohomology theory (this is rather uninteresting: the Thom space is just a suspension). (more…)

So, now with the preliminaries on connections and curvature established, and the Chern classes summarized, it’s time to see how they connect with one another. Namely, we want to say that, given a complex vector bundle, we can compute the Chern classes in de Rham cohomology by picking a connection — any connection — on it,  computing the curvature, and then applying various polynomials.

We shall start by warming up with a special case, of a line bundle, where the algebra needed is easier. Let ${M}$ be a smooth manifold, ${L \rightarrow M}$ a complex line bundle. Let ${\nabla}$ be a connection on ${L}$, and let ${\Theta}$ be the curvature.

Thus, ${\Theta}$ is a global section of ${\mathcal{A}^2 \otimes \hom(L, L)}$; but since ${L}$ is a line bundle, this bundle is canonically identified with ${\mathcal{A}^2}$. (Recall the notation that $\mathcal{A}^k$ is the bundle (or sheaf) of smooth $k$-forms on the manifold $M$.)

Proposition 1 (Chern-Weil for line bundles) ${\Theta}$ is a closed form, and the image in ${ H^2(M; \mathbb{C})}$ is ${2\pi i}$ times the first Chern class of the line bundle ${L}$. (more…)

So, I’m in a tutorial this summer, planning to write my final paper on the Kodaira embedding theorem, and I’ve been finding my total ignorance of complex algebraic geometry to be something of a problem. One of my goals next year is, coincidentally, to acquire a solid understanding of most of the topics in Griffiths-Harris. To start with, I’d like to spend a few posts on Chern-Weil theory. This gives an analytic method of computing the Chern classes of a complex vector bundle, and more generally a framework for the characteristic classes of a principal bundle over a Lie group. In fact, it tells you what the cohomology of the classifying space of a Lie group is (it’s a certain algebra of invariant polynomials on the Lie algebra), from which — by Yoneda’s lemma — you can associate cohomology classes to a principal bundle on any space.

Today, I’d like to review what Chern classes are like.

1. Introduction

To start with, we will need to describe what the Chern classes really are. These are going to be natural maps

$\displaystyle \mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z}),$

from the complex vector bundles on a space ${X}$ to the cohomology ring. In other words, to each vector bundle ${E \rightarrow X}$, we will have an element ${c(E) \in H^*(X; \mathbb{Z})}$. In order for this to be natural, we are going to want that, for any map ${f: Y \rightarrow X}$ of topological spaces,

$\displaystyle c(f^*E) = f^* c(E) \in H^*(Y; \mathbb{Z}).$

In other words, we are going to want the map ${\mathrm{Vect}_{\mathbb{C}}(X) \rightarrow H^*(X; \mathbb{Z})}$ to be functorial in ${X}$, when both are considered as contravariant functors in ${X}$. It turns out that each functor ${\mathrm{Vect}_{n, \mathbb{C}}}$ (of ${n}$-dimensional complex vector bundles) and ${H^k(X; \mathbb{Z})}$ is representable on the appropriate homotopy category. (more…)