# Donaldson Turns 60

Donaldson has opened up an entirely new area; unexpected and mysterious phenomena about the geometry of 4-dimensions have been discovered. Moreover, the methods are new and extremely subtle, using difficult nonlinear partial differential equations. On the other hand, this theory is firmly in the mainstream of mathematics, having intimate links with the past, incorporating ideas from theoretical physics, and tying in beautifully with algebraic geometry. It is remarkable and encouraging that such a young mathematician can understand and harness such a wide range of ideas and techniques in so short a time and put them to such brilliant use. It is an indication that mathematics has not lost its unity, or its vitality.” – Sir Michael Atiyah

Sir Simon K. Donaldson FRS is a British geometer. His honors include the Fields Medal in 1986 “primarily for his work on topology of 4-manifolds, especially for showing that there is a differential structure on Euclidean 4-space which is different from the usual structure,” the Shaw Prize in 2009 with C. H. Taubes “for their contributions to geometry in 3 and 4 dimensions,” knighthood in 2012 “for services to mathematics,” and the Breakthrough Prize in Mathematics in 2014 “for the new revolutionary invariants of 4-dimensional manifolds and for the study of the relation between stability in algebraic geometry and in global differential geometry, both for bundles and for Fano varieties.”

Donaldson received his BA in Mathematics from Cambridge in 1979 and his PhD from Oxford in 1983. His thesis “The Yang-Mills Equations on Kähler Manifolds” was supervised by Sir Michael Atiyah and Nigel Hitchin. Donaldson proved his celebrated diagonalizability theorem while still a graduate student.

This month marks his 60th birthday. A conference on symplectic geometry will be held at the Issaac Newton Institute, Cambridge to celebrate this occasion and various advances in symplectic geometry.

Below we mention some major contributions of Donaldson to the ancient science of geometry; in particular, to gauge theory and to Kähler-Einstein metrics. This is by no means a comprehensive account of the life and work of Simon Donaldson. I have tried my best to describe some of his contributions to mathematics without totally embarrassing and discrediting myself; for this to be somewhat successful, it was important to restrict attention to theorems I was more familiar with. I have also attempted to avoid overlap with other posts on this blog about gauge theory.

## Gauge Theory

Donaldson’s diagonalizability theorem is an application of physics to the geometry of 4-manifolds. Previous posts on this blog have discussed Freedman’s theorem, which states that for a closed, simply-connected topological 4-manifold $M$, the intersection form $H^2(M,\mathbb{Z}) \times H^2(M,\mathbb{Z}) \to \mathbb{Z}$ is (up to a binary choice) a complete invariant – every possible intersection form can be realized by some $M$, and at most two such manifolds share the same intersection form. However, smooth 4-manifolds are totally different.

Theorem [Donaldson]. Let $M$ be a simply-connected, smooth 4-manifold. If the intersection form of $M$ is definite, then it is diagonalizable. That is, if the intersection form is positive (resp. negative) definite, then it is equivalent to $Id$ (resp. $-Id$) over $\mathbb{Z}$.

Combined with Freedman’s theorem, this proves the existence of non-smoothable topological 4-manifolds (such as the $E_8$ manifold). See [5] for a full account.

Yang-Mills Theory and 4-Manifolds

The proof of Donaldson’s theorem makes use of Yang-Mills theory, a classical field theory detailed by Yang and Mills in their 1954 Nobel Prize-winning physics paper [10]. Fix a manifold $M$ and a Lie group $G$. If $P$ is a principal $G$-bundle over $M$, we may consider the space of all $G$-connections on $P$, which we denote by $\mathcal{A}$. In local coordinates, these may be thought of as $\mathfrak{g}$-valued one-forms on $M$. A gauge transformation is an automorphism of $P$; the group of gauge transformations acts on the space of connections and is written $\mathcal{G}$. The Yang-Mills action functional is defined on the space $\mathcal{A}/\mathcal{G}$ of connections modulo gauge by

$$YM(A) = \int_M |F_A|^2 d\mu.$$

Here $F_A$ is the curvature of the connection $A$ and may be thought of as a $\mathfrak{g}$-valued two-form on $M$. It can be shown that the critical points of the Yang-Mills functional are given by solutions to the Yang-Mills equations

$$d^\ast_AF_A = 0.$$

The Yang-Mills equations generalize Maxwell’s equations. For the Abelian group $U(1)$, the curvature of a connection $A$ is just $F_A = dA$. Here $A$ is called the electromagnetic potential and $F_A$ the electromagnetic field. A computation in local coordinates shows that $A$ is not gauge invariant but $F_A$ is. This corresponds to the physicist’s idea that the electromagnetic potential cannot be assigned a direct physical interpretation whereas the electromagnetic field can. The Yang-Mills equations for $U(1)$ are just Maxwell’s equations.

There is a lot of delicate structure here. Whenever ones takes quotients like $\mathcal{A}/\mathcal{G}$ one must worry about “stability”. Also the group $\mathcal{G}$ is infinite-dimensional, so the moduli space can be a bit wonky and a lot of hard functional analysis has to be done. It is this potent synthesis of analysis, topology, geometry, and physics that Atiyah reverently referenced in this post’s opening quotation.In four dimensions something special happens – the Yang-Mills equations have instanton solutions. For a smooth 4-manifold $M$, the Hodge star operator $\ast : \Omega^2 M \to \Omega^2 M$ is an isomorphism with eigenvalues $\pm 1$. Elements of the corresponding eigenbundles $$\Omega^2M = \Omega^2_+M \oplus \Omega^2_-M$$ are called self-dual and anti-self dual (ASD) forms, respectively. The curvature of a connection similarly breaks up into self-dual and anti-self dual parts $F_A = F^+_A + F^-_A$. (This is also related to the fact that $Spin(4) = SU(2) \times SU(2) = Spin(3) \times Spin(3)$.) When $\| F^+_A\|^2=0$ (resp. $\| F^-_A\|^2=0$) the connection $A$ is called self-dual (resp. ASD). The space of ASD instantons modulo gauge, the moduli space, is the star of Donaldson’s proof.

Donaldson-Thomas Invariants

If $M$ is a 3-manifold, one can define the Chern-Simons functional on $\mathcal{A}$ by

$$cs(A) = \int_M\text{Tr}(A \wedge F_A + \frac{2}{3}A \wedge A \wedge A).$$

It can be checked that the critical points of $cs$ are flat connections; e.g., connections $A$ such that the de Rham complex $$\Omega^0 \to\Omega^1 \to \Omega^2 \to \Omega^3$$ coupled to the connection on the adjoint bundle induced by $A$ is an elliptic complex.

The Casson invariant is a gadget of low-dimensional topology. In Taubes’ 1990 paper “Casson’s Invariant and Gauge Theory” he reintepreted the Casson invariant as an Euler characteristic of an infinite-dimensional space that counts the flat connections [8].

In the 1998 paper [3] of Donaldson and Thomas “Gauge Theory in Higher Dimensions” (of which there is a 2011 sequel by Donaldson and Segal [4]) they define a holomorphic Casson invariant. These invariants now go by the named of “Donaldson-Thomas invariants”. Donaldson-Thomas theory is an active area of research.

Heuristically, their construction goes by defining a holomorphic version of Chern-Simons functional. This requires manifolds with holomorphic volume forms – Calabi-Yau manifolds. In analogy to the real Chern-Simons map, the critical points of the “holomorphic Chern-Simons map” are connections on a complex bundle $E$ for which the Dolbeaut complex

$$\Omega^{0,0}\to \Omega^{0,1} \to \Omega^{0,2} \to \Omega^{0,3}$$

coupled to the induced connection on $End(E)$ is elliptic. Such connections are not flat, but rather correspond to holomorphic structures on $E$. On a Calabi-Yau threefold there are a finite number of critical values. So Donaldson-Thomas invariants count the number of holomorphic bundles over a Calabi-Yau threefold of fixed topological type. In more algebraic language, they count the Hilbert scheme $I_n(M,\beta)$ of ideal sheaves with holomorphic Euler characteristic $n$ and homology class $\beta$ on some three-dimensional Calabi-Yau complex variety $M$.

## Kähler-Einstein Geometry

If $M$ is a complex manifold, then its complex structure induces a field $J$ of automorphisms of the tangent bundle such that $J^2 = -Id$. For a metric $g$ on $M$ there is an associated form defined by $\omega_g(X,Y) = g(JX,Y).$ When $\omega$ is closed (that is, $d\omega = 0$), $g$ is called a Kähler metric.  When the metric is a constant multiple of the Ricci tensor, the metric is called Einstein because if $g = \lambda Ric(g)$ then the manifold satisfies Einstein’s equations. A Kähler-Einstein metric is a metric both Kähler and Einstein.

The Kobayashi-Hitchin Correspondence

Stability is a concept from algebraic geometry, originating with David Mumford’s Geometric Invariant Theory (GIT). GIT is essentially the study of quotients. Stability conditions are conditions ensuring the quotient is not something ridiculous that everybody hates.

For example, in the above discussion of gauge theory we did not want all connections, because the action of the gauge group did not really affect the things we cared about. Thus, we quotiented out by this symmetry. The space left standing is the “space of moduli” or in more common parlance the “moduli space”.

People like to think about moduli spaces of bundles. So there is some notion of a stable bundle. Kobayashi and Hitchin (independently) conjectured that the moduli spaces of stable vector bundles and Einstein-Hermitian vector bundles over a complex manifold were equivalent. This was proved by Donaldson for algebraic manifolds [2],[6], and for Kähler manifolds by Uhlenbeck and Yau [10].

The Calabi Conjecture for Fano Manifolds

The “Calabi Conjecture for Fano Manifolds”, often called the Yau-Tian-Donaldson conjecture, is an influential Kobayashi-Hitchin type conjecture that was proved by Chen, Donaldson, and Sun in 2012.

Poincaré’s uniformization theorem says that there is a unique metric with constant scalar curvature in each Kähler class on a Riemann surface. Calabi conjectured a far-reaching generalization of this theorem, as follows. Suppose the first Chern class of $M$ is a constant multiple of its Kähler form $c_1(M) = \lambda \omega \in H^2(M,\mathbb{Z}).$ Then Calabi conjectured there is a Kähler-Einstein metric cohomologous to $\omega.$

The cases $\lambda < 0$, $\lambda = 0$, and $\lambda > 0$ are quite different. The case $\lambda < 0$ was solved by Aubin and Yau. The case $\lambda = 0$ was solved by Yau in the 1970s.

Manifolds with $\lambda > 0$ are called Fano manifolds. In the spirit of Kobayshi-Hitchin, Yau-Tian-Donaldson conjectured that a Fano manifold will admit a Kähler-Einstein metric if and only if it is “stable” for some suitable notion of stability. The correct notion of stability is called “K-stability.” In a series of 2012 papers Chen, Donaldson, and Sun proved the following [1]:

Theorem [Chen-Donaldson-Sun]. A Fano manifold admits a Kähler-Einstein metric if and only if it is K-stable.

References

[1] X. X. Chen, S. K. Donaldson, S. Sun, Kähler-Einstein Metrics and Stability. arXiv:1210.7494. 2012.

[2] S. K. Donaldson, Anti-self-dual Yang-Mills connections on complex algrebraic surfaces and stable vector bundles, Proc. London Math. Soc. 3 (1985), 1-26.

[3] S. K. Donaldson and R. Thomas, Gauge theory in higher dimensions, The Geometric Universe, Oxford UP, 1998, pp. 31-47.

[4] S. K. Donaldson and E. Segal, Gauge theory in higher dimensions, II, Surveys in Differential Geometry 16 (2011), 1–41. arXiv:0902.3239.

[5] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford U.P., 1990.

[6] S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. Jour. 54 (1987), 231-47.

[7] G. Tian, On Kähler-Einstein metrics on certain Kahler manifolds with c_1(M) > 0. Invent, math. 89 (1987) 225-246.

[8] C. H. Taubes, Casson’s invariant and gauge theory, Jour. Differential Geometry 31 (1990), 363-430.

[9] K. K. Uhlenbeck and S-T. Yau, On the existence of hermitian Yang-Mills connections on stable bundles over compact Kahler manifolds, Commun. Pure Applied Math. 39 (1986), 257-93.

[10] C. N. Yang and R. Mills, Conservation is Isotopic Spin and Isotopic Gauge Invariance, Phys. Rev. 96, 191 (1954).