3Projective varieties
III Positivity in Algebraic Geometry
3.4 Kodaira dimension
Let
X
be a normal projective variety, and
L
a line bundle with
|L| 6
= 0. We
then get a rational map
ϕ
|L|
: X 99K P(H
0
(L)).
More generally, for each
m >
0, we can form
L
⊗m
, and get a rational map
ϕ
L
⊗m
. What can we say about the limiting behaviour as m → ∞?
Theorem
(Iitaka)
.
Let
X
be a normal projective variety and
L
a line bundle
on
X
. Suppose there is an
m
such that
|L
⊗m
| 6
= 0. Then there exists
X
∞
, Y
∞
,
a map
ψ
∞
:
X
∞
→ Y
∞
and a birational map
U
∞
:
X
∞
99K X
such that for
K 0 such that |L
⊗K
| 6= 0, we have a commutative diagram
X Im(ϕ
|L
⊗K
)
X
∞
Y
∞
ϕ
|L
⊗k
|
U
∞
ψ
∞
where the right-hand map is also birational.
So birationally, the maps ψ
K
stabilize.
Definition (Kodaira dimension). The Kodaira dimension of L is
K(X, L) =
(
−∞ h
0
(X, L
⊗m
) = 0 for all m > 0
dim(Y
∞
) otherwise
.
This is a very fundamental invariant for understanding
L
. For example, if
K(X, L) = K ≥ 0, then there exists C
1
, C
2
∈ R
>0
such that
C
2
m
K
≤ h
0
(X, L
⊗m
) ≤ C
1
m
K
for all m. In other words, h
0
(X, L
⊗m
) ∼ m
K
.