ROST’S DEGREE FORMULA

ALEXANDER MERKURJEV

Some parts of algebraic quadratic form theory and theory of simple algebras (with involutions) can be translated into the language of algebraic geometry.

Example 0.1. Let (V, q) be a quadratic form over a field F, Q the associated projective quadric in ℙ(V). Then q is isotropic iff Q(F) ≠ ∅.

1. Numbers n_{X} and n_{L}

Notation: “scheme” = separated scheme of finite type over a field F. “variety” = integral scheme over F, F (X) function field of X.

Let X be a scheme over F. For a closed point x ∈ X we define degree of x: deg(x) = [F (x) : F] and the number

n_{X} = gcd deg(x),

where the gcd is taken over all closed points of X.

Example 1.1. For an anisotropic quadric Q, n_{Q} = 2.

Example 1.2. For a Severi-Brauer variety S = SB(A), n_{S} = ind(A).

Proposition 1.3. (i) If for a scheme X over F, X(F) ≠ ∅, then n_{X} = 1; (ii) Let X and Y be two schemes. If there is a morphism f : Y → X, then n_{X} divides n_{Y}.

Proof. (ii) For a closed point y ∈ Y, the point x = f(y) is closed and deg(x) divides deg(y). □

Let L/F be a field extension, v a valuation on L over F with residue field F (v). We define degree of v: deg(v) = [F (v) : F] and the number

n_{L} = gcd deg(v),

where the gcd is taken over all valuations of finite degree.

Proposition 1.4. (i) Let X be a complete scheme over F. If X(L) ≠ ∅, then n_{X} divides n_{L}. (ii) Let X be a variety, L = F (X). If X is smooth, then n_{L} divides n_{X}. (iii) If a variety X is complete and smooth, then n_{X} = n_{L} where L = F (X). In particular, n_{X} is a birational invariant of a complete smooth variety X. (iv) If f : Y → X is a rational morphism of complete varieties with Y smooth, then n_{X} divides n_{Y}.

Date: June, 2001.

Proof. (i) Let O ⊂ L be the valuation ring. Since X is complete, a point in X(L) factors as

Spec L ^{}Spec O ^{f}^{}X.

Let m be the closed point in Spec O, x = f(m). Then F (x) ⊂ O/m = F (v) and therefore, deg(x) divides deg(v).

(ii) Let x ∈ X. Since x is smooth, there is a valuation v on L with residue field F (x), so that deg(v) = deg(x).

(iii) follows from (i) and (ii).

(iv) Let L = F (Y). Since X(L) ≠ ∅, by (i), n_{X} divides n_{L}. Finally, by (ii), n_{L} divides n_{Y}. □

2. Vector bundles and Chern classes

A vector bundle over a schemes X is a morphism E → X which is locally on X (in Zariski topology) is isomorphic to the projection X × 𝔸^{n} → X with linear transition functions.

Let f : V → W be a dominant morphism of integral schemes. We define degree of f by

deg(f) = \begin{cases}[F (V) : F (W)] & if dim V = dim W , \\ 0 & otherwise. \end{cases}

For a scheme X denote by CH_{n}(X) the Chow group of classes of cycles in X of dimension n. For a proper morphism f : Y → X there is push-forward homomorphism

f_{∗} : CH_{n}(Y) ^{}CH_{n}(X)

defined by

f_{∗}([V]) = deg(V → f(V)) · [f(V)].

The group CH_{0}(X) is generated by classes [x] of closed points x ∈ X. If X is complete, the push-forward homomorphism with respect to the structure morphism X → Spec F is called the degree map

deg : CH_{0}(X) ^{}ℤ, [x] ↦ deg(x) = [F (x) : F].

If X is a smooth scheme, the ring CH_{∗}(X) is a commutative ring with the identity 1 = [X].

For a vector bundle E → X there are operational Chern classes

c_{i}(E) : CH_{n}(X) ^{}CH_{n−i}(X), α ↦ c_{i}(E) ∩ α.

If X is a smooth variety, CH_{∗}(X) is a commutative ring and c_{i}(E) is the multiplication by c_{i}(E) ∩ [X]. The latter element we will also denote by c_{i}(E) Thus, c_{i}(E) ∈ CH_{d−i}(X) where d = dim X.

For a smooth X the Chern classes extend to a map

c_{i} : K_{0}(X) ^{}CH_{∗}(X).

In particular, for a vector bundle E → X, the classes c_{i}(−E) ∈ CH_{d−i}(X) are defined.

3. 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 scheme over F. The group G = ℤ/pℤ acts by cyclic permutations on the product

X^{p} = X × X × ⋯ × X.

The factor scheme 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 subscheme in C^{p}X, isomorphic to X. In particular, n\overline{X} = n_{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. Set

l_{X} = c_{1}(L_{X})^{pd} ∩ [C^{p}X \ \overline{X}] ∈ CH_{0}(C^{p}X \ \overline{X}).

Remark 3.1. The element l_{X} is known as the Euler class of the bundle L_{X}^{⊕pd}, i.e. l_{X} is the image of [X] under the composition

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

where s is the zero section of the vector bundle L_{X}^{⊕pd} and p^{∗} is the pull-back with respect to the (flat) morphism p : L_{X}^{⊕pd} → C^{p}X \ \overline{X}.

If X is projective, we have degree homomorphism

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

Thus, the degree of l_{X} defines an element η_{p}(X) ∈ ℤ/n_{X}ℤ.

Thus, for a projective scheme we have defined an invariant η_{p}(X) ∈ ℤ/n_{X}ℤ. Note that X is projective but not necessarily smooth scheme over F.

Lemma 3.2. p · l_{X} = 0 ∈ CH_{0}(C^{p}X \ \overline{X}). In particular,

p · η_{p}(X) = 0 ∈ ℤ/n_{X}ℤ.

Proof. Let

f : X^{p} \ X → C^{p}X \ \overline{X}

be the projection (of degree p). Since the linear bundle f^{∗}(L_{X}) is trivial, by the projection formula,

0 = f_{∗}(c_{1}(f^{∗}L_{X})^{pd} ∩ [X^{p} \ X]) = c_{1}(L_{X})^{pd} ∩ f_{∗}[X^{p} \ X] =
p · c_{1}(L_{X})^{pd} ∩ [C^{p}X \ \overline{X}] = p · l_{X}.
□

4. Degree formula

We prove two degree formulas due to M. Rost.

Theorem 4.1. (Regular Degree Formula) Let f : Y → X be a morphism of projective schemes of the same dimension d. Then n_{X} divides n_{Y} and

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

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,

n_{X} = n\overline{X} | nỸ | n\overline{Y} = n_{Y} .

We have open embedding

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

and proper morphism

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

of degree d^{p}.

Denote by L′ the restriction of the vector bundle L_{Y} on C^{p}Y \ Ỹ. Clearly, L′ is the inverse image of L_{X} with respect to i, L′ = i^{∗}(L_{X}). We have by projection formula,

i_{∗}j^{∗}(l_{Y}) = i_{∗}j^{∗}(c_{1}(L_{Y})^{pd} ∩ [C^{p}Y \ \overline{Y}]) =
i_{∗}(c_{1}(L′)^{pd} ∩ [C^{p}Y \ Ỹ]) = i_{∗}(c_{1}(i^{∗}L_{X})^{pd} ∩ [C^{p}Y \ Ỹ]) =

c_{1}(L_{X})^{pd} ∩ i_{∗}[C^{p}Y \ Ỹ] = deg(i) · (c_{1}(L_{X})^{pd} ∩ [C^{p}X \ \overline{X}]) = deg(f)^{p} · l_{X}. Note, that by Lemma 3.2,

deg(f)^{p} · l_{X} = deg(f) · l_{X}.

Finally, taking degree,

η_{p}(Y) = deg(l_{Y}) = deg(i_{∗}j^{∗}(l_{Y})) = deg(f) · η_{p}(X) ∈ ℤ/n_{X}ℤ.
□

Theorem 4.2. (Rational Degree Formula) Let f : Y → X be a rational morphism of projective varieties of dimension d with Y smooth. Then n_{X} divides n_{Y} and

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

Proof. By Proposition 1.4(iv), n_{X} divides n_{Y}. Let Y_{1} be closure of the graph of f in Y × X. Applying Theorem 4.1 to the birational isomorphism Y_{1} → Y (of degree 1), we get

η_{p}(Y_{1}) = η_{p}(Y) ∈ ℤ/n_{Y} ℤ.

On the other hand, applying Theorem 4.1 to the projection Y_{1} → X(of degree = deg(f)), we get

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

Thus,

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

Corollary 4.3. The class η_{p}(X) ∈ ℤ/n_{X}ℤ is a birational invariant of a smooth projective variety X.

Problem 4.4. How to define the invariant η_{p}(X) ∈ ℤ/n_{X}ℤ out of the function field L = F (X)? (Note that n_{X} = n_{L} by Proposition 1.4.)

Here is the main application:

Theorem 4.5. Let X and Y be two projective varieties with Y smooth. Assume that X has a rational point over the field F (Y). Then n_{X} divides n_{Y} and if η_{p}(Y) ≢ 0 (mod n_{X}), the following holds:

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

Proof. By assumption, there exists a rational morphism f : Y → X, hence n_{X} divides n_{Y} by Proposition 1.4(iv).

(i) Assume that m = dim(Y) − dim(X) > 0. Consider the composition

g : Y ^{f}^{}X ↪ X × ℙ_{F}^{m}.

Clearly, deg(g) = 0 and n_{X×ℙ_{F}^{m}} = n_{X}. By Theorem 4.2, applied to g,

η_{p}(Y) ≡ 0 (mod n_{X}),

a contradiction.

(ii) By the degree formula, applied to f, and Lemma 3.2, 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 4.6. The first statement of Theorem 4.5 shows that a variety Y cannot be “compressed” to a variety X of smaller dimension if

η_{p}(Y) ≠ 0 ∈ ℤ/n_{X}ℤ.

5. Computation of η_{2} for a quadric

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

Proposition 5.1.

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 U′ ⊂ 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 5.2. 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 tautological 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 5.3. (α|_{U})^{∗}(E) ≃ L′.

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

β : Ũ × 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 Ũ × 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}. □

At every point of “bad loci” C^{2}Q\U and Gr(2, V)\U′ the form q is isotropic. Hence the numbers n_{C^{2}Q\U} and n_{Gr(2,V)\U′} are even. Lemma 5.3 implies that

η_{2}(Q) ≡ deg(c_{1}(E))^{2d} (mod 2).

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). The first Chern class of the tautological vector bundle is negative of the class of a hyperplane section. Thus, the degree deg(c_{1}E)^{2d} is equal to (−1)^{d} times the degree of the subvariety i(Gr(2, V)) in the projective space ℙ(∧^{2}V). The later degree is known as the Catalan number

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

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

Theorem 5.4. 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 n_{X} is even and X has a point over F (Q). Then

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

Proof. Let Q → X be a rational morphism. Choose a subquadric Q′ ⊂ Q of dimension 2^{k} − 1 such that the composition f : Q′ ↪ Q → X is defined.

(i) Since η_{2}(Q′) ≠ 0 ∈ ℤ/2ℤ and hence in ℤ/n_{X}ℤ, by Theorem 4.5(1),

dim(X) ≥ dim(Q′) = 2^{k} − 1.

(ii) By Theorem 4.5(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). □

Corollary 5.5. (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 5.6. (Izhboldin) Let Q_{1} and Q_{2} be two anisotropic quadrics. If dim(Q_{1}) = dim(Q_{2}) = 2^{k} − 1 and Q_{2} is isotropic over F (Q_{1}), then Q_{1} is isotropic over F (Q_{2}).

6. Computing η_{2} for smooth varieties

For a smooth variety X let T_{X} be the tangent bundle over X.

Theorem 6.1. Let X be a proper smooth variety of dimension d > 0. Then the degree of the 0-cycle c_{d}(−T_{X}) is even and

η_{2}(X) = \frac{deg c_{d}(−T_{X})}{2} ∈ ℤ/n_{X}ℤ.

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 X^{2} along the diagonal X ⊂ 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}) :

[figure: ℙ(T_{X}); ⊂; W; ⊃; W \ ℙ(T_{X}); X; ⊂; X^{2}; ⊃; X^{2} \ 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 L_{can} over the blow-up W with the induced G-action. The group G acts by −1 on the restriction of L_{can} on the exceptional divisor ℙ(T_{X}). The restriction of L_{can} 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 L_{can} by −1. The new G-action of the restriction of L_{can} on the exceptional divisor ℙ(T_{X}) is trivial, hence L_{can} 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/n_{X}ℤ.

On the other hand, consider morphisms

[figure: ℙ(T_{X}); ⊂; i; W; p; W/G; X; q]

We have

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

In particular, p^{∗}c_{1}(L′)^{2d} = c_{1}(L_{can})^{2d} and hence

2 deg c_{1}(L′)^{2d} = deg c_{1}(L_{can})^{2d}.

The restriction of L_{can} on ℙ(T_{X}) is the canonical line bundle E_{can} with the sheaf of sections O_{ℙ(T_{X})}(1). Hence, by the projection formula (note that d > 0),

c_{1}(L_{can})^{2d} = i_{∗}(1) · c_{1}(L_{can})^{2d−1} = i_{∗}(c_{1}(E_{can})^{2d−1}) ∈ CH_{0}(W),

hence

deg c_{1}(L_{can})^{2d} = deg c_{1}(E_{can})^{2d−1}

The class

q_{∗}(c_{1}(E_{can})^{2d−1}) ∈ CH_{0}(X)

is known as the Segre class of T_{X} and coincides with c_{d}(−T_{X}). (In general, q_{∗}(c_{1}(E_{can})^{i+d−1}) = c_{i}(−T_{X}).)

Finally,

2 deg c_{1}(L′)^{2d} = deg c_{1}(L_{can})^{2d} = deg c_{1}(E_{can})^{2d−1} = deg c_{d}(−T_{X}).
□

Remark 6.2. The integer \frac{deg c_{d}(−T_{X})}{2} does not change under field extensions and therefore can be computed over an algebraically closed field.

Remark 6.3. 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 6.4. The number deg c_{d}(−T_{X}) does not change under field extensions and hence can be computed over algebraically closed fields.

Remark 6.5. 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.

7. Computing η_{2} for projective spaces

Proposition 7.1. 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)[E_{can}]+1 ∈ K_{0}(X). We can compute the total Chern class:

c(−T_{X}) = \frac{1}{c(E_{can})^{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 7.2. Let Y be a Severi-Brauer variety of dimension d. Then

v_{2}(deg c_{d}(−T_{Y})) = s_{2}(d), η_{2}(Y) = 2^{s_{2}(d)−1} + n_{Y} ℤ ∈ ℤ/n_{Y} ℤ.

Corollary 7.3. (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 2^{k} = n_{Y} divides n_{X}. If σ is isotropic over F (Y) then there is a rational morphism Y → X. By Corollary 7.2, η_{2}(Y) = 2^{k−1} + n_{X}ℤ is nontrivial in ℤ/n_{X}ℤ. A contradiction by Theorem 4.5(1). □

Remark 7.4. 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 7.5. Let p be any prime integer. One can show that for a Severi-Brauer variety Y of dimension d divisible by p − 1,

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

As in Corollary 7.3 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 n_{X} = p^{k}ℤ.

8. Fundamental class of a smooth variety

A partition α = (α_{1}, α_{2}, …, α_{k}) is a sequence of integers (possibly empty) α_{1} ≥ α_{2} ≥ ⋯ ≥ α_{k} > 0. The degree of α is the integer

|α| = α_{1} + α_{2} + ⋯ + α_{k}.

Denote by π(d) the set of all partitions α with |α| = d and by |π(d)| the number of all partitions in π(d).

Consider the polynomial ring ℤ[b_{1}, b_{2}, …] = ℤ[b] as a graded ring with deg b_{i} = i. For every partition α set

b_{α} = b_{α_{1}}b_{α_{2}} … b_{α_{k}}.

The monomials b_{α} form a basis of the polynomial ring over ℤ, and more precisely, the b_{α} with |α| = d form a basis of ℤ[b]_{d}.

Example 8.1. ℤ[b]_{0} = ℤ, ℤ[b]_{1} = ℤb_{1}, ℤ[b]_{2} = ℤb_{2} ⊕ ℤb_{1}^{2}, ℤ[b]_{3} = ℤb_{3} ⊕ ℤb_{1}b_{2} ⊕ ℤb_{1}^{3}.

Consider another polynomial ring ℤ[c_{1}, c_{2}, …] = ℤ[c] with similar grading deg c_{i} = i. The elements of ℤ[c] we call characteristic classes and the c_{n} - the Chern classes.

For any partition α we define the “smallest” symmetric polynomial

P_{α}(x_{1}, x_{2}, …) = ∑_{(i_{1},i_{2},...,i_{k})} x_{i_{1}}^{α_{1}}x_{i_{2}}^{α_{2}} … x_{i_{k}}^{α_{k}} = Q_{α}(σ_{1}, σ_{2}, …),

containing the monomial x_{1}^{α_{1}}x_{2}^{α_{2}} … x_{k}^{α_{k}} where the σ_{i} are the standard symmetric functions, and set

c_{α} = Q_{α}(c_{1}, c_{2}, …).

Example 8.2. 1. c_{n} = c_{(1,1,...,1)} (n units).

2. The classes s_{d} = c_{(d)} are called the additive characteristic classes. Note that Q_{(d)} is the Newton polynomial.

The characteristic classes c_{α} with |α| = d form a basis of ℤ[c]_{d}. We consider a pairing between ℤ[b]_{d} and ℤ[c]_{d} such that the b_{α} and c_{α} form dual bases. In particular, the additive class s_{d} corresponds to b_{d}.

For a smooth complete variety X of dimension d we define the fundamental class

[X] = ∑_{|α|=d} deg c_{α}(−T_{X})b_{α} ∈ ℤ[b]_{d}.

Thus, [X] is a homogeneous polynomial in the b_{i} of degree d. Note that [X] does not change under field extensions. The numbers deg c_{α}(−T_{X}) are called characteristic numbers. They are well known in topology. The class −T_{X} plays the role of the normal bundle of an embedding of X into ℝ^{n}.

Example 8.3. 1. [pt] = 1;

2. For a smooth projective curve X of genus g, [X] = 2(g − 1)b_{1}. In particular, [ℙ^{1}] = −2b_{1}.

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

4. [ℙ^{3}] = −4b_{3} + 20b_{1}b_{2} − 20b_{1}^{3}, [Q_{3}] = 6b_{3} − 10b_{1}^{3}.

Exercise 8.4. Show that [X × Y] = [X] · [Y] for every smooth complete varieties X and Y.

9. Formal group laws

Let R be a commutative ring. A (one-dimensional, commutative) formal group law over R is a power series

Φ(x, y) = x + y + ∑_{i,j≥1} r_{ij}x^{i}y^{j},

with r_{ij} ∈ R such that (1) Φ(Φ(x, y), z) = Φ(x, Φ(y, z)),

(2) Φ(x, y) = Φ(y, x).

Example 9.1. Φ(x, y) = x + y (additive group law), Φ(x, y) = x + y + rxy, r ∈ R.

Let f : R → S be a commutative ring homomorphism, Φ is a formal group law over R. Then f(Φ) is a formal group law over S. A formal group law Φ_{univ} over a ring L is called universal if for a any group law Φ over a ring R there is unique ring homomorphism f : L → R such that Φ = f(Φ_{univ}).

Proposition 9.2. A universal group law exists.

Proof. Consider the polynomial ring ℤ[a_{ij}], i, j ≥ 1, and let L be the factor ring modulo the ideal generated by elements encoded in the definition of a formal group law. □

The ring L is unique (up to canonical isomorphism) and it represents the functor

R ↦ {formal group laws over R}

from the category of commutative rings to the category of sets. The ring L is called the Lazard ring. Note that L is generated by the coefficients of Φ_{univ}.

We would like to find a convenient model for L.

Consider the power series

exp t = t + b_{1}t^{2} + b_{2}t^{3} + ⋯ ∈ ℤ[b][[t]]

and its formal inverse

log t = t + m_{1}t^{2} + m_{2}t^{3} + ⋯ ∈ ℤ[b][[t]].

Exercise 9.3. Prove that m_{d} = [ℙ^{d}]/(d + 1).

Consider the formal group law over ℤ[b]:

Φ = exp(log x + log y) = x + y + ∑_{i,j≥1} r_{ij}x^{i}y^{j}.

By the universal property, there is unique ring homomorphism L → ℤ[b] taking Φ_{univ} to Φ. The following algebraic result is well known.

Theorem 9.4. The homomorphism L → ℤ[b] is injective and its image is generated by the coefficients of Φ.

We shall identify the Lazard ring L with its image in ℤ[b]. Note that L is a graded subring in ℤ[b] generated by the coefficients a_{ij}.

Example 9.5.

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

Remark 9.6. The power series log is called the logarithm of the formal group law Φ.

It turns out that the Lazard ring is a polynomial ring ℤ[M_{1}, M_{2}, …] with deg M_{i} = i, where the M_{i} ∈ ℤ[b] can be chosen as follows:

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

Example 9.7.

M_{1} = 2b_{1} = −[ℙ^{1}],
M_{2} = 3b_{2} − 2b_{1}^{2} = −[ℙ^{2}],
M_{3} = 2b_{3} + 10b_{1}b_{2} − 204b_{1}^{3} = −[ℙ^{3}] + [Q_{3}],

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 of degree d set

r_{α} = r_{α_{1}}r_{α_{2}} … r_{α_{k}} ∈ ℤ.

Proposition 9.8. The factor group ℤ[b]_{d}/L_{d} is finite and isomorphic to ∐_{α} ℤ/r_{α}ℤ, where the coproduct is taken over all partitions α of degree d. In particular, L_{d} is a free group of rank |π(d)|.

Thus, ℤ[b]_{d}/L_{d} is a finite group of order

r = ∏_{|α|=d} r_{α} = ∏_{p prime} p^{[ \frac{d}{p−1} ]}.

This number coincides with the denominator of the Todd class td_{d}.

Example 9.9.

r_{1} = 2,
r_{2} = 12,
r_{3} = 24,
r_{4} = 720.

10. Fundamental classes and Lazard ring

Let L′ be the subgroup in ℤ[b] generated by the classes [X] for all smooth varieties X over F. Clearly, L′ is a graded subring in ℤ[b].

Theorem 10.1. If the base field F is infinite, L′ = L.

Remark 10.2. In topology the fundamental class of a manifold takes values in L, the cobordism group of the point.

We first prove that L ⊂ L′. The Lazard ring L is generated by the coefficients a_{ij} of the universal group law Φ. Hence it is sufficient to prove that a_{ij} ∈ L′.

Let H_{n,m} be a smooth hypersurface of type (1, 1) in ℙ^{n} × ℙ^{m}.

Proposition 10.3.

[H_{n,m}] = ∑_{i,j≥0} a_{ij}[ℙ^{n−i}][ℙ^{m−j}].

It follows from the formula by induction on n + m that a_{nm} ∈ L′.

We need to show now that L ⊂ L′. The idea (due to Stong) is to construct a fundamental class of a smooth complete variety in K-theory. We define the first Chern class of a linear bundle L over X by

c_{1}^{K}(L) = 1 − [L^{∨}] ∈ K_{0}(X)

and then extend the definition to get classes c_{α}^{K}(E) ∈ K_{0}(X) for arbitrary partition α and vector bundle E over X.

We define the fundamental class in K-theory for a smooth projective variety X of dimension d by

[X]^{K} = ∑_{α} f_{∗}(c_{α}^{K}(−T_{X}))b_{α} ∈ ℤ[b],

where f_{∗} : K_{0}(X) → K_{0}(pt) = ℤ is the push-forward homomorphism with respect to f : X → pt. The class [X]^{K} is a polynomial over ℤ of degree at most d.

Example 10.4.

[ℙ^{1}]^{K} = 1 − 2b_{1},
[ℙ^{2}]^{K} = 1 − 3b_{1} + (−3b_{2} + 6b_{1}b_{2}).

We compare two fundamental classes of X using the Chern character

ch : K_{0}(X) ^{}CH_{∗}(X) ⊗ ℚ

The Riemann-Roch theorem gives

Proposition 10.5. For a smooth projective variety X,

f_{∗}(c_{α}^{K}(−T_{X})) = (ch(c_{α}^{K})td^{−1})(−T_{X})

for all α. In particular, [X] = ([X]^{K})_{d}.

Denote by L_{d}^{K} the subgroup of ℤ[b] generated by the classes [X]^{K} for all projective smooth varieties X of dimension d. By Proposition 10.5, the restriction on L_{d}^{K} of the projection ℤ[b] → ℤ[b]_{d} is injective with image L′_{d}. Hence L_{d}^{K} is a free abelian group of rank at most |π(d)|.

Proposition 10.6. For every prime p and every integer d, there are varieties M_{α}^{p} for α ∈ π(d) such that the classes [M_{α}^{p}]^{K} in ℤ[b]/pℤ[b] are linearly independent.

It follows from Proposition that dimension of the image of L_{d}^{K} in ℤ[b]/pℤ[b] is at least |π(d)|. Hence, L_{d}^{K} is a free abelian group of rank exactly |π(d)| and hence L_{d}^{K} is generated by [M_{α}^{p}]^{K} for all p and α. Also, the map

L_{d}^{K}/pL_{d}^{K} → ℤ[b]/pℤ[b]

is an isomorphism, hence L_{d}^{K} is a direct summand of ℤ[b] (Hattori-Stong theorem).

Note that [M_{α}^{p}] generate L′. Computing classes [ℙ^{n}]^{K} and [H_{n,m}]^{K} in terms of the classes of generators [M_{α}^{p}]^{K} we get

Corollary 10.7. The ring L′ is generated by the classes [ℙ^{n}] and [H_{n,m}] for all n and m.

Finally one shows that the classes [ℙ^{n}] and [H_{n,m}] can be expressed in terms of the coefficients a_{ij} of the formal group law Φ, hence L′ ⊂ L.

Proposition 10.8. Let X be a smooth projective variety of dimension d = p^{k} − 1 where p is prime. Then deg s_{d}(−T_{X}) is divisible by p.

11. General degree formulas

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. Thus, as an abelian group, M(X) is generated by the classes [Y × Z] of varieties of dimension d, where dim Z > 0 and there is a morphism Y → X.

Conjecture 11.1. (General Degree Formula) For any morphism f : Y → X of proper smooth varieties of dimension d,

[Y] = deg(f) · [X] ∈ L_{d}/M(X)_{d}.

Remark 11.2. If f : Y → X is étale morphism, then T_{Y} = f^{∗}(T_{X}) and hence

[Y] = deg(f) · [X] ∈ L_{d},

i.e. the degree formula holds in a stronger form. In general, the difference [Y] − deg(f) · [X] can be non-trivial and is caused by the ramification of f.

Remark 11.3. In topology the statement of the conjecture is known as Quillen’s theorem.

The group L_{d} is a subgroup in ℤ[b]_{d} of finite index. Hence, the inclusion of dual groups

ℤ[c]_{d} = (ℤ[b]_{d})^{∗} ⊂ (L_{d})^{∗} = Hom(L_{d}, ℤ)

is an equality after tensoring with ℚ. Thus, the group (L_{d})^{∗} can be identified with the group of rational characteristic classes c ∈ ℤ[c]_{d}⊗ℚ such that c(L_{d}) ⊂ ℤ, i.e. c(−T_{X}) ∈ ℤ for every smooth complete X of dimension d. We call such rational characteristic classes c special.

Example 11.4. By Riemann-Roch theorem, for a smooth complete variety X,

deg td(T_{X}) = ∑_{i≥0} dim H^{i}(X, O_{X}),

thus, the inverse Todd class td^{−1} is special.

Let c ∈ ℤ[c]_{d} ⊗ ℚ be a special characteristic class. We can consider c as a homomorphism c : L_{d} → ℤ. Applying c to both sides of the degree formula, we get a degree formula

c[Y] = deg(f) · c[X] ∈ ℤ/c(M(X)_{d}).

Thus, to every special characteristic class c we have associated special degree formula.

Example 11.5. Let c = \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 Z > 0 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 (the Euler class) is multiplicative, hence

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

Note that the second factor is integer by Theorem 6.1. Clearly, deg c_{k}(−T_{Y}) is divisible by n_{Y}. Since there is a morphism Y → X, n_{X} | n_{Y} by Proposition 1.4 and hence deg c_{k}(−T_{Y}) is divisible by n_{X}. Thus c(M(X)_{d}) = n_{X}ℤ) and we get the degree formula (4.1) in the case of smooth varieties and p = 2.

16 A. MERKURJEV

Example 11.6. The additive integral class s_{d} = c_{(d)} vanishes on decomposable elements in L_{d}. Hence if d = p^{k} − 1, we get

\frac{s_{d}(Y)}{p} ≡ deg(f)\frac{s_{d}(X)}{p} mod I(X).

Recently, F. Morel and M. Levine prove the conjecture over fields of characteristic zero. They used heavily resolution of singularities.