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Transcript
Vector bundles 2
Paul Hacking
2/21/10
1
Sections of line bundles on Pn
Last time we defined the tautological line bundle L on projective space Pn
and proved that there are no global sections of L. Let H := L∗ be the dual
of L. We compute the global sections of H ⊗d .
(Some notation: If L is a line bundle on a complex manifold X then
L ⊗ L∗ is isomorphic to the trivial line bundle C × X. This follows from
the analogous fact for vector spaces: if V is a 1-dimensional vector space
then V ⊗ V ∗ = Hom(V, V ) is naturally isomorphic to C via idV 7→ 1. So
sometimes we write L−1 for L∗ , etc. The line bundle H ⊗d on Pn is often
denoted O(d).)
The line bundle H ⊗d over Pn can be obtained as the quotient of the
trivial line bundle (Cn+1 \ {0}) × C over Cn+1 \ {0} by the following action
of C× :
C× 3 λ : (x0 , . . . , xn , y) 7→ (λx0 , . . . , λxn , λd y).
For example, in case d = −1 (the tautological line bundle L = H −1 ) the
quotient map
(Cn+1 \ {0}) × C → L ⊂ Pn × Cn+1
is given by
(x0 , . . . , xn , y) 7→ ((x0 : · · · : xn ), y · (x0 , . . . , xn )).
So a global section s of H ⊗d corresponds to a holomorphic function
F : Cn+1 \ {0} → C
such that
F (λx0 , . . . , λxn ) = λd F (x0 , . . . , xn )
1
(1)
for λ ∈ C× . By Hartogs’ theorem [GH78, p. 7], F extends to a holomorphic
function F : Cn+1 → C with the same property (1). We can expand F in a
power series about 0 ∈ Cn+1 :
X
F =
ai0 ,...,in xi00 · · · xinn .
i0 ,...,in
P
The property (1) holds iff ai0 ,...,in is zero for
ij 6= d, that is, iff F is a
homogeneous polynomial of degree d. We deduce that the global sections
Γ(Pn , H ⊗d ) of H ⊗d are the homogeneous polynomials of degree d in the
homogeneous coordinates X0 , . . . , Xn on Pn . (In particular, there are no
global sections if d < 0).
2
Line bundles and divisors
There is a correspondence between line bundles and divisors on complex
manifolds. To explain this we first consider the case of Riemann surfaces.
Suppose X is a Riemann surface and P ∈ X a point. We can define a
line bundle L = O(P ) associated to P as follows. Let U1 = X \ {P }
and U2 = {|z| < δ} be a small disc in X centered at P . (That is, z is a
local coordinate at P and δ > 0 is sufficiently small so that z defines an
isomorphism of U2 onto the disc.) We glue L from the trivial line bundles
over U1 and U2 via the transition function
g12 = z : U1 ∩ U2 = U2 \ {P } → C× .
Notice that the sections of L over an open set U ⊂ X can be identified with
meromorphic functions f on U with at worst a simple pole at P (if P ∈ U ).
Indeed f is given by holomorphic functions f1 , f2 on U ∩ U1 , U ∩ U2 , such
that f2 = g12 f1 = zf1 on the overlap. So f := f1 is meromorphic on X, has
at worst a simple polePat P , and is holomorphic elsewhere.
Similarly, if D = ri=1 ni Pi , ni ∈ Z, is a formal sum of points of X (a
divisor on X) we can define a line bundle O(D) by
O(D) = O(P1 )⊗n1 ⊗ · · · ⊗ O(Pr )⊗nr .
If f is a meromorphic function at a point P ∈ X, let z be a local coordinate
at P and write
f = aν z ν + aν+1 z ν+1 + · · ·
near P , where ν ∈ Z, aν 6= 0. Then we say the order of vanishing of f at P
equals ν and write ordP (f ) = ν. Then for U ⊂ X the sections Γ(U, O(D))
2
of O(D) over U are identified with meromorphic functions f on U such that
the order of vanishing ordPi (f ) of f at Pi is at least −ni (if Pi ∈ U ) for each
i.
The general construction is as follows (see [GH78,P
p. 129–136] for more
details). Let X be a complex manifold and D =
ni Yi a formal sum
of codimension 1 complex subvarieties (a divisor ). For f a meromorphic
function on X, the associated principal divisor is
X
(f ) :=
ordY (f ) · Y
Y
Then there is an open covering {Ui } of X such that for each i the restriction
D|Ui is a principal divisor (fi ). We can define a line bundle O(D) associated
to D by specifying transition functions
gij = fj /fi : Uij → C× .
For open U ⊂ X, we can identify
Γ(U, O(D)) = {f meromorphic on U | (f ) + D|U ≥ 0}.
In particular, a meromorphic function f on X determines a meromorphic
section s of O(D), with divisor of zeroes minus poles
(s) = (f ) + D.
Note also that O(D0 ) ' O(D) iff D0 = D + (g) for some meromorphic
function g, the isomorphism being given by multiplication by g. Conversely,
if L is a holomorphic line bundle with meromorphic section s, then L ' O(D)
where D = (s).
3
The degree of a line bundle on a Riemann surface
Let X be a compactPRiemann P
surface and L a line bundle on X. The degree
of a divisor D =
ni Pi is
ni . The degree of L is the degree of (s)
for any nonzero meromorphic section s of L, that is, the number of zeroes
minus the number of poles of s, counted with multiplicities. Note that if t is
another meromorphic section then t/s = f is a meromorphic function, and
deg(f ) = 0 (because f defines a map f : X → P1 and the fibers over 0, ∞
have the same size when counted with multiplicities), so deg(t) = deg(s).
3
4
Tangent bundle
Let X be a complex manifold. The tangent bundle T X → X is the holomorphic vector bundle with fibres
∂
∂
TP X =
,...,
∂z1
∂zn C
where z1 , . . . , zn are local coordinates at P ∈ X. Sections of T X are holomorphic vector fields. Sections of ∧p T ∗ X are holomorphic p-forms. The
line bundle ∧n T ∗ X is called the canonical line bundle and denoted KX .
The transition functions of KX are the Jacobian determinants
gij = det D(φj ◦ φ−1
i ).
Example 4.1. The canonical line bundle of projective space. The form Ω =
dx1 ∧ · · · ∧ dxn is a nowhere zero holomorphic n-form on (X0 6= 0) ⊂ Pn ,
where xi = Xi /X0 . On (X1 6= 0) we have coordinates y0 , y2 , . . . , yn where
yi = Xi /X1 . Now x1 = y0−1 , xi = yi /y0 for i ≥ 2, so
−(n+1)
Ω = −y0−2 dy0 ∧ y0−1 dy2 ∧ · · · ∧ y0−1 dyn = −y0
dy0 ∧ dy2 ∧ · · · ∧ dyn .
So Ω has a pole of order n + 1 along the hyperplane (X0 = 0) ⊂ Pn . It
follows that
KPn ' H −(n+1) .
5
Riemann–Hurwitz, redux
Let f : X → Y be a nonconstant map between compact Riemann surfaces.
Then
X
KX = f ∗ KY ⊗ O( (eP − 1)P ).
(Note: If X → Y is a morphism of complex manifolds and V is a line bundle
on Y then f ∗ V is the vector bundle on X with fibres (f ∗ V )P = Vf (P ) .) This
follows from a local calculation: if z is a local coordinate at P ∈ X and w is
a local coordinate at f (P ) ∈ Y such that f is locally given by w = z e , then
f ∗ (dw) = d(f ∗ w) = d(z e ) = ez e−1 dz.
To relate this to the earlier (topological) form of Riemann–Hurwitz we
need to show
deg KX = 2g(X) − 2
4
There are many ways to prove this, perhaps the most basic is to reduce to
the usual Poincaré–Hopf theorem for Riemann surfaces, cf. [G89, p. 22]. We
give an argument using the topological version of Riemann–Hurwitz already
proved. There exists a nonconstant meromorphic function f on the Riemann
surface X (this is a hard theorem). The function f defines a map
f : X → P1
of some degree d. We know deg KP1 = −2, so using the two forms of
Riemann Hurwitz for f we obtain
X
X
deg KX = deg(f ∗ KP1 ) +
(eP − 1) = −2d +
(eP − 1) = 2g(X) − 2
as required.
6
The adjunction formula
Let X be a complex manifold and i : Y ⊂ X a complex submanifold. We
have an exact sequence of vector bundles on Y
0 → TY → TX |Y → NY /X → 0
(2)
where TX |Y = i∗ TX is the restriction of TX to Y and NY /X is the normal
bundle of Y ⊂ X (which is defined by the above exact sequence).
Lemma 6.1. If
0 → V1 → V → V2 → 0
is an exact sequence of vector spaces of dimensions r1 , r, r2 then there is a
natural isomorphism
∧r V ' ∧r1 V1 ⊗ ∧r2 V2
Proof. Let v1 , . . . , vr1 be a basis of V1 , v r1 +1 , . . . , v r a basis of V2 , and
vr1 +1 , . . . , vr lifts to V . Then the isomorphism is given by
v1 ∧ · · · ∧ vr 7→ v1 ∧ · · · ∧ vr1 ⊗ v r1 +1 ∧ · · · ∧ v r .
That it is independent of choices follows from
A B
det
= det A · det D.
0 D
5
By the lemma, the exact sequence of vector bundles (2) implies
∧n TX |Y ' ∧m TY ⊗ ∧n−m NY /X
where n = dim X and m = dim Y . Rearranging gives the adjunction formula
KY = KX |Y ⊗ ∧n−m NY /X .
If Y ⊂ X has codimension 1, then the ideal of holomorphic functions on
X vanishing on Y are (the sections of) the line bundle IY = O(−Y ). The
normal bundle of Y equals (IY |Y )∗ = O(Y )|Y . To see this, consider the dual
of the sequence (2)
0 → NY∗ /X → TX∗ |Y → TY∗ → 0.
The kernel of the map TX∗ |Y → TY∗ is locally generated by df where f is a
local equation of Y , and this defines an isomorphism
IY |Y → NY∗ /X ,
f 7→ df.
So in this case the adjunction formula becomes
KY = (KX ⊗ O(Y ))|Y .
Example 6.2. (Genus formula for plane curve, redux.) Let X = P2 and
Y = (F = 0) ⊂ X a plane curve of degree d. Then
KY = (KX ⊗ O(Y ))|Y = (H −3 ⊗ H d )|Y = H d−3 |Y .
In particular
deg KY = (d − 3)d
and using deg KY = 2g(Y ) − 2 we find
1
g(Y ) = (d − 1)(d − 2).
2
References
[G89]
P. Griffiths, Introduction to algebraic curves, AMS 1989.
[GH78] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley 1978.
6