DEGREE FORMULA

(poorly ordered collection of Markus’ e-mail messages^{1})

1. The ideal I

For an algebraic variety X over F denote by I(X) the ideal in ℤ generated by deg(x) for all x ∈ X_{(0)}.

Example 1.1. 1. Let X be anisotropic smooth projective quadric. Then I(X) = 2ℤ.

2. Let X be the Severi-Brauer variety of a central simple F -algebra A. Then I(X) = ind(A) · ℤ.

For a field extension L/F, denote by I(L) the ideal in ℤ generated by the degrees [F (v) : F] for all valuation v on L over F such that the residue field F (v) is a finite extension of F.

Proposition 1.2. (1) If for a variety X over F, X(F) ≠ ∅, then I(X) = ℤ;

(2) If there is a morphism f : Y → X, then I(Y) ⊂ I(X);

(3) Let X be an irreducible variety over F and L = F (X).

(a) If X is proper then I(L) ⊂ I(X).

(b) If X is smooth then I(X) ⊂ I(L).

(c) If X is proper and smooth, then I(X) = I(L). In particular, I(X) is a birational invariant of a proper smooth variety X.

(4) Let morphism f : Y → X be a birational isomorphism of irreducible proper varieties with X smooth, then I(Y) = I(X).

(5) If f : Y → X is a rational morphism of proper varieties with Y smooth, then I(Y) ⊂ I(X).

Proof. (3a) Let v be a valuation of L over F. Since X is proper, the valuation ring of v dominates a point x ∈ X. In particular, F (x) is a subfield of F (v) over F. Hence x is a closed point and deg x divides [F (v) : F].

(3b) Let x ∈ X be a closed point. Since x is a smooth point, there is a valuation v of L over F with the residue field F (x).

(4) Let L = F (X) = F (Y). By (2), I(Y) ⊂ I(X) and by (3a,c), I(X) = I(L) ⊂ I(Y).

(5) Let Y_{1} be the graph of f in the product Y ×X. We have two projections f_{1} : Y_{1} → Y and g : Y_{1} → X with f_{1} a birational isomorphism. By (4), I(Y_{1}) = I(Y) and by (2), I(Y_{1}) ⊂ I(X).

Date: May 2000. Text by Alexander Merkurjev, with footnotes by Markus Rost.

1and of Sasha’s considerations.

2. Definition of the invariant η_{p}(X)

Let p be a prime number, F a field. We assume that char(F) ≠ p and μ_{p} ⊂ F. We fix a primitive p-th root of unity ξ.

Let X be quasi-projective variety over F. The group G = ℤ/pℤ acts by cyclic permutations on the product

X^{p} = X × X × ⋯ × X.

The factor variety X^{p}/G we denote by C^{p}X. The image \overline{X} of the diagonal X ⊂ X^{p} under the natural morphism X^{p} → C^{p}X is a closed subvariety in C^{p}X, isomorphic to X. In particular, I(\overline{X}) = I(X).

Consider a G-action on the trivial linear bundle X^{p} × 𝔸^{1} over X^{p} by

(x_{1}, x_{2}, …, x_{p}, t) ↦ (x_{2}, …, x_{p}, x_{1}, ξt).

The projection X^{p} \ X → C^{p}X \ \overline{X} is unramified, hence the restriction of the factor vector bundle (X^{p} ×𝔸^{1})/G to C^{p}X \\overline{X} is a linear bundle over C^{p}X \\overline{X}. Denote it by L_{X}.

Let d = dim X. The image of the zero section of the vector bundle L_{X}^{⊕pd} of rank pd over C^{p}X \ \overline{X} defines an element

l_{X} ∈ CH_{pd}(L^{⊕pd}).

By homotopy invariance,

CH_{pd}(L_{X}^{⊕pd}) = CH_{0}(C^{p}X \ \overline{X}),

so that we will also assume that l_{X} ∈ CH_{0}(C^{p}X \ \overline{X}). If X is projective, we have the degree homomorphism

deg : CH_{0}(C^{p}X \ \overline{X}) ^{}ℤ/I(X).

Thus, the image of l_{X} defines an element

η_{p}(X) ∈ ℤ/I(X).

Note that X is projective but not necessarily smooth variety over F.

Unfortunately, the invariant η_{p}(X) is always of exponent p, even when ℤ/I(X) is large.

Lemma 2.1. p · l_{X} = 0 ∈ CH_{0}(C^{p}X \ \overline{X}). In particular,

p · η(X) = 0 ∈ ℤ/I(X).

Proof. The class l_{X} in CH_{pd}(L^{⊕pd}) is the direct image of the image of the zero section of the trivial vector bundle under

(X^{p} \ X) × 𝔸^{pd} → L^{⊕pd}

of degree p.

3. Degree formula

Theorem 3.1. (Regular Degree Formula) Let f : Y → X be a morphism of projective varieties of dimension d. Then I(Y) ⊂ I(X) and

η_{p}(Y) = deg(f) · η_{p}(X) ∈ ℤ/I(X).

Proof. Denote by Ỹ the inverse image of \overline{X} under C^{p}f : C^{p}Y → C^{p}X, so that \overline{Y} ⊂ Ỹ. In particular,

I(Y) = I(\overline{Y}) ⊂ I(Ỹ) ⊂ I(\overline{X}) = I(X).

We have an open embedding

j : C^{p}(Y) \ Ỹ ↪ C^{p}(Y) \ \overline{Y}

and a proper morphism

i : C^{p}(Y) \ Ỹ ^{}C^{p}(X) \ \overline{X}.

Denote by W the restriction of the vector bundle L_{Y}^{⊕pd} on C^{p}Y \ Ỹ. Clearly, W is the inverse image of L_{X}^{⊕pd} with respect to i. We have the following commutative diagram

[figure: CH_{pd}(L_{Y}^{⊕pd}); j^{∗}; CH_{pd}(W); i_{∗}; CH_{pd}(L_{X}^{⊕pd})]

||  ||  ||

[figure: CH_{0}(C^{p}Y \ \overline{Y}); j^{∗}; CH_{0}(C^{p}Y \ Ỹ); i_{∗}; CH_{0}(C^{p}X \ \overline{X})]

[figure: ℤ/I(Y); deg; ℤ/I(Ỹ); deg; ℤ/I(X).; deg]

The image j^{∗}(l_{Y}) ∈ CH_{pd}(W) is the class l′ of the image of the zero section of W. Then i_{∗} maps l′ to the image of the zero section of L_{X}^{⊕pd} and the degree of i|_{Z} is equal to deg(C^{p}f) = deg(f)^{p}, hence

i_{∗}j^{∗}(l_{Y}) = i_{∗}(l′) = deg(f)^{p} · l_{X}.

Finally, we can replace deg(f)^{p} by deg(f) in view of Lemma 2.1.

Remark 3.2. In fact, we have proved a stronger version of the degree formula on the level of Chow groups:

i_{∗}j^{∗}(l_{Y}) = deg(f) · l_{X} ∈ CH_{0}(C^{p}X \ \overline{X}).

Theorem 3.3. (Rational Degree Formula) Let f : Y → X be a rational morphism of projective varieties of dimension d with Y smooth. Then I(Y) ⊂ I(X) and

η_{p}(Y) = deg(f) · η_{p}(X) ∈ ℤ/I(X).

Proof. The inclusion I(Y) ⊂ I(X) holds by Proposition 1.2(5). Let Y_{1} be the graph of f in Y × X. Applying Theorem 3.1 to the birational isomorphism Y_{1} → Y (of degree 1), we get

η_{p}(Y_{1}) = η_{p}(Y) ∈ ℤ/I(Y) = ℤ/I(Y_{1}).

On the other hand, applying Theorem 3.1 to the projection Y_{1} → X(of degree = deg(f)), we get

η_{p}(Y_{1}) = deg(f) · η_{p}(X) ∈ ℤ/I(X).

Corollary 3.4. The class η_{p}(X) ∈ ℤ/I(X) is a birational invariant of a smooth projective variety X.

Problem 3.5. How to define the invariant η_{p}(X) ∈ ℤ/I(X) out of the function field E = F (X)? (Note that I(X) = I(E).)

Here is the main application:

Theorem 3.6. Let X and Y be two irreducible projective varieties with Y smooth. Assume that X has a rational point over F (Y). Then I(Y) ⊂ I(X) and if η_{p}(Y) ≠ 0 ∈ ℤ/I(X), the following holds:

(1) dim(X) ≥ dim(Y).

(2) If dim(X) = dim(Y), then Y has a closed point over F (X) of degree prime to p.

Proof. There exists a rational morphism f : Y → X. By Proposition 1.2(5), I(Y) ⊂ I(X).

(1) Assume that n = dim(Y) − dim(X) > 0. Consider the composition

g : Y ^{f}^{}X ↪ X × ℙ_{F}^{n} .

Clearly, deg(g) = 0 and I(X × ℙ_{F}^{n}) = I(X). By Theorem 3.3, applied to g, η_{p}(Y) = 0 ∈ ℤ/I(X), a contradiction.

(2) By the degree formula, applied to f, and Lemma 2.1, the degree deg(f) is not divisible by p. Hence the generic point of Y determines a point over F (X) of degree = deg(f).

Remark 3.7. The first statement of Theorem 3.6(1) shows that a variety Y cannot be “compressed” to a variety X of smaller dimension if η_{p}(Y) ≠ 0 ∈ ℤ/I(X).

One can give another definition of η_{p}(X) by using Chern class operations as defined in [2, Ch. 3] for arbitrary (not necessarily smooth) varieties. For any vector bundle E over a variety V we have Chern class operations

c_{p}(L)∩? : CH_{k}(V) ^{}CH_{k−p}(V), u ↦ c_{p}(E) ∩ u.

In particular, the operation c_{1}(L)∩? for a linear vector bundle L over V is the intersection with the Cartier divisor associated to L.

By [2, Ex. 3.3.2], the class of l_{X} in CH_{0}(C^{p}X \ \overline{X}) is equal to

c_{pd}(L_{X}^{⊕pd}) ∩ [C^{p}X \ \overline{X}].

By Whitney formula [2, Th. 3.2 (e)],

c_{pd}(L_{X}^{⊕pd}) = c_{1}(L_{X})^{pd}.

Thus, we can give an equivalent definition:

η_{p}(X) = deg(c_{1}(L_{X})^{pd} ∩ [C^{p}X \ \overline{X}]) ∈ ℤ/I(X).

One can give another proof of Theorem 3.1 using standard properties of Chern class operations (projection formula, inverse image under a flat morphism) given in [2, Th. 3.2].

4. Computation of η_{2} for a quadric

Let Q = Q(V, q) be anisotropic smooth projective quadric of dimension d, so that I(Q) = 2ℤ.

Proposition 4.1.

η_{2}(Q) = \begin{cases}1 + 2ℤ, & if d = 2^{k} − 1 for some k, \\ 2ℤ, & otherwise. \end{cases}

Proof. Each pair of distinct points in Q determines a line in ℙ(V), i.e. a plane in V. Thus we have a rational morphism

α : C^{2}Q ^{}Gr(2, V).

Clearly, α is a birational isomorphism! Indeed, let U be the open subvariety in C^{2}Q consisting of all pairs of points ([v], [u]) such that [v] ≠ [u] and the restriction of the quadratic form q on the 2-dimensional subspace generated by v and u is nondegenerate. Then the restriction α|_{U} is an open immersion identifying U with the open subvariety of Gr(2, V) consisting of all planes W ⊂ V such that the restriction q|_{W} is nondegenerate. The inverse rational morphism α^{−1} takes a plane W ⊂ V to the intersection ℙ(W) ∩ Q. If q|_{W} is nondegenerate, this intersection is an effective 0-cycle of degree 2, i.e. is a point of C^{2}X.

Remark 4.2. If dim(Q) = 1, i.e. if Q is a conic, α is an isomorphism between C^{2}Q and Gr(2, V) = ℙ(V^{∗}). In the split case this isomorphism looks as follows: C^{2}ℙ_{F}^{1} ≃ ℙ_{F}^{2} .

Let E be the canonical linear bundle over Gr(2, V) (the second exterior power of rank 2 tautological vector bundle). Denote by L′ the restriction of the linear bundle L_{Q} to the open subvariety U ⊂ C^{2}Q.

Lemma 4.3. (α|_{U})^{∗}(E) ≃ L′.

Proof. Let U′ be the inverse image of U under the natural morphism Q^{2} → C^{2}Q. We have the following morphism of vector bundles:

β : U′ × 𝔸_{F}^{1} → E, ([v], [u], t) ↦ (⟨v, u⟩, t \frac{v ∧ u}{b(v, u)})

where b is the polar form of q. The action of G = ℤ/2ℤ on U′ × 𝔸_{F}^{1} by ([v], [u], t) ↦ ([u], [v], −t) commutes with the trivial action of G on E. Hence β induces an isomorphism L′ → E over α|_{U}.

Clearly, 2ℤ = I(Q) = I(C^{2}Q \ U). Lemma 4.3 and the commutativity of the diagram

CH_{0}(Gr(2, V)) \overline{α|_{U}^{∗}_{}}CH_{0}(U)

[figure: ℤ; deg; ℤ/2ℤ; deg]

imply that

η_{2}(Q) = deg(c_{1}E)^{2d} + 2ℤ.

Let

i : Gr(2, V) ↪ ℙ(∧^{2}V)

be the canonical closed embedding. The vector bundle E on Gr(2, V) is the inverse image under i of the tautological vector bundle on ℙ(∧^{2}V). Thus, the degree deg(c_{1}E)^{2d} is equal to the degree of the subvariety i(Gr(2, V)) in the projective space ℙ(∧^{2}V). It is the Catalan number [2, Ex. 14.7.11]

\frac{1}{d + 1}\binom{2d}{d}.

One easily checks that this number is odd iff d = 2^{k} − 1 for some k.

Theorem 4.4. Let Q = Q(V, q) be a smooth projective quadric of dimension d ≥ 2^{k} − 1 and let X be a variety over F such that I(X) ⊂ 2ℤ and X has a point over F (Q). Then

(1) dim(X) ≥ 2^{k} − 1.

(2) If dim(X) = 2^{k} − 1, then Q has a point over F (X).

Proof. Let Q′ be a subquadric in Q of dimension 2^{k} − 1. Clearly, there is a rational morphism f : Q′ → X.

(1) Since η_{2}(Q′) = 1 + I(Q′) ≠ 0 ∈ ℤ/I(X), by Theorem 3.6(1),

dim(X) ≥ dim(Q′) = 2^{k} − 1.

(2) By Theorem 3.6(2), Q′ has a point of odd degree over F (X). By Springer’s Theorem, Q′ and hence Q has a point over F (X).

Remark 4.5. One can replace the condition that X has a point over F (Q) by the weaker one: X has an odd degree 0-cycle over F (Q).

Corollary 4.6. (Hoffmann) Let Q_{1} and Q_{2} be two anisotropic quadrics. If dim(Q_{1}) ≥ 2^{k} − 1 and Q_{2} is isotropic over F (Q_{1}), then dim(Q_{2}) ≥ 2^{k} − 1.

Corollary 4.7. (Izhboldin) Let Q_{1} and Q_{2} be two anisotropic quadrics. If dim(Q_{1}) = 2^{k} − 1 and Q_{2} is isotropic over F (Q_{1}), then Q_{1} is isotropic over F (Q_{2}).

5. Computing η_{2} for smooth varieties

Let X be a smooth proper variety. Denote by c′ the total characteristic class opposite to the Chern class c, i.e.

1 + c′_{1} + c′_{2} + ⋯ = (1 + c_{1} + c_{2} + …)^{−1}.

In other words, c′_{d}(E) = c_{d}(−E) for any vector bundle over X. The class c′_{d} is known as the Segre class (denoted s_{d} in [2, Ch. 3]; by the letter s_{d} we denote the additive class of degree d).

Denote by T_{X} the tangent bundle over X.

Theorem 5.1. Let X be a proper smooth variety of dimension d. Then the degree of the 0-cycle c′_{d}(T_{X}) = c_{d}(−T_{X}) is even and

η_{2}(X) = \frac{deg c′_{d}(T_{X})}{2} ∈ ℤ/I(X).

Proof. We would like first to compactify smoothly C^{2}X \ \overline{X} and extend the line bundle L_{X} to the compactification. Note that C^{2}X is smooth only if dim(X) = 1.

Let W be the blow up of the diagonal X in X^{2}. The G-action on X^{2} extends to one on W. The subvariety W^{G} coincides with the exceptional divisor

ℙ((T_{X} ⊕ T_{X})/T_{X}) = ℙ(T_{X}).

Since W^{G} is of codimension 1 in W, W/G is smooth and therefore can be taken as a smooth compactification of C^{2}X \ \overline{X}.

Now we would like to construct a canonical extension L′ of L_{X} to the whole W/G. There is a canonical linear bundle O_{W} (1) over the blow up W with the induced G-action. The group G acts by −1 on the restriction O_{ℙ(T_{X})} of O_{W} (1) on the exceptional divisor ℙ(T_{X}). The restriction of O_{W} (1) to the complement of the exceptional divisor W \ ℙ(T_{X}) ≃ X^{2} \ X is the trivial vector bundle with the trivial G-action. Now we modify the G-action on O_{W} (1) by −1. New G-action of the restriction of O_{W} (1) on the exceptional divisor ℙ(T_{X}) is trivial, hence O(1) descends to a linear vector bundle L′ on W/G which is a desired extension of L_{X} on C^{2}X \ \overline{X}.

Now we compute deg c_{1}(L′)^{2d}. At one hand, since L_{X} is the restriction of L′ on C^{2}X \ \overline{X}, we have

deg c_{1}(L′)^{2d} = deg c_{1}(L_{X})^{2d} = η_{2}(X) ∈ ℤ/I(X).

On the other hand, let

i : ℙ(T_{X}) ↪ W

be the closed embedding and

p : W ^{}W/G

the natural projection. We have

p^{∗}c_{1}(L′) = c_{1}(O_{W} (1)) = [ℙ(T_{X})] = i_{∗}(1) ∈ CH^{1}(W).

By the projection formula,

c_{1}(O_{W} (1))^{2d} = i_{∗}(1) · c_{1}(O_{W} (1))^{2d−1} = i_{∗}(c_{1}(O_{ℙ(T_{X})}(1))^{2d−1}) ∈ CH_{0}(W).

Let q : ℙ(T_{X}) → X be the natural morphism. The class

q_{∗}(c_{1}(O_{ℙ(T_{X})}(1))^{2d−1}) ∈ CH_{0}(X)

is the Segre class c′_{d}(T_{X}) [2, Ch. 3]. Finally,

2 deg c_{1}(L′)^{2d} = deg c_{1}(O_{W} (1))^{2d} = deg c_{1}(O_{ℙ(T_{X})}(1))^{2d−1} = deg c′_{d}(T_{X}).

Remark 5.2. If p > 2, the blow up W of the diagonal in X^{p} does not have smooth orbit space W/G and the linear bundle L_{X} cannot be extended to a linear bundle on W/G.

Remark 5.3. The number deg c_{d}(−T_{X}) does not change under field extensions and hence can be computed over algebraically closed fields.

Remark 5.4. The Theorem shows that the degree m of the cycle c_{d}(−T_{X}) is always even. Hence this cycle defines a cycle on the symmetric square C^{2}X of degree \frac{m}{2}. In the proof we construct this class “canonically” (it is c_{1}(L_{X})^{2d}). The degree formula follows from “canonical” nature of this class. One should study other cases of divisibility of characteristic numbers.

Example 5.5. Let Q be again a smooth projective quadric Q(V, q) and let i : Q ↪ ℙ(V) be the embedding. We have the following exact sequence of vector bundles

0 ^{}T_{Q}  ^{}i^{∗}T_{ℙ(V)}  ^{}i^{∗}O_{ℙ(V)}(2) ^{}0.

A nasty calculation gives the Catalan number again:

\frac{1}{2} c_{d}(−T_{Q}) = \frac{(−1)^{d}}{d + 1} \binom{2d}{d}.

Can one get this computation using Gr(2, V)?

6. Computing η_{2} for projective spaces

Proposition 6.1. Let X = ℙ_{F}^{d}. Then

deg c_{d}(−T_{X}) = (−1)^{d}\binom{2d}{d}.

Proof. It is known that [−T_{X}] = −(d + 1)[O_{X}(1)] + [O_{X}]. We can compute the total Chern class:

c(−T_{X}) = \frac{1}{c(O_{X}(1))^{d+1}} = \frac{1}{(1 + t)^{d+1}} .

Hence

deg c_{d}(−T_{X}) = \binom{−d − 1}{d} = (−1)^{d}\binom{2d}{d}.

Let v_{p} be the p-adic discrete valuation and s_{p}(a) be the sum of digits of a written in base p.

Corollary 6.2. Let X be a Severi-Brauer variety of dimension d. Then

v_{2}(deg c_{d}(−T_{X})) = s_{2}(d), η_{2}(X) = 2^{s_{2}(d)−1}.

Corollary 6.3. (Karpenko) Let A be a division algebra with orthogonal involution σ, Y the Severi-Brauer variety of A. Then the involution σ is anisotropic over F (Y).

Proof. Let X = I(A, σ) ⊂ Y be the involution variety, deg(A) = 2^{k}. Then I(X) ⊂ I(Y) = 2^{k}ℤ. If σ is isotropic over F (Y) then there is a rational morphism Y → X. By Corollary 6.2,

v_{2}(deg c_{d}(−T_{Y})) = s_{2}(2^{k} − 1) = k,

hence η_{2}(Y) = 2^{k−1} is nontrivial in ℤ/I(X). A contradiction by Theorem 3.6(1).

Remark 6.4. In fact, we proved that the Severi-Brauer variety of a division algebra of degree 2^{k} cannot be compressed to a variety X of smaller dimension with I(X) = 2^{k}ℤ.

Remark 6.5. Let p be any prime integer. We will see later (Corollary 10.6) that for a Severi-Brauer variety X of dimension d divisible by p − 1,

η_{p}(X) = p^{\frac{sp(d)}{p−1} −1} ∈ ℤ/I(X).

As in Corollary 6.3 one can prove that the Severi-Brauer variety of a division algebra of degree p^{k} cannot be compressed to a variety X of smaller dimension with I(X) = p^{k}ℤ.

7. Todd class

The total Todd class td is the rational characteristic class

td = 1 + \frac{1}{2}c_{1} + \frac{1}{12}(c_{1}^{2} + c_{2}) + \frac{1}{24}(c_{1}c_{2}) + ….

Lemma 7.1. [3, Lemma 1.7.3] The denominator of the Todd class td_{d} is equal to

∏_{p prime} p^{[ \frac{d}{p−1} ]}.

It follows from the RR Theorem that for a smooth proper variety X of dimension d,

td_{d}(X) =^{def} deg td_{d}(T_{X}) = χ(X) = ∑_{i=1}^{d}(−1)^{i} dim H^{i}(X, O_{X}) ∈ ℤ

is the Euler characteristic of X.

The structure morphism p^{X} : X → Spec F induces the direct image

p_{∗}^{X} : K_{0}(X) ^{}K_{0}(F) = ℤ.

We have p_{∗}([O_{X}]) = χ(X).

Denote by J(X) the image of K_{0}(X)^{(1)} under p_{∗}. Clearly, I(X) ⊂ J(X) and this is an equality if dim X = 1.

Theorem 7.2. For a morphism f : Y → X of proper smooth varieties of dimension d,

td_{d}(Y) = deg(f) · td_{d}(X) ∈ ℤ/J(X).

Proof. Clearly,

f_{∗}([O_{Y}]) − deg(f) · [O_{X}] ∈ K_{0}(X)^{(1)}.

We apply then p_{∗}^{X} and use p^{Y} = p^{X} ∘ f.

Example 7.3. Let Y be the Severi-Brauer variety corresponding to a central simple algebra of prime degree p. One has J(Y) = pℤ and td_{p−1}(Y) = 1. Hence Y cannot be compressed to a variety X of smaller dimension with J(Y) = pℤ.

Example 7.4. Let X be the product of n Severi-Brauer varieties corresponding to central simple algebras A_{1}, A_{2}, …, A_{n} of prime degree p. Assume that these algebras are linearly independent in the Brauer group of F. Then J(X) = pℤ. Let L/F be a field extension such that

1. tr. deg.(L/F) ≤ n(p − 1);

2. L splits all the A_{i}.

3. F_{sep}L/F_{sep} is purely transcendental.

Claim: The degree tr. deg.(L/F) is exactly n(p − 1), the field L can be embedded into F (X) over F and for any such an embedding the degree [F (X) : L] is prime to p.

Proof: Let Y be a projective variety such that L = F (Y). We may assume that dim(Y) = n(p − 1) by replacing Y by Y × ℙ^{k} for appropriate k. Since F (Y) splits all the A_{i}, there is a rational morphism f : Y → X. Replacing Y by the graph of this morphism we may assume that f is regular. Since Y is rational over F_{sep}, td_{n(p−1)}(Y) = 1. By the degree formula, deg(f) is not divisible by p.

In particular, X does not contain proper subvarieties rational over F_{sep}.

Example 7.5. Let L/F be a quadratic field extension and Q be a quaternion algebra over L. Consider X = R_{L/F} (C) where C is the conic over L corresponding to Q. Clearly td_{2}(X) = 1 since X is rational over F_{sep}. Assume that the algebra A = cor_{L/F} (Q) of degree 4 over F is a division algebra. We show that J(X) = 4ℤ. Indeed,

K_{0}(X) = K_{0}(F) ⊕ K_{0}(A) ⊕ K_{0}(Q).

The degree

K_{0}(X) ^{}K_{0}(F) = ℤ

is the identity on the first component and trivial on the others (look at the split case). On the other hand, the map to the generic point is given by the norms. Hence, the first component of any element in K_{0}(X)^{(1)} is divisible by 4.

8. Complex cobordisms

Consider the formal polynomial ring R = ℤ[c_{1}, c_{2}, …] generated by the Chern classes. We think of c_{d} having degree 2d, so that R is a graded ring. Denote by R_{d} the component of R of degree 2d. It is a free abelian group generated by c_{d}, …, c_{1}^{d}. We call the elements of R_{d} the integral characteristic classes of degree d. The rational characteristic classes of degree d are the elements of R_{d} ⊗ ℚ.

Let E be a vector bundle over a variety X of dimension d. Then for any c ∈ R_{d} ⊗ ℚ the value c(E) is a rational 0-cycle in CH_{0}(X) ⊗ ℚ. If X is proper, we can compute the degree deg c(E) ∈ ℚ.

Denote by R′_{d} the subgroup in R_{d} ⊗ ℚ consisting of all rational characteristic classes c such that deg c(−T_{X}) ∈ ℤ for all smooth proper varieties X of dimension d over F. Clearly, R_{d} ⊂ R′_{d}.

Example 8.1. We know that \frac{1}{2} c_{d} ∈ R′_{d}, where c is the Chern class, defines an element of R′_{d}/R_{d} of exponent 2. Let td′ be the inverse td^{−1} of the Todd class. The class td′_{d} is another example of an element of R′_{d}.

Let t ∈ R′_{d}. For a smooth proper variety X of dimension d we can consider the class t(X) = deg t(−T_{X}) + I(X) ∈ ℤ/I(X). If t ∈ R_{d} is an integral class, then t(E) ∈ CH_{0}(X) for any vector bundle E over X, hence deg t(E) ∈ I(X) and therefore t(X) = 0 ∈ ℤ/I(X). Thus, for any smooth proper variety X of dimension d we have a well defined group homomorphism

R′_{d}/R_{d} ^{}ℤ/I(X), t + R_{d} ↦ t(X).

Problem 8.2. Determine all classes t + R_{d} which admit the degree formula, i.e. for a morphism f : Y → X of smooth proper varieties,

t(Y) = deg(f) · t(X) ∈ ℤ/I(X).

Probably in some cases one has to enlarge the group I(X) (see the degree formula for the Todd class).

Example 8.3. The classes \frac{1}{2} c_{d} and td′_{d} admit the degree formula.

First, we would like to determine the factor group R′_{d}/R_{d}. Let Ω_{∗}^{U} = π_{∗}(MU) be the complex bordism ring. The Hurewicz ring homomorphism

h_{d} : Ω_{2d}^{U} ^{}H_{2d}(BU, ℤ)

is injective [5, Cor. 20.26]. Using duality between H_{∗}(BU, ℤ) and H^{∗}(BU, ℤ) one can determine h by the formula

[X] ↦ (t ↦ ⟨t(−T_{X}), σ_{X}⟩)

where t ∈ H^{2d}(BU, ℤ), t(−T_{X}) ∈ H^{2d}(X, ℤ) and σ_{X} ∈ H_{2d}(X, ℤ) is the fundamental class of X.

The ring H_{∗}(BU, ℤ) is a graded polynomial ring ℤ[b_{1}, b_{2}, …], deg b_{d} = 2d. The ring Ω_{∗}^{U} is also a polynomial ring ℤ[M_{1}, M_{2}, …] where the generators M_{i} can be chosen such that [4, p. 129]

h_{i}(M_{i}) = \begin{cases}p(b_{i} + \{decomposable terms\}), & if i = p^{k} − 1 \\ & for some prime p and integer k > 0, \\ b_{i} + \{decomposable terms\}, & otherwise. \end{cases}

Consider the following elements in H_{∗}(BU, ℤ):

x_{i} = \begin{cases}\frac{h_{i}(M_{i})}{p}, & if i = p^{k} − 1 \\ & for some prime p and integer k > 0, \\ h_{i}(M_{i}), & otherwise. \end{cases}

By [4, p. 130], H_{∗}(BU, ℤ) is the polynomial ring ℤ[x_{1}, x_{2}, …].

Also set

r_{i} = \begin{cases}p, & if i = p^{k} − 1 \\ & for some prime p and integer k > 0, \\ 1, & otherwise. \end{cases}

For any partition α: d = i_{1} + i_{2} + ⋯ + i_{k}, i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k} set

x_{α} = x_{i_{1}}x_{i_{2}} … x_{i_{k}} ∈ H_{2d}(BU, ℤ),
r_{α} = r_{i_{1}}r_{i_{2}} … r_{i_{k}} ∈ ℤ.

Proposition 8.4. ([4, Cor., page 129])

Coker h_{d} = ∐_{α}(ℤ/r_{α}ℤ) x_{α},

where the coproduct is taken over all partitions α of d.

Corollary 8.5. The exponent of the group Coker h_{d} coincides with the denominator of the Todd class td_{d}.

Proof. Let p be a prime integer, k = [\frac{d}{p−1}]. Clearly, for any partition α of d, v_{p}(x_{α}) ≤ k and there is a partition α : d = (p − 1) + (p − 1) + … with v_{p}(x_{α}) = k. The statement now follows from Lemma 7.1.

Now we dualize h over ℤ. The group H^{2d}(BU, ℤ) is canonically isomorphic to R_{d} since by [5, Cor. 16.11],

H^{∗}(BU, ℤ) = ℤ[c_{1}, c_{2}, …].

Thus, the dual of h_{d} is the embedding

R_{d} ↪ Hom(Ω_{2d}^{U}, ℤ)

being an isomorphism after tensoring with ℚ. This map takes a characteristic class t of degree 2d to a homomorphism

Ω_{2d}^{U} ^{}ℤ, [X] ↦ ⟨t(−T_{X}), σ_{X}⟩. (1)

If X is a smooth proper variety and we consider characteristic classes with values in Chow groups of X the homomorphism (1) takes [X] to deg t(−T_{X}). Thus, we can identify the group Hom(Ω_{2d}^{U}, ℤ) with R′_{d}.

The pairing

H_{2d}(BU, ℤ) ⊗ R′_{d} ^{}ℚ

induces an isomorphism

R′_{d}/R_{d} ≃ Hom(Coker h_{d}, ℚ/ℤ) = (Coker h_{d})^{∗}.

Thus, the structure of the finite group R′_{d}/R_{d} is similar to that in Proposition 8.4.

Example 8.6. Let d = 1. We have R_{1} = ℤc_{1}, M_{1} = 2b_{1} is the generator of Ω_{2}^{U} with M_{1} = −[ℙ^{1}] by [5, Ex. 16.46]. Hence R′_{1} = \frac{1}{2}ℤc_{1} and the class td_{1} = \frac{1}{2}c_{1} + R_{1} generates R′_{1}/R_{1}:

R′_{1}/R_{1} = (ℤ/2ℤ)td_{1} = (ℤ/2ℤ)(\frac{1}{2}c_{1}) = (ℤ/2ℤ)(\frac{1}{2}s_{1}) = (ℤ/2ℤ)(\frac{1}{2}c′_{1}),

where c_{1} = s_{1} = −c′_{1}. The generator admits the degree formula.

Example 8.7. Let d = 2. We have R_{2} = ℤc_{2} ⊕ ℤc_{1}^{2}. By [5, Ex. 16.46], the generators of Ω_{4}^{U} can be chosen as follows:

M_{2} = −[ℙ^{2}] = 3b_{2} − 6b_{1}^{2}, M_{1}^{2} = [ℙ_{1} × ℙ_{1}] = 4b_{1}^{2}.

Hence the group R′_{2}/R_{2} which is dual to Coker h_{2}, is cyclic of order 12,

R′_{2}/R_{2} = (ℤ/12ℤ)td′_{2},

where

td_{2} = \frac{1}{12}(c_{1}^{2} + c_{2}), td′_{2} = \frac{1}{12}(2c_{1}^{2} − c_{2}).

Also:

\frac{1}{2}c′_{2} + R_{2} = 6td_{2} + R_{2},

\frac{1}{3}s_{2} + R_{2} = 4td_{2} + R_{2}

both admit degree formulas. What about td_{2}?

Example 8.8. Let d = 3. We have R_{3} = ℤc_{3} ⊕ ℤc_{1}c_{2} ⊕ ℤc_{1}^{3}. Let Q_{3} be a smooth 3-dimensional quadric. One has

h(Q_{3}) = 6b_{3} − 10b_{1}^{3},
h(ℙ^{3}) = −4b_{3} + 20b_{1}b_{2} − 20b_{1}^{3}.

One can choose the following generators of Ω_{6}^{U}:

M_{3} = [Q_{3}] + [ℙ^{3}], M_{1}M_{2} = [ℙ^{1} × ℙ^{2}], M_{1}^{3} = [ℙ^{1} × ℙ^{1} × ℙ^{1}].

One finds

R′_{3}/R_{3} = (ℤ/2ℤ)(\frac{1}{2}c′_{3}) ⊕ (ℤ/2ℤ)(\frac{1}{2}c_{1}^{3}) ⊕ (ℤ/24ℤ)td′_{3}.

9. General problem

Let X be a projective variety of dimension d over F. For any characteristic class t ∈ R′_{d} denote by I_{t}(X) the smallest subgroup in ℤ containing I(X) such that for any morphism f : Y → X, where Y is a projective variety of dimension d, one has

t(Y) ≡ deg(f) · t(X) (mod I_{t}(X)).

Clearly, I_{t}(X) depends only on t modulo R_{d}. Denote by o_{t} the l.c.m. of the orders of t(X) in ℤ/I_{t}(X) for all projective X of dimension d. Thus, if o_{t} ≠ 1, there is a nontrivial degree formula involving the class t.

Problem 9.1. Determine the function

R′_{d}/R_{d} ^{}ℤ, t + R_{d} ↦ o_{t}.

Clearly, o_{t} divides the order of t + R_{d} in R′_{d}/R_{d}.

Example 9.2. Let d = 1. The group R′_{1}/R_{1} is cyclic of order 2 with the generator t = \frac{c_{1}}{2} . For a conic curve X one has t(X) = 1 + 2ℤ ∈ ℤ/2ℤ, hence o_{t} = 2.

Example 9.3. Let d = 2. The group R′_{2}/R_{2} is cyclic of order 12 with the generator t = td′_{2}. Example of a Severi-Brauer variety of a csa of degree 3 shows that 3|o_{t} and Example 7.5 shows that 4|o_{t}. Thus, o_{t} = 12.

10. Steenrod operations

Let p be a prime number. We consider the action of the Steenrod algebra modulo p on the Thom class T in the cohomology ring

H^{∗}(MU, ℤ/pℤ).

For an operation A, denote by c(A) the image of A(T) under the canonical isomorphism

H^{∗}(MU, ℤ/pℤ) ≃ H^{∗}(BU, ℤ/pℤ) = ℤ/pℤ[c_{1}, c_{2}, …].

Example 10.1. Let P^{i} be the Steenrod power. Then c(P^{i}) = (b^{−1})_{i(p−1)} where b is the total characteristic class defined by

b = (1 + t_{1}^{p−1})...(1 + t_{n}^{p−1})

where

c = (1 + t_{1})...(1 + t_{n})

is the total Chern class. In particular, b = c, the Chern class, if p = 2.

Example 10.2. Let q_{i} be the Quillen operation of degree p^{i} −1. Then c(q_{i}) = s_{p^{i}−1} is the additive class.

Let X be a smooth variety over a field F. In [1], P. Brosnan defines Steenrod operations

P^{i} : CH_{∗}(X) ^{}CH_{∗−i(p−1)}(X)/p

which satisfy the following properties:

• (functoriality with respect to proper morphisms) Let f : Y → X be a proper morphism of smooth varieties. Then the following diagram commutes [1, Prop. 8.11]:

CH_{∗}(Y) \overline{P^{i}_{}}CH_{∗−i(p−1)}(Y)/p

[figure: CH_{∗}(X); f_{∗}; P^{i}; CH_{∗−i(p−1)}(X)/p.; f_{∗}]

• Let dim(X) = d. The image of the the class [X] under

P^{i} : CH_{d}(X) ^{}CH_{d−i(p−1)}(X)/p

equals b_{i(p−1)}(−T_{X}) by [1, Prop. 8.4].

Theorem 10.3. (1) Let X be a smooth variety of dimension d divisible by p − 1. Then deg b_{d}(−T_{X}) is divisible by p, i.e. \frac{1}{p}b_{d} is a rational class in R′_{d}.

(2) For a proper morphism f : Y → X of smooth varieties of dimension d divisible by p − 1,

f_{∗}(b_{d}(−T_{Y})) ≡ deg(f) · b_{d}(−T_{X}) (mod p CH_{0}(X)).

Proof. Both statements follow from the properties above applied to the operation P^{i} for i = \frac{d}{p−1} and morphisms X → Spec F and f : Y → X.

In view of Example 10.1, this Theorem implies Theorem 5.1 in the case p = 2 for smooth projective varieties.

Remark 10.4. Let X be a smooth variety of dimension d divisible by p − 1. One can show that

η_{p}(X) = \frac{deg b_{d}(−T_{X})}{p} ∈ ℤ/I(X).

Thus, Theorem 10.3 implies Theorem 3.1 for smooth varieties.

We generalize now Proposition 6.1:

Proposition 10.5. Let X = ℙ^{d} where d = i(p − 1). Then

deg b_{d}(−T_{X}) = (−1)^{i}\binom{pi}{i}.

Corollary 10.6. Let X be a Severi-Brauer variety of dimension d divisible by p − 1. Then

η_{p}(X) = p^{\frac{sp(d)}{p−1} −1}.

Proof. Let d = i(p − 1). Then

v_{p}\binom{pi}{i} = \frac{s_{p}(d) + s_{p}(i) − s_{p}(pi)}{p − 1} = \frac{s_{p}(d)}{p − 1}.

11. General formula

We would like to define things algebraically and give topological explanations.

Let H_{∗}ℤ[b_{1}, b_{2}, …] be the polynomial ring. We assume that H_{∗} is graded in such a way that b_{i} is of degree i.

Explanation: In topology, it is the homology ring H_{∗}(BU, ℤ) = H_{∗}(MU, ℤ) (the difference is that the degree of b_{i} in topology is 2d).

For any partition α: d = i_{1} + i_{2} + ⋯ + i_{k}, i_{1} ≤ i_{2} ≤ ⋯ ≤ i_{k} set

b^{α} = b_{1} … b_{i_{1}}b_{i_{1}+1}^{2} … b_{i_{2}}^{2}b_{i_{2}+1}^{3} … b_{i_{k}}^{k}.

The elements b^{α} for a basis of the polynomial ring over ℤ, and more precisely, the b_{α} with |α| = d form a basis of H_{d}.

Consider another polynomial ring H^{∗} = ℤ[c_{1}, c_{2}, …] with the same grading deg(c_{i}) = i. For any partition α as above we define the symmetric polynomial

P_{α} = ∑ X_{1}^{α_{1}}X_{2}^{α_{2}} … X_{k}^{α_{k}}

and set

c_{α} = P_{α}(c_{1}, c_{2}, …, c_{k}).

The c_{α} with |α| = d form a basis of H^{d}.

We consider a pairing between H_{d} and H^{d} such that the b^{α} and c_{α} form dual bases.

Explanation: H_{∗} is the cohomology ring H^{∗}(BU, ℤ) = H^{∗}(MU, ℤ) with the standard pairing with the homology ring.

Let

exp(x) = x + b_{1}x^{2} + b_{2}x^{3} + …

be the formal power series over the ring H_{∗} and

log(x) = x + m_{1}x^{2} + m_{2}x^{3} + …

be the formal inverse of exp. Clearly,

H_{∗} = ℤ[m_{1}, m_{2}, …]

is a graded polynomial ring. For example,

m_{1} = −b_{1},
m_{2} = 2b_{1}^{2} − b_{2},
m_{3} = −5b_{1}^{3} + 5b_{1}b_{2} − b_{3}, …

Consider the formal power series in two variables

μ(x, y) = exp(log x + log y)

over the ring H∗. Clearly, μ is a commutative formal group law over H_{∗}. The map exp is an isomorphism of the additive group law x + y and the group law μ. Denote the coefficient of μ by a_{ij} ∈ H_{i+j−1}, i.e.

μ(x, y) = x + y + ∑_{i≥1,j≥1} a_{ij}x^{i}y^{j}.

For example,

a_{11} = 2b_{1},
a_{12} = 3b_{2} − 2b_{1}^{2},
a_{13} = 4b_{3} − 8b_{1}b_{2} + 4b_{1}^{3},
a_{22} = 6b_{3} − 6b_{1}b_{2} + 2b_{1}^{3}, …

Denote by L the graded subring in H_{∗} generated by the coefficients a_{ij}. By definition, the group law μ is defined over L. It is the universal formal group law. The ring L is called the Lazard ring.

Let X be an algebraic proper smooth variety over F of dimension d. Denote by [X] the element in H_{d} such that

⟨c_{α}, [X]⟩ = deg c_{α}(−T_{X})

for any partition α with |α| = d. It turns out that [X] ∈ L_{d} (algebraic proof?).

Example 11.1. [ℙ^{d}] = (d + 1)m_{d}.

Explanation: The Lazard ring is identified with the complex cobordism ring Ω_{∗}^{U}. The map X ↦ [X] is the injective Hurewicz homomorphism

h : Ω_{∗}^{U} ^{}H_{∗}.

The formula above looks as follows:

⟨c_{α}, h[X]⟩ = ⟨c_{α}(−T_{X}), σ_{X}⟩,

where σ_{X} is the fundamental class of X in H_{2d}(X, ℤ).

Let X be an algebraic proper smooth variety over F of dimension d. We define the (graded) ideal M(X) in L generated by the classes [Y] ∈ L for all smooth proper varieties Y with dim(Y) < d and such that there exists a morphism Y → X.

Conjecture 11.2. (General Degree Formula^{2}) For any morphism f : Y → X of proper smooth varieties of dimension d,

[Y] = deg(f) · [X] ∈ L_{d}/M(X)_{d}.

Let t : L_{d} → ℤ be a homomorphism. Applying t to the degree formula, we get a degree formula

t[Y] = deg(f) · t[X] ∈ ℤ/t(M(X)_{d}).

In section (8) we identified such homomorphisms t with rational characteristic classes in R′_{d}.

Example 11.3. Let t = \frac{1}{2}c_{d} (c is the Chern class). The group M(X)_{d} is generated by the classes [Y ×Z] of varieties of dimension d, where k = dim(Y) < d and there is a morphism Y → X. Let p and q be two projections of Y × Z on Y and Z respectively. Then

T_{Y ×Z} = p^{∗}(T_{Y}) ⊗ q^{∗}(T_{Z}).

The highest Chern class is multiplicative, hence

\frac{1}{2} deg c_{d}(−T_{Y ×Z}) = deg c_{k}(−T_{Y}) · \frac{1}{2} deg c_{d−k}(−T_{Z}).

Clearly, deg c_{k}(−T_{Y}) ∈ I(Y). Since there is a morphism Y → X, deg c_{k}(−T_{Y}) ∈ I(X). Thus t(M(X)_{d}) = I(X) and we get the degree formula 3.1 in the case of smooth varieties and p = 2.

Similar argument works for the class t = \frac{1}{p}b_{d} to give Theorem 3.1 for any prime p.

Example 11.4. Let t = td′_{d}. Since the Todd class is multiplicative, similar argument gives t(M(X)_{d}) = J(X) and one obtains the degree formula in Theorem 7.2.

[Y] = deg(f) · [X] ∈ L/M(X),

2More generally one can expect that for any morphism f : Y → X of proper smooth varieties (not necessarily of the same dimension) with X irreducible one has where deg(f) denotes the class of the generic fibre.

12. Landweber-Novikov operations

Let E be a ℂ-oriented spectrum. We have a natural isomorphism

E^{∗}(BU) ≃ E^{∗}(MU).

The ring E^{∗}(BU) is the polynomial ring over E^{∗}(pt) in the “Chern classes” c_{1}, c_{2}, . . .. The corresponding elements in E^{∗}(MU) we denote s_{1}, s_{2}, . . .. For any partition α of d one can define in a standard way the elements s_{α} ∈ E^{2d}(MU).

Now set E = MU; we get the elements s_{α} ∈ MU^{2d}(MU) corresponding to the Conner-Floyd classes c_{α} ∈ MU^{2d}(BU). The class s_{α} can be considered as an operation

MU ^{}S^{2|α|}MU.

In particular, s_{α} acts on L_{∗} = MU^{−∗}(pt) by endomorphisms

L_{n} ^{}L_{n−|α|}.

Example 12.1. The operations s_{α} act nicely on the classes of projective spaces:

s_{α}[ℙ^{n}] = ⟨c_{α}, b^{−n−1}⟩[ℙ^{n−|α|}],

where b = ∑_{i=0}^{∞} b_{i}.

Example 12.2. The action of s_{α} on L_{∗} extends (uniquely) to an action on H_{∗}:

s_{α}(b) = ∑_{i≥0}⟨c_{α}, b_{i}⟩b^{i+1},

s_{α}(xy) = ∑_{(β,γ)=α}(s_{β}x)(s_{γ}y).

Fix a dimension n. For any partition α choose an element

x_{α} ∈ MU^{2n−2|α|}(pt) = L_{|α|−n} ⊂ H_{|α|−n}.

We view the multiplication by x_{α} as an operation

MU ^{}S^{2n−2|α|}MU.

We can consider the infinite sum

∑_{α} x_{α}s_{α} (2)

as an operation

MU ^{}S^{2n}MU.

Theorem 12.3. (Landweber-Novikov) Any cohomology operation on the spectrum MU is of the form (2) for uniquely determined elements x_{α}.

Conjecture 12.4. For any smooth proper variety X, the ideal M(X) in L is invariant under Landweber-Novikov operations.^{3}

3One may hope to prove these conjectures using Voevodsky’s stable homotopy theory of schemes.

References

[1] P. Brosnan, Steenrod operations in chow theory, http://www.math.uiuc.edu/Ktheory/0370/ (1999).

[2] William Fulton, Intersection theory, Springer-Verlag, Berlin, 1984.

[3] Friedrich Hirzebruch, Topological methods in algebraic geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Translated from the German and Appendix One by R. L. E. Schwarzenberger, With a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, Reprint of the 1978 edition.

[4] Robert E. Stong, Notes on cobordism theory, Princeton University Press, Princeton, N.J., 1968, Mathematical notes.

[5] Robert M. Switzer, Algebraic topology—homotopy and homology, Springer-Verlag, New York, 1975, Die Grundlehren der mathematischen Wissenschaften, Band 212.