This is an introduction to a GIT construction of the moduli space of stable vector bundles on curves, presented at the GIT seminar. Our main sources are [1] and [2].

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Fix a smooth projective algebraic curve (over ) of genus . Unlike the case of line bundles, it has been observed that in general all vector bundles are not classifiable. For example, one can construct a family of vector bundles of rank parametrized by the disk where all the fibers apart from the origin are mutually isomorphic, but not isomorphic to the fiber at the origin ([1, 11.32]). This "jump phenomenon" illustrates that the set of all vector bundles on curves is not even separated. In other words, even the coarse moduli space does not exist. To construct well-behaved moduli spaces of vector bundles, Mumford's geometric invariant theory hints at finding stable conditions on vector bundles and restricting our attention to *stable* vector bundles.

Definition 1
The *slope* of a vector bundle is the ratio . is called *stable* (resp. *semistable*) if every subbundle satisfies (resp. ). Equivalently, is stable (resp. semistable) if every quotient bundle satisfies (resp. ).

By definition, every line bundle is stable. The following implications partially explain that why stable bundles are "nice".

Proof
The definition of stability implies that every nonzero endomorphism is an isomorphism, hence is a scalar by looking at one fiber.
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The moduli space of stable vector bundles over of rank and degree was first given by Mumford [3] and Seshadri [4]. Later, Gieseker gave a different construction which generalized to higher dimensions. Simpson invented a more natural and general method using Grothendieck's Quot scheme which also extends to singular curves and higher dimensions (see [5]).

is simply the Picard variety we have constructed. We have the natural map sending to its determinant bundle . We fix a line bundle and study the fiber of this map. In other words, we are going to construct the space of stable vector bundles using GIT.

Similarly to the case of Picard varieties, we will assume so that Riemann-Roch brings us some convenience.

Proof
By assumption, every quotient line bundle of has degree . So there is no nonzero morphism as . Hence by Serre duality.
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Proof
Since for any , , we know that as is semistable (tensoring with a line bundle does not change stability). By the exact sequence We know that is surjective. By Nakayama's lemma we know that is surjective.
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Now let us concentrate on the case . Similarly to the case of Picard varieties, we will associate to each isomorphism class of vector bundles of rank 2 a -orbit of a matrix and study its stability. Let be a vector bundle of rank 2 with and generated by global sections. Let and be a basis. Since is generated by global sections, we have a surjection The pairing induces a map which is injective since is generated by global sections. The composition map is given by the matrix Let be the set of skew-symmetric matrices with entries in a vector space . Then and we call it the *Gieseker point* of . The different choices of the marking correspond to the -orbit of under the action Moreover, is isomorphic to the image of , hence one can recover from its Gieseker points. So we have proved:

Since is a vector bundle of rank 2, the matrix have rank 2 over the function field . Denote to be set of matrices having rank over . is a subvariety of and the image of the above map lies in it.

To apply GIT to construct , we need to study the stability of under the action of . This consists of two steps: for ,

Step 1 is semistable (stable) if and only if is semistable (stable).

Step 2 Every semistable is the Gieseker point for some .

Assuming these two steps, we can construct as a GIT quotient immediately.

Proof
The only thing to check is that the orbits in are free. Suppose . We then have the following commutative diagram This gives an endomorphism of . Since is stable, we know that is simple. Hence with . Namely , which acts trivially on .
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For the first step, we need the following observation. This phenomenon did not appear in the case of line bundles.

Proof
Let , . We may assume generates over . Since is a line bundle, we know contains a top left block consisting of only zeros. Write Consider a 1-parameter subgroup Then If , then . Letting , we know is unstable. If , then . Letting , we obtain the matrix is not stable since it does not have finite stabilizer. So is not stable.
¡õ

Definition 2
Let be a vector bundle. We call *-semistable* (resp. *-stable*) if (resp. ) for any subbundle .

So we have shown:

In fact, the converse is also true:

We omit the proof of this fact since it is a bit long (see [1, 10.70]). The key idea is to construct -semi-invariant polynomials using the Pfaffians or the radical vectors of skew symmetric matrices depending on whether is even or odd.

The condition for -semistability is already quite similar to semistability. One can use Riemann-Roch to prove the following, which finishes the proof of the first step.

Proof
By Riemann-Roch, for any subbundle, Since , we know that Since the left-hand-side is , -(semi)stability implies (semi)stability. Now suppose . Since , we know that , hence . Riemann-Roch implies that . Therefore . The converse is proved.
¡õ

Now let us come to the second step.

Proof
If not, by a suitable change of basis, we may assume has only zeros in the first row and column. Then choosing the 1-parameter subgroup . Then goes to 0 as . Then is unstable, a contradiction.
¡õ

Proof
Since is skew-symmetric, it has even rank, so over . Let be the image of , then is a vector bundle of rank 2.

Let us first show . Since is semistable, we know that by the last lemma. Suppose , then by Serre duality, we have a nonzero morphism , which gives a map . Let , then as . So is unstable, a contradiction.

Next we need to show that . The map is skew-symmetric and vanishes on , hence induces a sheaf morphism . From we know that . So .

Finally we conclude is semistable as is the Gieseker point of generated by global sections and . ¡õ

Remark 1
For general rank , Gieseker similarly considered the action of on the space and deduced the stability condition.

is smooth and if is nonempty (when , or , coprime, or , ), then . Moreover, is a fine moduli space if and only if are coprime. The dimension of is as expected. In particular, if it is nonempty.

Example 1
For , , Grothendieck's theorem asserts that each vector bundle on is a direct sum of line bundles. So there is no stable vector bundle of rank 2 and is empty.

Example 2
For , an elliptic curve, Atiyah's theorem asserts that for any line bundle of odd degree, there exists a unique isomorphism class of stable vector bundles of rank 2 and is a single point. For any line bundle of even degree, there is no such stable bundle and is empty.

Example 3
Let be a curve of genus . Its canonical bundle has degree 2. Take , then . By Riemann-Roch, , so defines an embedding , whose image is the intersection of four quartics. A stable bundle has . An element of is a skew-symmetric matrix of linear forms in which has rank 2 on , hence its Pfaffian is a linear combination of the four quartics. The ring of semi-invariants of has 15 generators and is the linear span of the four quartics . Moreover, the stable moduli is the complement , where is known as the *Kummer quartic surface*.

[1]An introduction to invariants and moduli, Cambridge University Press, 2003.

[2]Vector Bundles on Algebraic Curves, 2002, http://www.mimuw.edu.pl/~jarekw/EAGER/Lukecin02.html.

[3]Projective invariants of projective structures and applications, Proc. Internal. Congr. Math., 1962, 526--530.

[4]Space of Unitary Vector Bundles on a Compact Riemann Surface, The Annals of Mathematics 85 (1967), no.2, pp. 303-336.

[5]The geometry of moduli spaces of sheaves, University Press, 2010.