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**Contents:**

The first Chern class c1 L completely classifies the topology of complex line bundlesor U 1 -bundles on any base spaceX. In other words, the total spaces of two line bundles are topologicallyequivalent if and only if their first Chern classes agree.

In addition, each element of H2 X,Z is realizedas the first Chern class of some line bundle. If X is now a compact, orientable four-dimensional manifold and E is a complex vector bundle on X ,then the only non-vanishing topological invariants are c1 E and ch2 E. The topology of an SU 2 -bundle on a compact, oriented four-dimensional manifold iscompletely classified by the topological charge k.

One should not get the impression that the Chern classes necessarily determine the topology of the bundle;this fails in general. In order to introduce the Yang-Mills functional and the Yang-Mills equations, we need further Let X, g be an m-dimensional, oriented, Riemannian manifold with Riemannian metricg. An action functional will require integrating over spacetime, so we will further assumeX is compact. Toprovide some perspective, let us briefly describe an analogy in basic Hodge theory.

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Hodge Theory in Riemannian GeometryIt is well-known that the de Rham cohomology groups of a smooth m-dimensional manifold X are isomor-phic to the singular cohomology groups Hp X,R with real coefficients, and are finite-dimensional groupsifX is compact. IfX additionally has a Riemannian metric g, then we can introduce the Hodge star operatoron differential forms? The conclusion follows by a simple computation making use of the Leibnizrule as well as the sign in?

This is also called a functional derivative or a varia-tional derivative.

In other words,there are no genuine critical points of E. However, we can look for critical points corresponding to variations in only certain directions.

One may hear harmonicforms described as zero modes of the Laplacian which by the discussion above, is equivalent to minimizingthe functional E. Generalization to Yang-Mills TheoryThe reason for reviewing Hodge theory above is that in some sense, Yang-Mills theory provides a beautifulnon-linear generalization of the Hodge-de Rham theory to differential forms on a compact Riemannianmanifold valued in an adjoint or endomorphism bundle.

As above, let X, g be a smooth compact orientableRiemannian manifold of dimension m. Of course the adjoint bundle adP is constructed as an associated vectorbundle, as we have seen.

We want an analogous inner product to 2. However, given an innerproduct on the fiber, to get a well-defined associated vector bundle, one must have the inner product invariantwith respect to the chosen representation.

We refer to such an inner product as ad-invariant. It is this condition whichrequires that G be a compact Lie group. In physics parlance, we refer to this as the gauge invariance of an action functional. With respect to the innerproduct 2. We may also call this the functionalderivative or variational derivative of SYM. We now want to ask what constraint must a G-connection dA, or its corresponding curvature FA, satisfy ifit is to locally minimize the value of the functional? Recall that equation 2. In order for 2.

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Together with the Bianchi identity 2. Therefore, Crit SYM inherits a well-defined action by the gauge group. This leadsus to a discussion of instantons. However, we did not place any restrictions on the dimension of X. It turns out, that four-dimensional Riemannian manifolds X, g hold a special place in Yang-Mills theory. It is in such a casewhere one can study instantons, which we will define to be a certain class of Yang-Mills connections. In fact, one unifying theme of my thesis is that insome of these examples, instantons bridge remarkable connections between differential geometry, algebraicgeometry, enumerative geometry, and physics.

In such a case, applying the operator twice yields theidentity? Lemma 2.

follow The decomposition 2. The above discussion, including the definition of self-duality and anti-self-duality, holds as well forbundle-valued forms. On any Riemannian manifold of dimension 2n with n even, the Hodge star operator induces a similardecomposition on the middle-dimensional forms. We will refer to a connection as SD or ASD if its curvature two-form hasthat property, as defined above. Recall from 2. There are two cases to consider Recall that the Yang-Mills equations 2. Let X be a compact, orientable four-dimensional Riemannian manifold and let E, h be a Hermitian vector bundle with structure group G, assumed to be either U n or SU n.

If E, h is a Hermitian vector bundle on a compact, oriented Riemannian four-manifoldX and if the topological charge k vanishes, then an instanton on E is a flat connection. Some features of the generaltheory for example, the infinite-dimensional affine space of connections are nicer in the more restrictivesetting.

Indeed, we will see that there is a unique connection compatible with the extra structure in such away that we can translate a space of such connections to a space of holomorphic structures on a Hermitianvector bundle. We will introduce the Hermitian Yang-Mills equation, which arises in the study of instantonsas well as D-branes in string theory. By the Donaldson-Uhlenbeck-Yau theorem, irreducible connectionssolving the Hermitian Yang-Mills equation correspond to stable holomorphic vector bundles.

This sectionis a short preview in some sense of our study in the following chapter on stability of coherent sheaves. Wewill not provide full details, and we will often refer to the literature for proofs. Some great resources forvarious parts of this material are [27, 34, 52, 70].

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In fact, much of what follows in this brief survey is modeledon the wonderful book [70]. Equivalently,it is a complex vector bundle whose transition functions are holomorphic. As shown in [8, Theorem 5. One should think that simpleholomorphic vector bundles are those with the minimal number of global holomorphic endomorphisms.

We define Ahol E to be the space of holomorphic structures on the underlying complexvector bundle E, and A shol E to be the corresponding space of simple holomorphic structures on E. Recall from Definition 2. The complex gauge transformationsalso act in a well-defined way on A shol E. We want to identify holomorphic structures related by gaugetransformations.

We make the same definition for simple holomorphic structures. Having laid some foundations on holomorphic structures on bundles, we will now see that a connec-tion dA on a complex vector bundle E, determines a holomorphic structure if an integrability condition issatisfied. As we defined in 2. If dA is an integrable connection then by Definition 2. Indeed, it is natural to stay in the Hermitian settingbecause we have the following important converse [52, Proposition 4.

Theorem 2. We will call such a dA the Chern connec-tion. For all Hermitian metrics h, by part of Proposition 1. We regard this as the precise correspondence between integrable Hermitian connections andholomorphic structures on bundles over a complex manifold. We will be primarily interested in simple holomorphic bundles, as these are more amenable to pa-rameterization in a moduli problem.

Equivalently, the Chern connection associated to a simple holomorphic structureis irreducible. The converse is not true, but it will be true for irreducible connections solving the HermitianYang-Mills equation.

Moreover, connections related by gauge However, just as with Yang-Mills con-nections and instantons, we do not want to consider all integrable Hermitian connections, but only thosesatisfying a condition known as the Hermitian Yang-Mills equation. Let E, h be a Hermitian vector bundle onX , and let E be the holomorphic bundle correspondinguniquely to Chern connection dA on E, h via 2.

This condition is called the Hermitian Yang-Mills equation. This definition is often given instead for the metric h, and is called a Hermitian-Einstein metric if the Her-mitian Yang-Mills equation is satisfied. Both sides of 2. VolJ X. However, if the connection is additionally Hermitian Yang-Mills, then we do get a map [70, Corollary 2. Slope stable holomorphic vector bundles are in particular simple, and the content of the following Donaldson-Uhlenbeck-Yau theorem is that the image of the above map is precisely the locus of stable holomorphicbundles.

Therefore, the Donaldson-Uhlenbeck-Yau theorem is a relationship between differential geometry specialconnections on Hermitian bundles and algebraic geometry stable holomorphic bundles.