Atiyah’s work on holomorphic vector bundles and gauge theories

By Simon Donaldson

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:

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Then the basic fact is:

Proposition 1.

Given holomorphic bundles there is a -to- correspondence between equivalence classes of extensions Equation 1 and the sheaf cohomology group such that the trivial extension defined by corresponds to .

In Reference 4 Atiyah writes that this result “follows from the general theory of fibre bundles” and refers to unpublished notes of Grothendieck.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 . Here we are writing for the smooth -forms with values in . Using the Leibnitz rule, one gets an extension to operators

with . Fixing , the cohomology of the resulting complex gives the sheaf cohomology .

Returning to the extension class, this time we use the fact that a sequence of bundles can be split. So we can choose a global (not necessarily holomorphic) splitting map and a corresponding projection . Then the composite is a map from sections of to -forms with values in . One finds that this commutes with multiplication by smooth functions so is given by a tensor which gives a Dolbeault representative for the extension class. Said in another way, if we identify with as a bundle then has a 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 . This can be described in two ways. For the first, for a point let be the set of -jets of sections of at . By definition, such a -jet is an equivalence class of local holomorphic sections where is equivalent to if and the derivative of at vanishes. (Recall that the derivative of a section of a vector bundle is not intrinsically defined in general but it is defined at a zero of the section.) Then we have an exact sequence of vector spaces

Letting vary, we get an exact sequence of bundles over

and the Atiyah class is the extension class in

A more general approach uses principal bundles, and this was the point of view Atiyah took in Reference 4. Let be a complex Lie group, and let be a principal -bundle. Then there is an exact sequence of vector bundles over :

Here is the trivial bundle with fibre the Lie algebra which can be identified with the vertical tangent vectors along the fibres of using the group action. All three bundles in this sequence are equivariant with respect to the action of , and this implies that the sequence is pulled back from an exact sequence of vector bundles over

where is the bundle over associated to the adjoint representation of on its Lie algebra. Then there is an extension class in . In the case when , so is the frame bundle of a vector bundle , we have and we get the same class as before. The different approaches correspond to two ways of viewing 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 -invariant horizontal subbundle . In the first approach a splitting of the sequence Equation 3 gives a map . From the definition of the jet space, a section of induces a section of and is a covariant derivative on sections of . While connections always exist, holomorphic connections need not, and one of the main results of Atiyah in Reference 4 is:

Proposition 2.

A bundle admits a holomorphic connection if and only if the Atiyah class is zero.

For another point of view on the Atiyah class, we consider a connection, or covariant derivative, on which is compatible with the holomorphic structure in the sense that the component of equals . (In the principal bundle approach this is the same as saying that the horizontal subspaces are complex subspaces of .) The connection has curvature and since the (0,2) part of vanishes and . The Bianchi identity implies that , so defines a class in which is another representation of the Atiyah class.

This makes a connection with Chern–Weil theory. Let be an invariant polynomial of degree on the Lie algebra of . (In other words, is a polynomial function on matrices with : these are just symmetric functions in the eigenvalues.) Then the combination of and wedge product defines a map from the tensor product of copies of to . If we have any class , we combine the map above with the product on cohomology to get . Suppose, for simplicity, that is a compact Kähler manifold. Then we have a Hodge decomposition of the cohomology and

So can be viewed as a class in the topological cohomology of . Applying this to , we get the usual Chern–Weil construction for characteristic classes of . This is clear from the representation of the Atiyah class by the curvature. We can choose a connection compatible with a Hermitian structure on the bundle. The curvature has type and writing out the recipe above, using the Dolbeault description where the product on cohomology is induced by wedge product on forms, gives exactly the Chern–Weil construction. (The whole discussion extends to general structure groups and that is the context in which Atiyah worked.)

The curvature of a holomorphic connection has type , so when is a Riemann surface, a holomorphic connection is flat. Using this, Atiyah obtained alternative proofs of some results of Weil from his 1938 paper Reference 71, which began the study of higher rank vector bundles over Riemann surfaces. In the classical case of line bundles of degree a standard approach is to view sections as automorphic functions on the universal cover, transforming according to a multiplier. In other words, the line bundle is endowed with a flat connection with structure group and arises from a representation . Weil extended this to higher rank bundles and flat bundles with structure group . He showed that an indecomposable bundle over a compact Riemann surface admits a flat connection if and only if it has degree (i.e., first Chern class) equal to . For a bundle over we have the Serre duality:

So a flat connection exists if any only if the pairing between the Atiyah class and every holomorphic section of vanishes. If the bundle is indecomposable (i.e., cannot be written as a nontrivial direct sum), then any holomorphic section of can be written as , where is nilpotent. (For otherwise the eigenspaces would give a decomposition.) By the discussion of Chern–Weil theory above, the pairing between and is the degree of , so one has to see that the pairing of with the nilpotent sections vanishes. Suppose, for simplicity that has rank and there is a rank subbundle . Then we can choose a connection on that preserves . The pairing is given by

and the integrand vanishes since preserves .

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 of rank . Then the projective bundle is a ruled surface so the study becomes part of the general study of algebraic surfaces. The starting point is the fact that any such bundle contains rank subbundles . For if is any positive line bundle over and is sufficiently large, the tensor product has a nonzero holomorphic section. This section might vanish at some points of , but it is still true that the image of the section defines a rank subbundle of . This is a special feature of dimension : in a local trivialisation about a point in , the section is defined by a vector-valued function and we take the fibre of the subbundle over to be the line spanned by the first nonvanishing derivative. This fact means that we can write as an extension

where and are line bundles and by the theory discussed in the previous subsection the bundle is determined by and an extension class in . Multiplying the extension class by a nonzero scalar just corresponds to scaling the inclusion map and does not change the rank 2 bundle. So if we leave out the trivial extension, the data is where . Thus, using the theory of extensions, Atiyah was able to 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 has a multitude of descriptions of this form.

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.

Definition 1.

For a rank bundle over a compact Riemann surface , let be the maximal degree of a rank subbundle of .

(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 . Recall that the line bundles over are just the powers of the Hopf bundle and that for while for . By taking the tensor product with a line bundle, we may reduce to the case when , so is an extension

but has no subbundle of strictly positive degree. Take the tensor product of Equation 6 with and the long exact sequence in cohomology, which runs

(Here, and later, we write for .) If , there is a section of which lifts to a section of since . The image of this section is a line subbundle of , and the section can be written as a section of composed with inclusion map, so the degree of is strictly positive which contradicts our hypothesis. Thus we see that . Now the extension class of Equation 6 lies in which vanishes for , so we conclude that the extension splits and . (The definition of implies that in fact .) In terms of ruled surfaces the result states that the bundles over are the Hirzebruch surfaces for , which Atiyah refers to as the classical result “that every rational normal ruled surface can be generated by a 1-1 correspondence between two rational normal curves lying in skew spaces”.

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 of degree and an extension class in . Assume that the extension does not split, so is determined by a point in the projective space which we denote by . The Serre dual of is so we have , and the linear system defines a map .

Now take a nontrivial line bundle of degree and consider the possible existence of an extension

where the last term is fixed by considering the determinant of . Taking a tensor product with , this is equivalent to considering extensions

where . The existence of such an extension is equivalent to a nonvanishing section of and from Equation 7 we have an exact sequence

So we need a section of which maps to zero in the -dimensional vector space . A section of defines a positive divisor, say for points . Now we have . So

Atiyah shows that this section maps to in if and only if the point is in the linear span of . So, starting with and , we consider the curve and find all the configurations of points on this curve whose linear span contains . For each such configuration we define another line bundle by Equation 8 and this gives all the descriptions of the rank 2 bundle by extensions with a subbundle of maximal degree.

Atiyah used these techniques in Reference 3 to classify all rank 2 bundles over curves of genus and , extending the classical case of genus that we saw above. Postponing the discussion of genus , we consider now genus . There are various cases, depending on the first Chern class of the bundle and the invariant . By taking the tensor product with a line bundle, it suffices to consider bundles with equal to or . The most interesting case, for reasons we will see more of later, is when and . So, after tensoring by a line bundle, we have an extension Equation 7 where the line bundle has degree . Thus has degree ; the Riemann–Roch formula shows that has dimension and is a projective line. Since the dimension of this vector space is we have a canonical identification between the projective space and its dual, so, in this special situation, we can regard the extension data as the zero divisor of a section of . Conversely, any divisor defines a degree line bundle and an extension class. So for each triple of points we have a rank vector bundle , say. The curve is hyperelliptic, with an involution . Applying his criterion, Atiyah shows that a different triple defines a bundle projectively equivalent to if and only if the new triple is given by applying to two of the points: for example . In this way he constructs a three-dimensional moduli space parametrising these projective bundles.

To see this moduli space more explicitly, we start with the symmetric product of triples . This is a fibre bundle over the Jacobian with fibre . There is a -to- map from to given by the identifications described above. This map takes the fibres in the symmetric product to the bundles constructed using a fixed line bundle but varying extension data. We have a -to- map from to which induces an -to- map from to . This map factors through the space , so we have

where is -to-. We have six branch points of the double covering . Each of these points defines a plane in . The 3-fold is the double cover of branched over these six planes .

We now turn to Atiyah’s results on bundles over elliptic curves (genus ). The rank 2 case was covered in Reference 3, and the later paper Reference 5 gave a complete classification for all ranks. Slightly before, Grothendieck classified bundles of all ranks over : they are all direct sums of line bundles, and the proof follows the same lines as the rank case discussed above, using induction on the rank.

Let be a compact Riemann surface of genus and fix a line bundle of degree over . For integers with , let be the set of isomorphism classes of indecomposable bundle of rank and degree . The most straightforward case is when are coprime, so we begin with that. Atiyah proved

Proposition 3.

There is a unique way to define indecomposable bundles (up to isomorphism) for coprime rank and degree such that

;

;

if , there is an exact sequence

Atiyah’s construction of the follows the Euclidean algorithm for the pair . To illustrate this, consider the case of . Following the third bullet, we want to build this as an extension

so we first have to construct . Following the second bullet, we have , so we have to construct . The three bullets together state that this should be an extension

These extensions Equation 10 are classified by which is one dimensional, so there is indeed a unique indecomposable