# Atiyah’s work on holomorphic vector bundles and gauge theories

## Abstract

The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.

## Part 1. Early work

Michael Atiyah began his research career as an algebraic geometer, writing a series of papers in the period 1952–1958, mostly about bundles over algebraic varieties. Many of the themes in these papers reappear 20 years later when Atiyah’s interests turned to Yang–Mills theories. In the commentary on Volume 1 of his collected works Atiyah wrote:

I was fascinated by classical projective geometry … at the same time great things were happening in France and I was an avid reader of the

Comptes Rendus, following the developments in sheaf theory.

The language and style of these early papers of Atiyah are an interesting mixture of classical and modern.

### 1.1. Extensions and the Atiyah class

*Extensions* of vector bundles form a theme running through much of Atiyah’s work. Let be a complex manifold. A holomorphic vector bundle is a complex vector bundle in the ordinary sense such that the total space is a complex manifold, and the projection and structure maps are all holomorphic. Alternatively, the bundle is defined by a system of holomorphic transition functions

with respect to an open cover The definition of a subbundle . is the obvious one, and we get a holomorphic quotient and an exact sequence:

The new feature, compared with the theory of topological or bundles, is that this exact sequence need not split: it is not usually true that is isomorphic to a direct sum We define an equivalence relation on short exact sequences like .Equation 1 by saying that two are equivalent if there is a commutative diagram:

Then the basic fact is:

In Reference 4 Atiyah writes that this result “follows from the general theory of fibre bundles” and refers to unpublished notes of Grothendieck. In the notation used above we do not distinguish between a holomorphic vector bundle and its sheaf of local holomorphic sections. There is a -to- correspondence between vector bundles and sheaves of locally free modules over the structure sheaf of holomorphic functions on and a version of Proposition ,1 holds for locally free sheaves of modules in general. To define the extension class from the axioms of sheaf cohomology we apply to the sequence to get

and this has a long exact cohomology sequence with coboundary map

The extension class is where , is the identity on each fibre of .

We can see this more explicitly using either Čech or Dolbeault representations of the cohomology. For the first we use the fact that the extension can be split locally, so there is an open cover over which we have holomorphic splitting maps

On an intersection we have two splitting maps and ,

where Thus . is a cochain in the Čech complex associated to the sheaf and one finds that it is a cocycle and that its cohomology class is independent of choices. ,

The Dolbeault description of the extension class is important in complex differential geometry. To set this up, we recall that a holomorphic vector bundle can be viewed as a bundle equipped with a -operator

satisfying the Leibnitz rule

with

Returning to the extension class, this time we use the fact that a sequence of *matrix* representation:

With this background in place we begin our review of some of Atiyah’s applications of the theory. In Reference 4 he introduced what is now called the *Atiyah class* of a bundle

Letting

and the Atiyah class

A more general approach uses principal bundles, and this was the point of view Atiyah took in Reference 4. Let

Here *vertical* tangent vectors along the fibres of

where *connections*. From the principal bundle point of view a connection is given by a splitting of the exact sequence Equation 4, that is to say a *horizontal* subbundle *holomorphic* connections need not, and one of the main results of Atiyah in Reference 4 is:

For another point of view on the Atiyah class, we consider a

This makes a connection with Chern–Weil theory. Let

So

The curvature of a holomorphic connection has type

So a flat connection exists if any only if the pairing between the Atiyah class and every holomorphic section of

and the integrand vanishes since

### 1.2. Bundles over curves

In this subsection we discuss the papers Reference 3 and Reference 5 of Atiyah which are more detailed investigations of holomorphic bundles, mainly over algebraic curves (or compact Riemann surfaces).

The first paper Reference 3 is focused on vector bundles

where *reduce* questions about rank 2 bundles to questions about line bundles and cohomology. The essential difficulty is that the line subbundle is not unique, so any bundle

To reduce this multitude, Atiyah considers subbundles of *maximal* degree. As we will see, this is an idea that goes a long way, so we will make a definition.

(It is straightforward to show that this is well-defined.)

Let us illustrate the utility of this notion by classifying rank 2 bundles over the Riemann sphere

but

(Here, and later, we write *Hirzebruch surfaces*

One of Atiyah’s main results in Reference 3 determines when two extensions define isomorphic rank 2 bundles over a Riemann surface. Start with one description which we can take to be

for a line bundle

Now take a nontrivial line bundle

where the last term is fixed by considering the determinant of

where

So we need a section of

Atiyah shows that this section maps to *the point *. So, starting with

Atiyah used these techniques in Reference 3 to classify all rank 2 bundles over curves of genus *two* of the points: for example

To see this moduli space more explicitly, we start with the symmetric product

where

We now turn to Atiyah’s results on bundles over elliptic curves (genus

Let

Atiyah’s construction of the

so we first have to construct

These extensions Equation 10 are classified by