Surfaces

Report on Meeting 3

Remark 1. Last time, we saw that there was an intersection theory that made the collection CH(X) of algebraic cycles modulo rational equivalence into a graded ring, with natural ring homomorphisms f^{∗} : CH(Y) → CH(X) associated to any morphism f : X → Y and natural graded group homomorphism f_{∗} : CH(X) → CH(Y) when f is proper. Moreover, the maps f^{∗} and f_{∗} are related by a projection formula.

Definition 2. Let X be a noetherian scheme. Define K_{0}(X) to be the quotient of the free abelian group generated by isomorphism classes of vector bundles on X by the relations:

[F] = [E] + [G] whenever there is an exact sequence of vector bundles 0 → E → F → G → 0. Define G_{0}(X) in the same way, by replacing “vector bundle” with “coherent sheaf” everywhere in the previous sentence. If X is nonsingular, then the obvious map K_{0}(X) → G_{0}(X) is an isomorphism. Both K_{0}(X) and G_{0}(X) can be made into rings, with the multiplication induced by the tensor product of sheaves.

Remark 3. If f : X → Y is any morphism of noetherian schemes, then pulling back vector bundles is an exact functor defining a natural ring homomorphism f^{∗} : K_{0}(Y) → K_{0}(X). If f is proper, then we can also define a natural group homomorphism f_{∗} : G_{0}(X) → G_{0}(Y) as follows. Note that this takes more care, first of all because f_{∗} of a vector bundle need not be a vector bundle. The hypothesis of properness assures us that all the higher direct images R^{i}f_{∗}(F) of a coherent sheaf F are themselves coherent. (This is a sheaf-theoretic version of Serre’s Finiteness Theorem.) However, we still need to be careful because f_{∗} is not an exact functor. As it turns out, we get a well defined group homomorphism by taking

f_{∗}[F] = ∑_{i≥0}(−i)^{i}[R^{i}f_{∗}(F)].

When X and Y are both nonsingular, we use the natural map to idenify the G-groups with the K-groups, and then the two homomorphisms are related by the projection formula f_{∗}(x.f^{∗}y) = f_{∗}(x).y.

Remark 4. On a strictly formal level, we see that the theory of algebraic cycles looks a lot like the K-theory of vector bundles. We also know that there is a direct connection between line bundles and divisors on X. The next step is to see that higher dimensional vector bundles are also related to algebraic cycles. First, we look at some additional properties of algebraic cycles.

Proposition 5. (The Exactness Property) Let Y be a nonsingular irreducible closed subscheme of X, with open complement U. Then there is an exact sequence

CH(Y)⟶^{i_{∗}}CH(X)⟶^{j^{∗}}CH(U) → 0.

Proof: Let s be the codimension of Y in X. You can separate the points of codimension r in X into two pieces: those in Y (of codimension r − s) and those in U. Now the result follows from a diagram chase when you write down the exact sequences defining the r^{th} grade pieces of the Chow groups.

Proposition 6. (The Homotopy Property) For every positive integer m, we have natural ring isomorphisms CH(X) → CH(X × A^{m}), induced by pulling back along the projection map.

Proof: By induction, we can reduce to the case m = 1. let π : X × A^{1} → X be the projection. Any prime cycle of codimension r on the product is either completely contained in a fibre, or else it projects onto a cycle of the same dimension in X. From this observation, it suffices to check that cycles contained in fibres are trivial. But that is just the statement that P ic(A^{1}) = 0.

Corollary 7. For all m, one has CH(P^{m}) = Z[h]/h^{m+1}, where h is the class of a hyperplane.

Proof: By induction on m, with the result trivial for m = 0, which is just a point. Take a hyperplane section and use the exactness property to write

CH(P^{m−1}) → CH(P^{m}) → CH(A^{m}) → 0.

By the homotopy property, one sees that CH(A^{m}) is isomorphic to Z (in degree 0). The result follows once you check that no linear subvariety of P^{m} can be rationally equivalent to 0, since the degree of a rationally trivial cycle on projective space is 0.

Proposition 8. Let E be a locally free sheaf of rank m on X. Let P(E) be the associated projective bundle. (So, there is a morphism π : P(E) → X whose fibres are isomorphic to P^{r−1}. Let ξ ∈ CH^{1}(P(E)) be the class of the divisor corresponding to the line bundle O_{P(E)}(1). Then π^{∗} makes CH(P(E)) into a free module over CH(X) with basis 1, ξ, …, ξ^{r−1}.

Remark 9. This proposition generalizes the previous computation on projective space. It is important because it allows us to make the following definition.

Definition 10. Let E be a locally free sheaf of rank m on X, and let P(E) be the associated projective bundle. For each i = 0, …, r, we define the i^{th} Chern class c_{i}(E) ∈ CH^{i}(X) by the requirement that c_{0}(E) = 1 and

∑_{i=0}^{r}(−1)^{i}π^{∗}(c_{i}(E)).ξ^{r−i} = 0

in CH^{r}(P(E)).

Definition 11. Define the total Chern class

c(E) = c_{0}(E) + c_{1}(E) + ⋯ + c_{r}(E),

and the Chern polynomial

c_{t}(E) = c_{0}(E) + c_{1}(E)t + ⋯ + c_{r}(E)t^{r}.

Theorem 12. The Chern classes are characterized by the following properties:

(C1) If E = O_{X}(D) is the line bundle associated to a divisor D, then c_{t}(E) = 1 + Dt. (C2) Iy f : Y → X and E is locally free on X, then for all i,

c_{i}(f^{∗}E) = f^{∗}c_{i}(E).

(C3) If 0 → F → E → G → 0 is an exact sequence of locally free sheaves, then

c_{t}(E) = c_{t}(F)c_{t}(G).

Proof: (i) Follows trivially from the definitions; (ii) follows from the naturality of the construction of projective bundles. For the rest of the proof, you start by observing that the pullback of E to P(E) is an extension of a line bundle by a bundle of rank r − 1. Then you can show inductively that there exists a morphism π : Y → X such that π^{∗}(E) has a filtration whose quotients are line bundles L_{i} and such that π^{∗} is injective. In this case, one computes the Chern classes on Y, where

c_{t}(E) = ∏ c_{t}(L_{i}) = ∏(1 + D_{i}t).

Proposition 13. Let E and F be locally free sheaves on X. Formally, write

c_{t}(E) = ∏_{i=1}^{r}(1 + a_{i}t) c_{t}(F) = ∏_{i=1}^{s}(1 + b_{i}t).

Then one has

c_{t}(E ⊗ F) = ∏_{i,j}(1 + (a_{i} + b_{j})t),

c_{t}(∧^{p}E) = ∏_{1≤i_{1}≤···≤i_{p}≤r}(1 + (a_{i_{1}} + ⋯ + a_{i_{p}})t),

c_{t}(E^{∧}) = c_{−t}(E).

Definition 14. Let E be a locally free sheaf on X, and write its Chern polynomial as above. Define the exponential Chern character to be

ch(E) = ∑_{i=1}^{r} e^{a_{i}}.

Remark 15. The Chern character is a ring homomorphism ch : K_{0}(X) → CH(X). By property (C2), associated to every morphism f : X → Y, there is a commutative diagram

K_{0}(Y) ⟶^{ch} CH(Y)
↓^{f^{∗}}  ↓^{f^{∗}}
K_{0}(X) ⟶^{ch} CH(X).

Remark 16. Now let’s consider the special case of the morphism f : X → Spec(k) for a nonsingular projective variety over a field k. We can easily see that ch : K_{0}(k) = Z → CH(k) = Z is the isomorphism that takes a finite dimensional k-vector space to its dimension. The natural f^{∗} maps, being ring homomorphisms, are then completely uninteresting, since they simply take 1 to 1. Since f is proper, however, we also have f_{∗} maps. On the K-theory level, the map is given by

f_{∗}[E] = ∑_{i≥0}(−1)^{i}[R_{∗}^{i}E] = ∑_{i≥0}(−1)^{i}[H^{i}(X, E)],

and so ch(f_{∗}(E)) = χ(E) is just the Euler characteristic. On the level of algebraic cycles, f_{∗} only takes on a nozero value at a zero cycle, where it maps zero cycles to their degrees. Thus, if we can understand how ch behaves compared to the two different f_{∗} maps, then we will get a formula that computes the Euler characteristic of a vector bundle in terms of the degree of some associated algebraic zero cycle.

Definition 17. Define the Todd class of E to be

td(E) = ∏_{i=1}^{r} \frac{a_{i}}{1 − e^{−a_{i}}} .

Remark 18. One can compute (tediously) that

ch(E) = r + c_{1}(E) + \frac{1}{2}(c_{1}^{2}(E) − 2c_{2}(E)) + ⋯

and

td(E) = 1 + \frac{1}{2}c_{1}(E) + \frac{1}{12}(c_{1}^{2}(E) + c_{2}(E)) + · · · .

Theorem 19. (Grothendieck-Hirzebruch-Riemann-Roch) Let E be a locally free sheaf on a nonsingular projective variety of dimension n. Then

χ(E) = deg[ch(E).td(T )]_{n}.

Remark 20. When X is a curve, we get

χ(E) = deg[c_{1}(E) + \frac{r}{2}c_{1}(T )]

If E = O_{X}(D), this becomes

χ(E) = deg(D) + (1/2)deg(−K) = deg(D) + 1 − g.

Remark 21. When X is a surface and E = O_{X}(D), the formula becomes

χ(D) = deg((1 + D + \frac{1}{2}D^{2}).(1 − \frac{1}{2}K + \frac{1}{12}(K^{2} + c_{2})) = \frac{1}{2}(D^{2} − d.K) + \frac{1}{12}K^{2} + c_{2}).

By taking D = 0, we get both the previous statement of the Riemann-Roch theorem for surfaces, along with Noether’s Formula:

χ(O_{X}) = \frac{1}{12}(K^{2} + c_{2}).