I’m supposed to give a talk on this subject for one of my courses, so I consider this post as a “pre-exposition.” I learned from and heavily used the great exposition “Vector bundles on curves” by Montserrat Teixidor I Bigas in this post. I wrote up the pre-requisites here.
In 1957, Atiyah in this famous paper “Vector bundles over an elliptic curve” classified indecomposable vector bundles of arbitrary rank and degree. Briefly, every vector bundle (locally free sheaf) is decomposed uniquely (up to the order) to the direct sum of indecomposable vector bundles and the set of isomorphism classes of indecomposable vector bundles of a fixed rank and degree is isomorphic to the Jacobian of the curve which the latter is isomorphic to the curve itself. The isomorphisms are canonical for vector bundles of degree zero and for higher degree, they depend on the choice of a line bundle of degree one (a base point on the curve) [“Vector bundles on curves” by Montserrat Teixidor I Bigas, section 4.]
An elliptic curve is a smooth projective curve of genus one over an algebraically closed field
One may assume that
Throughout, the words vector bundle
and locally free sheaf
are used interchangeably.
Denote by the set of isomorphism classes of indecomposable vector bundles of rank
and degree
where the degree of a vector bundle
of rank
is defined as the degree of the associated locally free sheaf
which is
One can also show that is equal to the degree of its determinant.
Case 1: Vector bundles of degree zero
For every positive integer there exists a unique (self-dual) indecomposable vector bundle
of rank
and degree zero with only one section. It turns out that any indecomposable vector bundle of rank
and degree zero is isomorphic to
for a unique line bundle
of degree zero. Therefore, the set (moduli space) of indecomposable vector bundles of degree zero on
i.e.
is (canonically) isomorphic to the moduli space of degree zero line bundles on
which in turn by definition is the Jacobian of
Case 2: Vector bundles of non-negative degree (general case)
In general, let be a fixed line bundle of degree one, (which corresponds to fixing a base point) on
then there is an isomorphism
sending
Note that
If there is an isomorphism
sending
to
where
is given by the following extension with
By these two operations, we can assume that and
Moreover, if
we will have
and if
then
Note that non of these operations changes
Thus, we can construct a sequence of isomorphisms
with
s.t.
and
This is the sequence of positive numbers, so it will terminate when
and
Hence, we have established the isomorphism
and by case 1, is isomorphic to the Jacobian of
(and is isomorphic to the curve itself.)
That being said, the above isomorphism is completely determined up to the choice of a line bundle of degree one.
So far, we have achieved what we wanted. Let’s now, try to dig more. Denote by the element in
corresponding to
in
Proposition 1: a) Every vector bundle
of rank
and degree
can be written as
for some line bundle
of degree zero.
b) If
then
where
run over the set of line bundles of order (in the Picard group)
c) If
then
d) If
then
e) If
then
Here is some results involving stability and semi-stability of vector bundles on
Proposition 2: a) An indecomposable vector bundle of degree zero is semi-stable not stable.
b) Indecomposable vector bundles are semi-stable and they are stable if and only if
For detailed and complete description of vector bundles on an elliptic curve, especially when the ground field is of look at Atiyah’s original paper.
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