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Dusa McDuff and Symplectic Geometry

Felix Schlenk

Communicated by Notices Associate Editor Chikako Mese

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Dusa McDuff has led three mathematical lives. In her early twenties she worked on von Neumann algebras and established the existence of uncountably many different algebraic types of -factors.⁠Footnote1 After an inspiring six months studying with Gel’fand in Moscow and a two-year post-doc in Cambridge studying topology, she wrote a “second thesis” while working with Graeme Segal on configuration spaces and the group-completion theorem, and investigated the topology of various diffeomorphism groups, in particular symplectomorphism groups. This brought her to symplectic geometry, which was revolutionized around 1985 by Gromov’s introduction of -holomorphic curves.

1

For a version of this text with more than 20 references see arXiv:2011.03317.

In this short text I will describe some of McDuff’s wonderful contributions to symplectic geometry. After reviewing what is meant by “symplectic” I will mostly focus on her work on symplectic embedding problems. Some of her other results in symplectic geometry are discussed at the end. More personal texts about Dusa can be found in 11b. Parts of this text overlap with the “Perspective” in 11b written jointly with Leonid Polterovich.

1. Symplectic

There are many strands to and from symplectic geometry. The most important ones are classical mechanics and algebraic geometry. I do not list these strands but refer you to 17b and to my 2018 survey in the Bulletin. Here, I simply give the definition.

Definition 1.

Let be a smooth manifold. A symplectic form on is a nondegenerate closed 2-form . A diffeomorphism of is symplectic (or a symplectomorphism) if .

The nondegeneracy condition implies that symplectic manifolds are even-dimensional. An example is with the constant differential -form

Other examples are surfaces endowed with an area form, their products, and Kähler manifolds.

If you begin your first lecture on symplectic geometry like this, you may very well find yourself alone the following week. You may thus prefer to start in a more elementary way. Let be a closed oriented piecewise smooth curve in . If is embedded, assign to  the signed area of the disc bounded by , namely or , as in Figure 1.1.

Figure 1.1.

The sign of the signed area of an embedded closed curve in .

Graphic for Figure 1.1.  without alt text

If is not embedded, successively decompose into closed embedded pieces as illustrated in Figure 1.2, and define as the sum of the signed areas of these pieces.

Figure 1.2.

Splitting a closed curve into embedded pieces.

Graphic for Figure 1.2.  without alt text
Definition 2.

The standard symplectic structure of  is the map

A symplectomorphism of is a diffeomorphism that preserves the signed area of closed curves:

A symplectic structure on a manifold is an atlas whose transition functions are (local) symplectomorphisms, and a symplectomorphism of  is then a diffeomorphism that preserves this local structure.

The standard symplectic structure of  is thus given by assigning to a closed curve the sum of the signed areas of the curves obtained by projecting to the coordinate planes . And a symplectic structure on a manifold is a coherent way of assigning a signed area to sufficiently local closed curves. The equivalence of the two definitions follows from

Darboux’s Theorem.

Around every point of a symplectic manifold there exists a coordinate chart  such that .

The group of symplectomorphisms of a symplectic manifold is very large. Indeed, for every compactly supported smooth function each time- map  of its Hamiltonian flow is a symplectomorphism. Symplectomorphisms of this form are called Hamiltonian diffeomorphisms. The Hamiltonian flow is the flow generated by the vector field  implicitly defined by

For one has , where is the usual complex structure on .

2. Symplectic Embedding Problems

By Darboux’s Theorem, symplectic manifolds have no local invariants beyond the dimension. But there are several ways to associate global numerical invariants to symplectic manifolds. One of them is by looking at embedding problems. Take a compact subset  of . By a symplectic embedding we mean the restriction to  of a smooth embedding of an open neighborhood of  that is symplectic, . In this case we write . For every , the largest number such that the dilate  symplectically embeds into is then a symplectic invariant of . Seven further reasons to study symplectic embedding problems can be found in my Bulletin article.

2.1. The Nonsqueezing Theorem

Now take the closed ball of radius centered at the origin of . (The notation reflects that symplectic measurements are 2-dimensional.) By what we said above, there are very many symplectic embeddings . However, none of them can make the ball thinner, as Gromov proved in his pioneering 1985 paper.

Nonsqueezing Theorem.

  only if   .

The identity embedding thus already provides the largest ball that symplectically fits into the cylinder of infinite volume. While there are many forms of symplectic rigidity, this theorem is its most fundamental manifestation. The theorem shows that some volume-preserving mappings cannot be approximated by symplectic mappings in the -topology.

In 95a, Lalonde and McDuff generalized the Nonsqueezing Theorem to all symplectic manifolds.

General Nonsqueezing Theorem.

For any symplectic manifold of dimension ,

Every good tool in symplectic geometry can be used to prove the Nonsqueezing Theorem. However, the technique of -holomorphic curves used by Gromov is the most influential one, and also the most important tool in McDuff’s work.

An almost complex structure  on a manifold  is a smooth collection , where is a linear endomorphism of  such that . The “almost” indicates that such a structure does not need to be a complex structure, i.e., does not need to come from a holomorphic atlas. Not all symplectic manifolds admit complex structures, but they all admit almost complex structures. A -holomorphic curve in an almost complex manifold  is a map  from a Riemann surface  to  such that

This equation generalizes the Cauchy–Riemann equation defining holomorphic maps . In this text, the domain of a -holomorphic curve will always be the usual Riemann sphere, namely the round sphere  whose complex structure  rotates a vector by . Even in this case, it is usually impossible to write down a -holomorphic curve for a given . But this is not a problem, since one usually just wants to know that such a curve exists.

Now assume that . Choose so large that after a translation the image of  is contained in . Compactifying the disc to the sphere  with its usual area form of area  and taking the quotient to the torus , we then obtain a symplectic embedding

where is the split symplectic structure on the product . We will see that . Since was arbitrary, Gromov’s theorem then follows.

Denote by the usual complex structure on , and let be an almost complex structure on  that restricts to on and that is -tame, meaning that for all nonzero . Such an extension exists, since -tame almost complex structures can be viewed as sections of a bundle over  whose fibers are contractible.

Lemma.

There exists a -holomorphic sphere through that represents the homology class of .

This existence result follows from Gromov’s compactness theorem for -holomorphic curves in symplectic manifolds. The key point for the proof of the compactness theorem is that is -tame, implying that -holomorphic curves cannot behave too wildly. The compactness theorem implies the lemma because the class of  is primitive in .

Figure 2.1.

The geometric idea of the proof.

Graphic for Figure 2.1.  without alt text

The Nonsqueezing Theorem readily follows from the lemma: the set

is a proper 2-dimensional complex surface in  through . By the Lelong inequality from complex analysis, the area of  with respect to the Euclidean inner product is at least . Using also that is symplectic we can now estimate

This proof of the Nonsqueezing Theorem also works, for instance, for closed symplectic manifolds  for which the integral of  over spheres vanishes. In the general case, however, there was trouble with holomorphic spheres of negative Chern number. By now, this trouble has been overcome thanks to the definition of Gromov–Witten invariants for general closed symplectic manifolds by the work of several (teams of) authors. McDuff helped to clarify the approach of Fukaya–Oh–Ono–Ohta in her joint work with Wehrheim; see 17c19. In 95a, however, Lalonde and McDuff circumvented all technical issues by using a chain of beautiful geometric constructions that reduced the General Nonsqueezing Theorem essentially to the case proved by Gromov. The most important and influential of these is a multiple folding construction. Its simple version was used in 95a to show how the General Nonsqueezing Theorem implies that for all symplectic manifolds Hofer’s metric on the group of compactly supported Hamiltonian diffeomorphisms is nondegenerate and hence indeed a metric; see §3.3. These two theorems were the first deep results in symplectic geometry proven for all symplectic manifolds.

2.2. Ball packings

We next try to pack a symplectic manifold by balls as densely as possible. Taking the open ball as target, let

be the percentage of the volume of that can be filled by symplectically embedded equal balls. Then:

The lower line gives the capacities

that are related to the packing numbers by . This table was obtained for by Gromov, for and a square by McDuff and Polterovich 94b, and for all  by Biran. This result is a special case of the following algebraic reformulation of the general ball packing problem

Ball Packing Theorem.

An embedding 2.2 exists if and only if

(i)

(Volume constraint)

(ii)

(Constraint from exceptional spheres) for every vector of nonnegative integers that solves the Diophantine system

and can be reduced to by repeated Cremona moves.

Here, a Cremona move takes a vector with to the vector

where , and then reorders .

Before discussing the proof, we use the theorem to obtain table 2.1. If is a solution of DE, then

that is,

For , this equation has finitely many solutions , and so DE has finitely many solutions for . They are readily computed:

and all these vectors reduce to by Cremona moves. For instance, for the problem the strongest constraint comes from the solution , that gives .

On the other hand, if is a solution of DE with , then

Hence the constraint is weaker than the volume constraint , and so .

The proof of the Ball Packing Theorem is a beautiful story in three chapters, each of which contains an important idea of McDuff. The original symplectic embedding problem is converted to an increasingly algebraic problem in three steps.

The starting point is the symplectic blow-up construction, that goes back to Gromov and Guillemin–Sternberg, and was first used by McDuff 91a to study symplectic embeddings of balls. Recall that the complex blow-up of at the origin  is obtained by replacing  by all complex lines in  through . At the topological level, this operation can be done as follows. First remove from an open ball . The boundary of is foliated by the Hopf circles , namely the intersections of with complex lines. Now is obtained by replacing each such circle by a point. The boundary sphere  becomes a 2-sphere  in  of self-intersection number , called the exceptional divisor. The manifold is diffeomorphic to the connected sum .

This construction can be done in the symplectic setting. If one removes from , then there exists a symplectic form on  such that outside a tubular neighborhood of the exceptional divisor  and such that is symplectic on  with . Given a symplectic embedding into a symplectic 4-manifold, we can apply the same construction to in to obtain the symplectic blow-up of by weight .

There is also an inverse construction. Given a symplectically embedded -sphere in a symplectic -manifold of area , one can cut out a tubular neighborhood of  and glue back , to obtain the “symplectic blow-down” of .

Now let be the smooth manifold obtained by blowing up the complex projective plane  in points. Its homology is generated by the class of a complex line and by the classes of the exceptional divisors. Let be their Poincaré duals. Every symplectic form on  defines a first Chern class , namely the first Chern class of any -tame almost complex structure. If we take a symplectic form  on  constructed as above via symplectic ball embeddings (that always exist if the balls are small enough), then is Poincaré dual to the class . Define to be the set of classes represented by symplectic forms with .

Compactifying to  with its usual Kähler form integrating to  over a complex line, and using that the classes  can be represented by symplectic -spheres, McDuff and Polterovich 94b obtained

Step 1.

There exists an embedding if and only if the class lies in .

Our symplectic embedding problem is thus translated into a problem on the symplectic cone . In Kähler geometry, the problem of deciding which cohomology classes can be represented by a Kähler form has a long history and is quite well understood. The solution of our symplectic analogue, however, needs different ideas and tools: call a class exceptional if and , and if can be represented by a smoothly embedded -sphere.

If is a symplectic form on  with , then any -symplectic embedded -sphere represents an exceptional class. Seiberg–Witten–Taubes theory implies that the converse is also true: every exceptional class  can be represented by an -symplectic embedded -sphere. This implies one direction of

Step 2.

lies in if and only if and for all exceptional classes.

This equivalence is remarkable. Of course, a necessary condition for a class  with positive square to have a symplectic representative is that evaluates positively on all classes that can be represented by closed symplectically embedded surfaces (and in particular on spheres). But the equivalence says that this is also a sufficient condition, and that it is actually enough to check positivity on spheres.

To prove the other direction in Step 2, one starts with an embedding of tiny balls of size and then changes the symplectic form on in class in such a way that these balls look large. This can be done with the help of the inflation method of Lalonde–McDuff 94a96a.

Inflation Lemma.

Let be a closed symplectic -manifold, and assume that the class with can be represented by a closed connected embedded -holomorphic curve  for some -tame . Then the class has a symplectic representative for all .

Figure 2.2.

The ray in the symplectic cone provided by the Inflation Lemma.

Graphic for Figure 2.2.  without alt text

I explain the proof for the case . In this case, the normal bundle of  is trivial, and we can identify a tubular neighborhood of  with , where is a disc. Pick a radial function with support in  that is non-negative and has . Let be the closed 2-form on  that equals on and vanishes outside of . Then and the forms are symplectic for all . Indeed,

where for the middle term we used that is symplectic.

Now take an embedding of tiny balls. By Step 1 we know that the class has a symplectic representative . We wish to inflate this form to a symplectic form in class . A first try could be to inflate directly in the direction to get up to . But this does not work, because