The set Pic(X) of isomorphism classes of line bundles on X is an Abelian group
with respect to 0. A line bundle is called trivial if it is isomorphic to the line bundle
O := {Ox, can)
where can is the isomorphism
* can can *
a%Ox ^ Ox ~ alOx.
Of course the isomorphism class of O is the identity element of Pic(X). There is a
natural exact sequence of groups
- Pic{X) - Pic(X)
where the left arrow is defined by
the middle arrow is defined by
and the right arrow is induced by
In this book a key role will be played by 2 types of line bundles on corre-
spondences: flat line bundles and pluricanonical line bundles; cf. the Definitions
DEFINITION 1.6. Assume R0 c R is a subring, X = (X,X,Ji,a2) is a corre-
spondence over R, and A is a locally constant ^—valued function on X. Define
the line bundle
(1.17) F(A):=(0,A)
A : a2Ox ^ Ox - O^ - ^ i ^ x -
We call A the multiplier of F(A). A line bundle on X is called RQ— flat if it isomorphic
to a line bundle of the form F(A). In particular O = F(l). Clearly F(Ai)®F(A2) ^
1.7. If X is smooth over R and etale then the canonical line bundle
is the line bundle
(1.18) K = K
= (#*,/?)
where Kx is the canonical line bundle on X and (3 is the composition
The line bundle K - 1 is called the anticanonical line bundle. For v G Z the line
bundle K1" is called the v—canonical line bundle (or a pluricanonical line bundle).
If L = (L, (3) is a line bundle on X then one can define the line bundle L o n
on X o n by L o n := (L,f3on) where j3on is the obvious "n—fold product" of various
pull-backs of (3. For an RQ— flat line bundle L ~ F(A) with A G i?£ we have
o n
If K = K x is the canonical bundle on X then it is easy to check
(1.19) K ^ K
o n .
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