Intersection theory

For a proper morphism of varieties

f : X \to Y

, a Cartier divisor

D

Y

, and an algebraic cycle

α

X

, pullback of

D

, intersection with

α

, and proper pushforward satisfy the displayed identity in the Chow group. In particular, for a generically finite morphism of surfaces of degree

r

(f^{*} D)^{2} = r D^{2}

. For a curve

C

contracted to a point,

f_{*} C = 0

, so

f^{*} D \cdot C = 0

Intersection number of a Cartier divisor with a curve ( $D \cdot C$ )

The intersection number

D \cdot C

is the degree of the restriction of

O_{X} (D)

to the complete integral curve

C

. On a smooth surface, proper effective intersections are sums of local intersection multiplicities.

Self-intersection of an algebraic curve ( $C^{2} = de g O_{C} (C)$ )

For a proper integral Cartier curve

C

on a smooth algebraic surface, its self-intersection is

de g O_{C} (C)

. If

C

is smooth, this is the degree of its normal line bundle. It is defined over any algebraically closed field, without requiring an associated real or complex manifold.

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Algebraic cycle

k

-cycle on a scheme

X

is a finite integer linear combination of integral closed subschemes of dimension

k

. Their free abelian group is denoted

Z_{k} (X)

Rational equivalence

Rational equivalence is generated by principal divisors

[div (r)]

of nonzero rational functions on integral subvarieties one dimension larger than the cycles.

Chow group ( $A_{k} (X)$ )

The Chow group is

A_{k} (X) = Z_{k} (X) / Rat_{k} (X)

, the group of

k

-dimensional algebraic cycles modulo rational equivalence.

Chow class ( $[Z] \in CH^{r} (X)$ )

The equivalence class of an algebraic cycle modulo rational equivalence. The Chow group consists of these classes with addition induced by adding cycles.

Cycle class map ( $cl : CH^{r} (X) \to H^{2 r} (X, Z)$ )

For a smooth complex projective algebraic variety, the cycle class map sends an algebraic cycle of codimension

r

to the Poincare dual of its homology class. Rational equivalence gives the same cohomology class, so the map descends to the Chow group. It respects intersection products, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of Chow classes.

Cycle class ( $cl (Z)$ )

The cycle class of an algebraic cycle is its image under the cycle class map. For a codimension-

r

cycle on a smooth complex projective variety, it belongs to

H^{2 r} (X, Z)

and is the Poincare dual of its fundamental homology class. The map factors through the Chow group, but a Chow class retains finer information than its cohomological cycle class.

Chow ring ( $A^{*} (X)$ )

For a smooth variety, intersection of cycles gives the graded Chow ring

A^{*} (X)

External product of Chow classes ( $α ⊠ β = π_{X}^{} α \cdot π_{Y}^{} β$ )

For smooth projective varieties

X, Y

, pull back Chow classes from the two factors and take their intersection product on

X \times Y

. This gives the external product and a graded map

CH^{*} (X) \otimes CH^{*} (Y) \to CH^{*} (X \times Y)

. Unlike the cohomological Künneth theorem, it need not generate every class of the product.

Chow Künneth failure for an elliptic self-product

E \times E

for a complex elliptic curve, the two factor curves and the diagonal have intersection matrix

011101110

, of determinant two. Their cohomology classes are therefore independent. The degree-one image of the external product of Chow classes has cohomology contained in the span of the two factor classes. The cycle class map thus shows that the diagonal is outside that image, proving failure of surjectivity.

Localization sequence for Chow groups

For a closed immersion

j : Z ↪ X

with open complement

i : U ↪ X

, restriction and proper pushforward give an exact sequence

A_{k} (Z) j_{*} A_{k} (X) i^{*} A_{k} (U) \to 0.

Cellular decomposition of a scheme

A cellular decomposition filters a scheme by closed subschemes whose successive differences are disjoint unions of affine spaces. The closures of the cells generate its Chow groups.

Chow ring of projective space ( $A^{*} (P^{r})$ )

For the hyperplane class

H

A^{*} (P^{r}) ≅ Z [H] / (H^{r + 1}),

and

A_{k} (P^{r}) ≅ Z

is generated by a linear

P^{k}

Projective bundle formula for Chow groups

For a rank-

e + 1

vector bundle

E \to X

and

p : P (E) \to X

, powers of

ζ = c_{1} (O (1))

give isomorphisms

⨁_{i = 0}^{e} A_{k - e + i} (X) \sim A_{k} (P (E)) .

Homotopy invariance of Chow groups

For a rank-

r

vector bundle

E \to X

, flat pullback is an isomorphism

A_{k} (X) ≅ A_{k + r} (E)

Chern class ( $c_{i} (E)$ )

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula

c (E) = c (E^{'}) c (E^{''})

for a short exact sequence.

Chern class axioms in the Chow ring ( $c_{i} (E) \in CH^{i} (X)$ )

For a vector bundle

E

of rank

r

on a smooth algebraic variety, algebraic Chern classes have

c_{0} (E) = 1

, vanish for

i > r

, commute with pullback, and obey the Whitney sum formula for Chern classes for every short exact sequence of vector bundles. A line bundle

O (D)

has total class

1 + [D]

. The projective-bundle splitting construction makes pullback injective and supplies a filtration by line bundles, so these axioms determine all classes as elementary symmetric functions of their first classes. In particular,

c_{i} (E^{*}) = (- 1)^{i} c_{i} (E)

and

c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)

in the ordinary Chow ring.

Chern classes of a smooth hypersurface cotangent bundle ( $c (Ω_{X}^{1}) = (1 - h)^{n + 1} / (1 - d h)$ )

For a smooth degree-

d

projective hypersurface

X \subset P^{n}

, put

h = c_{1} (O_{X} (1))

. The cotangent Euler sequence in homogeneous coordinates gives

c (Ω_{P^{n}}^{1} ∣_{X}) = (1 - h)^{n + 1}

. The Conormal exact sequence for Kähler differentials and Whitney sum formula for Chern classes therefore yield

c (Ω_{X}^{1}) = (1 - h)^{n + 1} / (1 - d h)

. In codimension

i \leq n - 1

, its coefficient is

c_{i} (Ω_{X}^{1}) = \sum_{j = 0}^{i} (- 1)^{j} (j n + 1) d^{i - j} h^{i}

. Higher classes vanish by rank and dimension; the formal quotient is truncated in the Chow ring.

Todd class ( $td (E) = \prod_{j} \frac{x _{j}}{1 - e ^{- x_{j}}}$ )

Here the

x_{j}

are the formal Chern roots of a vector bundle. The degree-four term is

(c_{1}^{2} + c_{2}) /12

. Applied to the tangent bundle of a compact complex surface, Riemann–Roch for the structure sheaf gives the Noether formula.

Chern class in complex cobordism ( $c_{j}^{M U} (E) \in M U^{2 j} (X)$ )

The complex orientation of complex cobordism defines Chern classes of actual complex bundles by the splitting principle and elementary symmetric expressions in the first Chern classes of line summands. For an actual rank-

n

complex bundle, its top Chern class is its Euler class in a generalized cohomology theory. Chern classes extend stably by the Whitney formula; this stable extension must not be confused with the Euler class of a merely stably complex real bundle.

Projective bundle relation in complex cobordism ( $x^{n} - p^{} c_{1} (E) x^{n - 1} + \dots + (- 1)^{n} p^{} c_{n} (E) = 0$ )

The relation uses

x = c_{1} (S)

for the tautological line on the projectivization of

E

. It follows from the rank of the complementary bundle and Whitney multiplication. The alternating signs are ring subtraction; they do not assert that the First Chern class of a dual line is the additive negative in complex cobordism, where duality uses the formal-group inverse.

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Chern number ( $⟨ c_{i_{1}} \dots c_{i_{s}}, [M]⟩$ )

On a compact oriented

2 r

-manifold without boundary, an integral product of Chern classes of total complex degree

r

pairs with the fundamental class to give a Chern number. A unitary connection supplies Chern-Weil theory representatives, but the integer is independent of that connection. The Second Chern number on a four-manifold is one important example; the integral of the First Chern class over a closed surface is another.

Whitney sum formula for Chern classes ( $c (E \oplus F) = c (E) c (F)$ )

The Total Chern class of a direct sum of complex vector bundles is the product of their total Chern classes. In each degree,

c_{k} (E \oplus F) = \sum_{i + j = k} c_{i} (E) ⌣ c_{j} (F)

. The degree-zero class is one, classes beyond the bundle rank vanish, and a trivial bundle has total class one.

Projective bundle definition of Chern classes ( $h^{r} + π^{} c_{1} (E) h^{r - 1} + \dots + π^{} c_{r} (E) = 0$ )

For a rank-

r

complex vector bundle, let

h = c_{1} (S^{*})

for the tautological line

S

on its projective bundle. The Leray-Hirsch theorem gives the free basis

1, h, \dots, h^{r - 1}

over base cohomology. The uniquely determined coefficients in the displayed monic relation define the integral Chern classes. The line normalization uses the Euler class of the canonically oriented underlying real two-plane bundle, and uniqueness makes this definition intrinsic and natural under pullback. No choice of a trivializing cover or classifying map remains in the definition.

Total Chern class ( $c (E)$ )

The total Chern class is

c (E) = 1 + c_{1} (E) + c_{2} (E) + \dots

Euler sequence

On projective space, the Euler sequence is

0 \to O \to O (1)^{\oplus (m + 1)} \to T P^{m} \to 0.

Cotangent Euler sequence in homogeneous coordinates ( $0 \to Ω_{P^{n}}^{1} \to O (- 1)^{\oplus (n + 1)} \to O \to 0$ )

For a degree-zero rational function regular on an open subset of projective space, each derivative

\partial f / \partial X_{i}

is a regular homogeneous rational function of degree

- 1

. These derivatives define a derivation with values in

O (- 1)^{\oplus (n + 1)}

, hence a map from the Kähler differential sheaf. The last map is

(g_{i}) \mapsto \sum_{i} X_{i} g_{i}

. On

X_{i} \neq = 0

, put

y_{j} = X_{j} / X_{i}

and use frame

X_{i}^{- 1}

. The derivative of

y_{j}

has coordinate vector with

1

j

- y_{j}

i

and zero elsewhere. Those vectors form a free basis of the kernel of

(b_{i}) \mapsto b_{i} + \sum_{j \neq = i} y_{j} b_{j}

, proving exactness directly in every characteristic.

Vanishing of global sections of the once-twisted projective cotangent bundle ( $H^{0} (P^{n}, Ω^{1} (1)) = 0$ )

Dualize the Euler sequence on complex projective space and tensor by the hyperplane line bundle to obtain

0 \to Ω^{1} (1) \to O^{n + 1} \to O (1) \to 0

. On global sections the last map sends

(c_{i})

\sum_{i} c_{i} X_{i}

, an isomorphism onto the linear forms. Left exactness of sheaf cohomology therefore gives the asserted vanishing. No higher-cohomology assumption is needed for this final kernel calculation.

Degree of a projective scheme

For a pure

k

-dimensional projective scheme

X \subseteq P^{m}

, its degree is

k!

times the leading coefficient of its Hilbert polynomial, equivalently

\int_{X} H^{k}

This extends degree of an algebraic variety to retain scheme multiplicities.

Normal bundle

For a smooth closed embedding

i : X ↪ Y

, the normal bundle fits into

0 \to TX \to i^{*} T Y \to N_{X / Y} \to 0

Normal bundle of a smooth plane conic ( $N_{C / P^{3}} ≅ O (2) \oplus O (4)$ )

Embed a smooth plane conic in

P^{3}

through a plane. The plane equation and conic equation form a regular sequence of degrees one and two, so its normal sheaf is

O_{C} (1) \oplus O_{C} (2)

. Under

C ≅ P^{1}

, hyperplane degree is two, hence this normal bundle is

O_{P^{1}} (2) \oplus O_{P^{1}} (4)

. The normal exact sequence splits as well because

H^{1} (P^{1}, O (2)) = 0

Normal bundle of a linear complex-projective embedding ( $ν (CP^{k} \subset CP^{n}) ≅ (γ_{k}^{*})^{\oplus (n - k)} ≅ \overset{γ}{ˉ}_{k}^{\oplus (n - k)}$ )

Near a projective subspace, transverse directions are linear maps from the tautological line to the complementary coordinate vector space. This describes the actual complex normal bundle. The conjugate line and the dual line are complex-isomorphic using a Hermitian metric; neither is the original tautological line as a complex bundle in general.

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Normal bundle of a transverse intersection ( $ν (L_{1} \cap L_{2} \subset M) ≅ ν (L_{1} \subset M) ∣_{L_{1} \cap L_{2}} \oplus ν (L_{2} \subset M) ∣_{L_{1} \cap L_{2}}$ )

For a transverse intersection, the quotient map

TM ∣_{L} \to ν_{1} ∣_{L} \oplus ν_{2} ∣_{L}

is onto and has kernel

T L

. This gives the displayed isomorphism of actual normal bundles. Complex structures on the two normal bundles therefore induce a complex structure on the intersection's normal bundle.

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Self-intersection number ( $[X] \cdot [X]$ )

For an oriented submanifold of half the dimension of its ambient manifold, the self-intersection number is the intersection number with a transverse perturbation. It equals the Euler number of the normal bundle.

Self-intersection of the diagonal of a curve ( $Δ_{C}^{2} = 2 - 2 g (C)$ )

For a smooth projective algebraic curve

C

, the normal bundle of its diagonal in

C \times C

TC

: on the diagonal the quotient of

TC \oplus TC

by the diagonal subbundle is identified with

TC

by taking the difference. The self-intersection formula therefore gives

Δ_{C}^{2} = de g TC = 2 - 2 g (C)

Self-intersection formula

For a regular closed embedding

i : X ↪ Y

of codimension

d

, the self-intersection formula is

i^{*} i_{*} (α) = c_{d} (N_{X / Y}) \cap α

Euler number equals self-intersection ( $⟨ e (ν_{S}), [S]⟩ = [S] \cdot [S]$ )

For an oriented closed surface smoothly embedded in an oriented four-manifold, a transverse section of the normal bundle gives a homologous small push-off. Its signed zeros are exactly the signed intersections with the original surface. This identifies the evaluation of the Euler class with the self-intersection number.

Framing of an embedded sphere