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What Is...A Pseudoholomorphic Curve?

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What Is...A Pseudoholomorphic Curve? ?WHAT IS... a Pseudoholomorphic Curve? Simon K. Donaldson The terminology pseudoholomorphic curve (or from C to C. The concept extends to the case of maps J-holomorphic curve) was introduced by Gromov in to almost-complex manifolds. Let M be a differentiable 1986. The notion has transformed the field of sym- manifold of dimension 2n. An almost-complex struc- plectic topology and has a bearing on many other ture J on M is a family of linear maps Jx : TMx → TMx, 2 areas such as algebraic geometry, string theory, and with Jx = −1, on each tangent space TMx of M, vary- 4-manifold theory; we will return to these later. ing smoothly with x ∈ M. Thus the tangent spaces are We are all familiar with the notion of a “curve”— made into complex vector spaces. Any complex mani- say a plane curve—at the elementary, and perhaps im- fold has a natural almost-complex structure, but the precise, level of ordinary calculus. We can specify a converse is not true if n>1: there is an integrability plane curve in two different ways: either as the set condition which characterises these special almost- of solutions of an equation f (x, y)=0 or via a para- complex structures. Many manifolds M which admit metrisation x = x(t),y = y(t). For example, we can almost-complex structures do not have any complex- 2 2 specify a circle by the equation x + y =1or by the manifold structure at all. parametrisation x = cos t,y = sin t. Another familiar A pseudoholomorphic curve is just the natural mod- concept is that of a “family” of curves, for example, ification of the notion of a holomorphic curve to the the family of lines in the plane. case when the ambient manifold is almost-complex. The theory of curves has, of course, been developed That is, we consider a Riemann surface Σ, an almost- extensively both in differential geometry and algebraic complex manifold (M,J), and a differentiable map geometry. The relevant branch of the classical theory f : Σ → M such that for each σ ∈ Σ the derivative for us here is that of “complex” or “holomorphic” curves. In the simplest situation, we replace the real dfσ :(T Σ)σ → TMf (σ ) variables x, y above by complex variables z,w and con- is complex-linear with respect to the given complex sider complex curves in the complex plane. Thus the structures on the tangent spaces. We can spell out same equation z2 + w 2 =1, for example, describes more concretely what a pseudoholomorphic map such a complex curve. Or we can consider para- amounts to in the case when we take Σ = C and let M metrised complex curves z = z(τ),w = w(τ) where be Cn , with some general almost-complex z(τ),w(τ) are holomorphic functions of a complex structure J. It turns out, purely as a matter of linear variable τ. More generally we may consider complex algebra, that the R-linear maps n → n with curves in complex manifolds: parametrised by holo- J : C C 2 − morphic maps from Riemann surfaces. J = 1 can be neatly parametrised by an open set × What is a holomorphic map? Think of the simplest of n n complex matrices µ =(µαβ). Thus our al- case of a map f from C to C: a holomorphic function. most-complex structure is represented by a matrix- ∈ n The condition of holomorphicity is characterised by valued function µ( z ) of z C . A pseudoholomor- the Cauchy-Riemann equation phic curve is given by a solution of the system of partial differential equations ∂f =0. ∂z ∂zα ∂zβ This expresses the fact that the derivative of f, in the + µ ( z ) =0, ∂τ αβ ∂τ sense of multivariable calculus, is a complex linear map β Simon K. Donaldson is a Royal Society Research Professor which can be thought of as a deformation of the at Imperial College. His email address is s.donaldson@ ordinary Cauchy-Riemann equations, for the vector- ic.ac.uk. valued function z(τ). 1026 NOTICES OF THE AMS VOLUME 52, NUMBER 9 The passage to almost-complex manifolds allows I = f ∗(ω) us to move from the classical setting of holomorphic Σ curves in complex manifolds to a much wider, more flexible, world. Crucially, many aspects of the theory in two ways. On the one hand, the pointwise compati- bility between the structures means that I is essentially do not change greatly when we extend our ideas in the area of the image of f, measured in the Riemannian this way. We can express this by the slogan the local metric g. On the other hand, the condition that ω is theory of pseudoholomorphic curves is closely akin to closed means that I is a topological (homotopy) invari- that of holomorphic curves. Here, local can have two ant of the map f. So the areas of pseudoholomorphic meanings: either that we are studying the situation curves, in this situation, are controlled by straightfor- locally in the manifold M or locally in the space of ward topological data. This allowed Gromov to prove a maps. It is crucial here that we are considering curves, partial compactness theorem for the moduli spaces. rather than higher-dimensional objects. For any pair For example, consider as before the maps from the of almost-complex manifolds M,N the notion of a Riemann sphere to the complex projective plane. If we (pseudo)holomorphic map f : N → M makes sense, allow large and arbitrary deformations of the standard but if the real dimension of N is greater than 2, this almost-complex structure, then we cannot say much, be- is not a very useful concept. For example, on a generic cause the pseudoholomorphic curves may degenerate almost-complex manifold N of dimension greater in some very complicated way as we deform the struc- than 2 there are no nonconstant (pseudo)holomorphic ture and perhaps “disappear”. But if we restrict to almost- functions, even locally—this is exactly the source of complex structures compatible with a symplectic form, the integrability condition for complex-manifold the curves cannot degenerate, because their area is con- structures. trolled, and in fact Gromov showed that in this case the Our more precise form of our slogan is the state- curves must persist, however large the deformation. ment that if Σ is a compact Riemann surface, there is These two properties—the Fredholm theory and a nonlinear Fredholm theory which describes the compactness—lay the foundations for Gromov’s the- deformations of a given pseudoholomorphic curve ory, in which the pseudoholomorphic curves are used → f : Σ (M,J) . This means, roughly, that the defor- as a tool in symplectic topology. The curves have been mations are parametrised by a finite-dimensional used in two main ways. The first way is as geometric M manifold or moduli space , whose dimension can probes to explore symplectic manifolds: for example be computed from standard topological data. More- in Gromov’s result (later extended by Taubes) on the over, again roughly, the moduli space will deform uniqueness of the symplectic structure on the com- smoothly with variations in the almost-complex struc- plex projective plane, proved by sweeping out the ture J or the Riemann surface structure on Σ. For manifold by “lines” (i.e., the pseudoholomorphic curves example, suppose we take M to be the complex pro- of the same topological type as lines in the standard jective plane with its standard complex-manifold case). The second way is as the source of numerical structure and Σ to be the Riemann sphere. Then any invariants: Gromov-Witten invariants. In the simplest “line” (in the sense of projective geometry) in M, case, where our moduli space has dimension zero together with a choice of parametrisation, gives a and consists of a finite set of points, we might get an pseudoholomorphic curve. Thus the moduli space M integer invariant by counting these points. This sec- is a bundle over the dual plane with fibre PGL(2, C) — ond direction has been developed most extensively in the group of Möbius maps. The nonlinear Fredholm the years following Gromov’s paper. The theory of theory tells us that if we deform the almost-complex Floer homology is based on pseudoholomorphic curves structure slightly, while we probably cannot describe with boundary lying on a Lagrangian submanifold. This the pseudoholomorphic maps explicitly, we get a leads on to the notion of the Fukaya category. In four moduli space of the same general character. dimensions, Taubes discovered that the Gromov- Gromov’s insight was that the local understanding Witten invariants coincide with the Seiberg-Witten of the pseudoholomorphic maps furnished by the invariants, defined in a completely different way. In Fredholm theory extends to good global theory in the the case when the manifold M is in fact a complex situation where the almost-complex structure on M manifold, say an algebraic variety, the invariants are is compatible with a symplectic structure. Recall that related to classical enumerative problems in alge- a symplectic structure is given by an exterior 2-form braic geometry. The same invariants also appear in ω satisfying two conditions. One is pointwise and topological string theory, arising from Feynman inte- algebraic: at each point ω is a nondegenerate skew- grals. This has provided completely new insights and symmetric form on the tangent space of M. The other uncovered wonderful and intricate algebraic struc- is more global and differential geometric: the form ω tures in the invariants such as quantum cohomology.
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