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Transcript
THE ADJUNCTION FORMULA FOR LINE BUNDLES
C. BRUSCHEK
Abstract. We give a short report on the adjunction formula for line bundles
as it can be found in Griffiths-Harris “Principles of Algebraic Geometry”.
Theorem 1. Let X be a smooth projective variety and Y ⊆ X a smooth subvariety
of codimension 1 in X. Then
KY = (KX ⊗ Y ) |Y
(1)
where KZ denotes the canonical class of a smooth variety Z.
1. Preliminaries
First recall the notion of vector bundles:
Definition 1.1. A smooth complex vector bundle of rank k over M is a smooth
manifold E together with a surjective smooth map π : E → M of smooth manifolds
such that:
(1) for all x ∈ M the fiber over x, i.e., Ex := π −1 (x), is a k-dimensional
complex vector space.
(2) for all x ∈ M there is a neighbourhood U ⊆ M of x and a diffeomorphism
ϕU : π −1 (U ) → U × Ck such that proj1 ◦ ϕU = π|U .
Vector bundles of rank 1 are called line bundles.
Example. A cylinder over S 1 is vector bundle of rank 1, thus a line bundle. Since
it is of the form M × Ck , we call it a trivial bundle. The Moebius-stripe is a vector
bundle, but not a trivial bundle.
The restriction ϕU |Ex : Ex → Ck is an isomorphism of C-vector spaces. For any
k
two trivializations ϕU , ϕV of M at x, the composition ϕU ◦ ϕ−1
V : (U ∩ V ) × C →
k
(U ∩ V ) × C is of the form
(x, z) 7→ (x, gU V (x) · z), gU V (x) ∈ Glk (C),
i.e., it is C-linear in the second component. We call gU V the transition function of
E at x relative to the trivializations ϕU and ϕV . Let M = ∪α Uα be a covering of
M and denote by ϕα a trivialization on Uα for all α. This gives transition functions
(gαβ )α,β . These fulfill the following properties:
(∗)
for all x ∈ Uα ∩ Uβ : gαβ (x) = gβα (x)−1
for all x ∈ Uα ∩ Uβ ∩ Uγ : gαβ (x)gβγ (x)gγα (x) = 1.
Construction. Let M = ∪α Uα be a covering of M and let gαβ : Uα ∩ Uβ → Glk (C)
be given such that the conditions (∗) are fulfilled, then this data defines (up to
isomorphism) a unique vector bundle: Set
1
2
C. BRUSCHEK
!
a
E :=
Uα × C
k
/∼ ,
α
where (x, z)β ∼ (x, gαβ (x)z)α for all x ∈ Uα ∩ Uβ and ϕα : [Uα × Ck ] → Uα × Ck ,
[(x, z)α ] 7→ (x, z).
Here are some important constructions with fibre bundles. They are all given by
fibre-wise operations:
• Let E, F be vector bundles over M with trivializations ϕα , ψα (resp. transition functions gαβ ∈ Glk (C), hαβ ∈ Gll (C)). Then the tensor product of
E and F is defined as
E ⊗ F :=
a
Ex ⊗C Fx
x∈M
with trivializations ϕα ⊗ ψα and transition functions
gαβ ⊗ hαβ ∈ Gl(Ck ⊗ Cl ).
Vr
`
Vr
Vr
• The wedge-product of E:
E := x∈M
Ex with { ϕα } as trivializaVr
tions and transition functions
gαβ .
∗
• The dual
bundle
of E: E :=
> −1
tions gαβ
.
`
x∈M
Ex∗ with {ϕ−1
α } and transition func-
Definition 1.2. Let E be a vector bundle over M ; F ⊆ E is called a subbundle of
E if (i) F ⊆ E is a submanifold and (ii) for all x ∈ M : Fx := F ∩ Ex ⊆ Ex is a
C-subvectorspace.
In this setting the quotient bundle E/F is defined by (E/F )x = Ex /Fx (transition
functions are a bit more involved).
2. Divisors and Line Bundles
The set of line bundles over a smooth manifold X has the structure of a group;
multiplication is given by “⊗” and inverses by the dual bundle construction “(−)∗ ”,
the unit element is the trivial bundle. Let L and L0 be two line bundles on X. Then
−1
0
L ⊗ L0 is given by {gαβ · gαβ
} and L∗ is given by {gαβ
}. Indeed the inverse is given
as stated above:
−1
L ⊗ L∗ given by {gαβ gαβ
} = {idαβ }.
Thus L ⊗ L∗ is the trivial bundle. There is a fundamental correspondence between
divisors on and linebundles over X. Let D be a divisor on X with local defining
functions fα ∈ M∗ (Uα ), X = ∪α Uα . Then
fα
∈ O∗ (Uα ∩ Uβ ),
fβ
are holomorphic and non-zero on Uα ∩ Uβ . Moreover, it’s easy to see
gαβ :=
i.e., the gαβ
that
gαβ gβγ gγα = 1
and
THE ADJUNCTION FORMULA FOR LINE BUNDLES
3
gαβ gβα = 1.
In other words the gαβ define transition functions of a line bundle. We denote
the line bundle associated to D by L(D). This constructions gives the following
correspondencies:
• L(D + D0 ) = L(D) ⊗ L(D0 ).
• If D = (f ) for f ∈ M(X), then L(D) is trivial.
• as an extension of the last statement: L(D) is trivial iff D = (f ) for some
meromorphic function f on X.
• D ∼ D0 , then L(D) = L(D0 ).
Note that by the first statement we get a group homomorphism
L(−) : DivX → PicX
.
3. Proof of the Adjunction Formula
Let X be a smooth projective variety over C, Y ⊆ X a smooth submanifold. The
Normal bundle NY on Y is defined as
NY := TX |Y /TY ,
i.e., for x ∈ Y we have NY,x = TX,x /TY,x . Thus we get an exact sequence of vector
bundles
0 → NY → TX |Y → TY → 0
and further
0 → TY∗ → TX |∗Y → NY∗ → 0.
On each fibre this gives us an exact sequence of C-vectorspaces of dimension n − 1,
n and 1.
Lemma 3.1. Let 0 → U → V → W → 0 be an exact sequence of C-vectorspaces
of dimension a, a + b, b, then
a+b
^
V =
a
^
U⊗
b
^
W.
In our situation the last lemma gives
(2)
n
^
TX |∗Y =
n−1
^
TY∗ ⊗
^
NY∗ .
By definition the left-hand side of equation (2) is KX |Y and the factor on the far
right is just NY∗ , which can be multiplied to the left by means of (−)∗ . This gives
(3)
KY = KX |Y ⊗ NY .
It remains to prove that NY = L(Y )|Y . For this note that the conormal bundle
NY∗ is the subbundle of TX |∗Y of cotangent vectors, which vanish on TY . Since
Y ⊆ X is of codimension 1, it is locally given by functions fα ∈ O(Uα ), X = ∪Xα .
4
C. BRUSCHEK
Thus L(Y ) is given by transition functions gαβ = ffαβ . Moreover fα |Uα ∩Y = 0 and
dfα ∈ Γ(Y ∩ Uα , NY∗ ). But Y is smooth, so dfα =
6 0 on Y ∩ Uα . Moreover on
Uα ∩ Uβ ∩ Y we have
(4)
(5)
(6)
Therefore the dfα ∈
NY∗ ⊗ L(Y ), i.e.,
dfα
= d(gαβ fβ )
= dgαβ · fβ + gαβ · dfβ
= gαβ dfβ .
Γ(Uα , NY∗ )
fit together to give a global non-zero section of
NY∗ ⊗ L(Y )|Y = 1
the trivial bundle. Note that in our notation equations (4) show that the transition
−1
functions for NY∗ are given by gβα that is by gαβ
. Hence, NY∗ = L(−Y )|Y . Together
with equation (3) this gives
KY = (KX ⊗ L(Y )) |Y .
Appendix A. hallo