Symplectic 4-Manifolds
Dusa McDuff
Department of Mathematics, State University of New York at Stony Brook
Stony Brook, NY 11794, USA
§1. Introduction
A symplectic structure on a 2«-dimensional manifold Kis a 2-form œ which is closed
and non-degenerate, that is, dco = 0 and œ" does not vanish. The main example is
the standard form
COQ = dx1 A dx2 + dx3 A dx 4 + • • • + dx2n-1 A dx2n
on R 2 ". Darboux showed that every symplectic form is locally diffeomorphic to m0.
Thus symplectic manifolds, in contrast to Riemannian manifolds, are all locally
isomorphic. We are concerned here with questions of their global structure.
When n = 1 a symplectic form is just an area form, and the corresponding global
topology is well-understood. For example, the existence problem is trivial: clearly,
if V is compact, orientable and connected, it has a symplectic structure in each
non-zero class in H2(V; R). The uniqueness problem is almost as easy, since Moser
[M] showed that an area form (or, in higher dimensions, a volume form) is determined up to isotopy by its cohomology class. (Two forms co0 and co 1 on V are said
to be isotopie if there is a family gt of diffeomorphisms of V such that g0 = id. and
QÌ^o) = œi- They are symplectomorphic if g*(co0) = co1 for some diffeomorphism
gx) However, in dimensions >2, these questions are far from being understood.
This paper will describe selected topics in the 4-dimensional case, concentrating
on my own work. As we shall see, there are situations in which a symplectic form
is determined up to symplectomorphism by its cohomology class. For example,
Gromov's celebrated uniqueness theorem says that this holds for symplectic forms
on (CP2 which are non-degenerate on some embedded 2-sphere in the homology
class of CP 1 . However, this does not solve the problem of describing all symplectic
structures on CP 2 since it is unknown whether this condition holds for every
symplectic form on CP 2 .
Almost all the results mentioned below use Gromov's technique of
J-holomorphic spheres. They exploit the almost complex structures associated to
a symplectic structure, and so emphasise the similarities between Kahler, complex
and symplectic geometry. However, symplectic geometry cannot always be derived
from Kahler (or even complex) geometry. For example, there are many non-Kähler
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
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Dusa McDuff
symplectic 4-manifolds. (There are even symplectic 4-manifolds which admit no
complex structure: see [FGG].) There is also an example in [McD7] of a symplectic
4-manifold which has a disconnected "symplectically convex" boundary. By contrast, any complex manifold with pseudo-convex boundary must have connected
boundary. So far, the geometric significance of a symplectic manifold being Kahler
is not well understood. It would be very interesting to have some answers to the
following questions:
1.1
Is every simply-connected symplectic 4-manifold Kahler?
This is not true in dimensions > 10 by [McDl].
1.2 Is every symplectic form on a 4-manifold determined up to symplectomorphism
(or even isotopy) by its cohomology class?
This is not true in dimension 8, and in dimension 6 there are examples of symplectomorphic but non-isotopic forms: see [McD2].
1.3
Which 4-manifolds admit a symplectic structure?
Very little is known about this unless V is a fibration or the form co is invariant
under a non-trivial S1 action (see [Au, AH]).
§ 2. /-Holomorphic Curves in Almost Complex 4-Manifolds
Because the linear symplectic group deformation retracts onto the unitary group,
there is a homotopy class of almost complex structures J associated to a symplectic
form co. (Recall that an almost complex structure J on F is an automorphism of the
tangent bundle TV of V such that J2 — —Id. Thus TV may be considered as a
complex n-dimensional vector bundle, with J corresponding to multiplication by
i.) When n = 1, every almost complex structure is integrable, that is, it comes from
an underlying complex structure' on V. This fact is one of the reasons why Jholomorphic curves have such nice properties.
Gromov realised that one could get a handle on the geometry of a symplectic
manifold by considering properties of a more restricted family of almost complex
structures on V, namely the set /(œ) of all J such that co(v, Jv) > 0 for all non-zero
ve TV. Such J are said to be co-tatne. Recall from [Gl]' that the set /(co) is
non-empty arid contractible.
A map / from a Riemann surface (S, J0) to (F, J) is said to be J-holomorphic
if / satisfies the generalized Cauchy-Riemann equation df o J0 = J o df. Because
this equation is elliptic, the space MP(J, A), of all J-holomorphic maps / which
have a fixed compact domain S and represent a fixed homology class A, is a finite
dimensional manifold for generic J. Moreover, if J is cò-tame, the quotient of this
space by the automorphism group G — Aut(S) of (5, J0) is either compact, or has a
nice compactification. In fact, as the following proposition shows, homological
conditions are often enough tö guarantee compactness. For simplicity, we state a
result about rational curves, i.e. curves C = f(S) which are ./-holomorphic images
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543
of spheres. The first Chern class of the complex vector bundle (TV, J) will be denoted
by c. A proof may be found in [GÌ] or [McD4].
Theorem 2.1 [ G l ] . Suppose that Cû is integral (i.e. [GL>] e H2(V, Z)) and that co(A) = 1.
Then, for generic co-tame J, the moduli space Mp(J, A)/G of rational A-curves is a
compact manifold of dimension 2(c(A) + n — 3). Moreover, the bordism class of the
evaluation map
e(J):Mp(J,A)xGS2-^V
given by (f, z)\-^f(z) is independent of J e /(co).
This result already has many interesting consequences: see [Gl, 2] for example.
In dimension 4 the theory is much sharper because, as the following results show,
the geometric behaviour of the curves is governed by their homology classes.
Theorem 2.2. Positivity of Intersections ([Gl]). Let C and C be distinct closed
J-holomorphic curves in the almost complex 4-manifold V, Then C and C have only
a finite number of intersection points. Each such point x contributes a number kx > 1
to the algebraic intersection number C • C. Moreover, kx = 1 iff the curves C and C
intersect transversally at x.
In particular, the curves C and C are disjoint iff C • C = 0. This result is almost
obvious if neither curve is singular at the point of intersection. The best way to
prove the result in general is to perturb the curves so that their intersections avoid
the singular points: see [McD8], The next result gives a homological criterion for
a curve to be embedded. By analogy with the integrable case, we define the virtual
genus g(C) of a closed curve C in an almost complex 4-manifold to be the number
g(C) = 1 + \(C - C — c(C)). If C is an embedded copy of a closed Riemann surface
S, it is easy to check that the virtual genus g(C) equals the genus g0 ofS. Conversely:
Theorem 2.3 [McD5]. Let C cz V4 be the J-holomorphic image of a closed Riemann
surface S of genus g0. Then g(C) is an integer which is greater than or equal to g0,
with equality if and only if C is embedded.
The crucial point here is that one can define for each point x e C = Im / a local
self-intersection number L. Int.(/, x) of Im / at x. One proves that this is always
>0, and is strictly > 0 iff x is a singular point off. In fact, it is not hard to see that
one can choose local coordinates near x G V so that the lowest order terms in the
Taylor expansion of / when regarded as a polynomial in z and z involve only z.
More precisely, one can suppose that / : D -> C 2 has the form
z !-• (zk, zm) + terms in z, z of order > m,
where m > k > 1 and k does not divide m. (Here, D is the unit disc in C, and we
identify a neighbourhood of x in V with a neighbourhood of {0} in C 2 in such a
way that J corresponds to an almost complex structure on C 2 , which equals the
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standard almost complex structure J 0 at {0}.) If m and k are mutually prime, it is
easy to see that L. Int.(/, x) = L. Int.(/ 0 , x) = (m — l)(/c — 1), where/ 0 ( z ) = (**> zm)In general, the proof of Theorem 2.3 involves perturbing/, using the techniques of
Nijenhuis and Woolf. A more detailed analysis of the generalized Cauchy-Riemann
equation shows that the jet of the J-holomorphic map at a critical point is dominated
by its J 0 -holomorphic part. More precisely,
Theorem 2.4 [McD8]. Let f : (D, 0) -> (C 2 , 0) be a J-holomorphic map, where J is an
almost complex structure on C 2 which equals the standard structure J 0 at {0}. Then
there is a J0-holomorphic mapf0 such that for sufficiently small e > 0 the knots formed
by intersecting f(D) and fQ(D) with the sphere S3(s) of radius e about {0} are isotopie.
§ 3. Applications to Symplectic 4-Manifolds; Uniqueness Results
In order to apply this theory one needs to know that the bordism class of the
evaluation map e(J) of Theorem 2.1 is non-trivial. For example, suppose that there
is a symplectically embedded 2-sphere C in V4". It is not hard to see that C can be
parametrized in such a way that it is J-holomorphic for some co-tame J. If in
addition the self-intersection number p = CC is > — 1 one can show that that the
corresponding moduli space is generically non-empty. Thus V contains a 2(p -f- 1)dimensional family of (unparametrized) curves which may be used to analyse the
structure of V.
The case p — — 1 is rather special since then, by Theorem 2.2, there is at most
one J-holomorphic curve in each homology class. Such an embedded 2-sphere is
said to be an exceptional sphere. By analogy with the complex case, a symplectic
4-manifold which contains no exceptional spheres is said to be minimal.
The following results describe all minimal symplectic 4-manifolds which contain
a symplectic 2-sphere with non-negative self-intersection.
Theorem 3.1 [McD6]. Let (V, co) be a closed symplectic 4-manifold (V, co) which
contains a symplectically embedded 2-sphere C with C • C = 0. Suppose further that
(V — C, œ) is minimal. Then (V, co) is ruled, that is, there is a fibration n : F-> M
whose fibers are symplectically embedded 2-spheres, one of which we may assume to
be C. Moreover the symplectic form co is determined up to symplectomorphism by its
cohomology class.
Sketch of proof. Let A be the homology class of C. One first shows that, for any
co-tame J, the set of y4-curves may be compactified by adding "^-cusp-curves". These
must consist of two exceptional curves joined at one point, and, because A-A = 0,
must be disjoint from all J-holomorphic v4-curves by Theorem 2.2. Therefore, if J
is chosen so that C is J-holomorphic, the minimality of V — C implies that the
moduli space of ,4-curves is compact. Using Theorem 2.2 again, one sees that there
is at most one >4-curve through each point. It follows that the evaluation map e(J)
has degree 1, and that there is exactly one curve through each point. Now observe
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that these curves are embedded by Theorem 2.3. With a little more work, one can
show that they form the fibers of a fibration.
To prove the uniqueness statement, one first reduces to the case V = S2 x S2
by some cutting and pasting. Next one constructs a symplectic section of V in the
class B = [pt. x 5 2 ]. We are now in a situation considered by Gromov. Since
B-B = 0, the B-curves also fiber V Thus there is a symplectomorphism from (V, co)
to (S2 x S2, Q) where Q is non-degenerate on all the slices pt. x S2 and S2 x pt. The
result now follows from:
Lemma 3.2 [K]. Suppose that oo is a symplectic form on the product M' x M" of two
compact 2-manifolds which is non-degenerate on all the slices M' x pt. and pt. x M".
Then co is isotopie to a product form.
Proof. This is a straightforward calculation.
An explicit description of all possible symplectic ruled 4-manifolds is given
in [McD6] §4. Basically, one gets all the Kahler ruled surfaces considered as
symplectic manifolds, i.e. one forgets the complex structure, since this is not relevant.
However, note that when F is a non-trivial bundle the cohomology classes a = [co]
which can be realised in this way satisfy the condition (a(C))2 < a2(V), and so do
not include every class a with a2(V) > 0. Note also that (V, co) is itself minimal unless
V is the non-trivial bundle over S2, in which case (V, co) is (DP2 blown up at one
point with a standard Kahler form.
There is a corresponding and even simpler result when the sphere has positive
self-intersection. It is the symplectic analogue of the fact that the only minimal
complex surfaces which contain a rational curve of positive self-intersection are (DP2
and (CP1 x CP 1 .
Theorem 3.3 [McD6]. Let (V, co) be a closed minimal symplectic 4-manifold (V, co)
which contains a symplectically embedded 2-sphere C with p = C • C > 0. Then (V, co)
is symplectomorphic either to (DP2 with its standard Kahler form or to S2 x S2 with
a product form. In the former case p must equal I or 4 and C is either (DP1 or a quadric,
and in the latter case p is even, and C is the graph of a holomorphic self-map of S2.
We will see in § 4 below that, just as in the complex case, one can get rid of
exceptional spheres by blowing them down. Thus these two theorems classify all
symplectic 4-manifolds which contain a symplectically embedded 2-sphere C with
C • C > 0, modulo the question of the uniqueness of blow ups. They generalize and
sharpen uniqueness results which Gromov obtained in [ G l ] for the manifolds
S2 x S2 and (DP2. One expects that there are corresponding results for manifolds
which contain a symplectically embedded 2-manifold M of higher genus, provided
that M • M (or equivalently, c(M)) is sufficiently large to guarantee that the index
(i.e. formal dimension of the corresponding moduli space) is positive. But it is
unknown what happens if M • M is small. For example, if co is a symplectic form on
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T 2 x S2 which is non-degenerate on one slice T2 x pt., it is not known whether co
must be symplectomorphic to a product form.
Note. These uniqueness results all derive from considering families of J-holomorphic
spheres, and so apply only to 4-manifolds which contain such spheres. In some
circumstances it is better to consider J-holomorphic discs whose boundary is
constrained to lie on a 2-manifold M which is totally real, i.e. J(TXM) n TXM = {0},
except at isolated points. This approach yields the sharpest results when M is a
2-sphere which is contained in a J-convex hypersurface: see [E, B]. Note that M
need not be symplectic.
§ 4. Blowing Up and Down, and Embeddings of Balls
In the category of complex manifolds, there is a well-known operation of blowing
up a point in which one replaces a point x by the space of all complex lines through
x. More formally, one cuts out a neighbourhood of x which is biholomorphic to C"
and glues back in a copy of the total space L of the canonical line bundle over (DP'1"1
by identifying the complement of the zero section in L with the deleted neighbourhood C" — {0} of x. Thus the point x is replaced by an "exceptional divisor" which
is a copy of (DP"'1 with normal bundle L. When n = 2, this divisor is simply an
embedded rational curve C with CC = —1. The converse process of "blowing
down" replaces an exceptional divisor by a point.
In the symplectic case, when one blows down an exceptional divisor P one gets
not a point but a ball B(X) whose radius X is related to the cohomology class
of co by the formula nX2 = co(A), where A is the positive generator of H2(P; Z).
Conversely, in order to "blow up a point" of (V, co) one chooses a symplectic
embedding gofB = B(X) into V, cuts out the interior of the ball g(B) and then forms
the blow up V by identifying the boundary sphere g(dB) with (DPn~x via the Hopf
map and smoothing the result: see [McD3]. It is not hard to check that co induces
a symplectic form co on V. Note that V is diffeomorphic to the manifold obtained
by putting any integrable tame almost complex structure on a neighbourhood of
some point x e V, and then blowing V up at x. The point of the above description
is that it gives the construction of co.
Clearly, by successive blowing down of a finite sequence of exceptional divisors
one can make any manifold minimal, i.e. such that it does not contain any exceptional divisors. When n = 2, one can use the theory of J-holomorphic curves to
control the exceptional divisors and hence can show:
Theorem 4.1 [McD3]. Every symplectic 4-manifold (V, co) covers a minimal symplectic
manifold (V, oo') which may be obtained from V by blowing down a finite collection
of disjoint exceptional curves. Moreover, (V, co') is determined up to symplectomorphism by the homology classes of the blown down curves.
Another important question is that of the uniqueness of blow ups. Not much is
known about this except when one blows up one point of the standard (DP2. In this
Symplectic 4-Manifolds
547
case, the resulting manifold (X, co) is the non-trivial S2 bundle over S2. Moreover,
it is not hard to show that X contains a symplectically embedded 2-sphere C with
C • C = 0. Hence the theory of § 3 applies, and one can show that all cohomologous
forms are symplectomorphic. As a corollary, one finds:
Theorem 4.2 [McD3]. For each X < 1, the space of symplectic embeddings of the ball
B(X) into the open unit ball in C 2 is connected.
The analogous statement is true if the target space is the standard (DP2. However,
this result is surprisingly delicate, and it is unclear if this holds for any other target
space, even those as simple as S2 x S2 or a convex subset of C 2 . This question is of
interest because of the following result.
Proposition 4.3 [McD3]. Let Xk be the complex surface obtained by blowing (DP2 up
at k points, and let S be a fixed copy of (DP1 in Xk (disjoint from the blown up points).
Then, there is a non-Kähler structure on Xk which is non-degenerate on S iff there are
X1,...,Xk such that the space of symplectic embeddings of the disjoint union \\{ B(Xf)
into B(l) is disconnected.
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[AH]
[B]
[E]
[FGG]
[Gl]
[G2]
[K]
[McDl]
[McD2]
[McD3]
[McD4]
[McD5]
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