UPDATE added 14.08.2018: A gap was recently discovered in the paper that this post is about, and it has not yet been fixed. You can still learn some interesting things from this post, but should be aware that a few of its claims may be premature. I have discussed the precise nature of the gap in a more recent post.
Let’s talk some more about transversality.
“So… the inability of transversality to exist, or the inability to create the situation in which transversality is a potential is not unattainable. This is of course different than to say, transversality is possible or even to say, transversality is not impossible. And if all of this seems funny it is because it is, in fact, comedy, by definition, through the irreducibility of nothing.” Excerpted from a transhumanist(?) essay I found when I googled the word “transversality”.
Most of my readers know that transversality is a stressful topic in symplectic topology. Moduli spaces of pseudoholomorphic curves are generically nice smooth objects… except for when they aren’t, which is generally the case if they contain multiple covers, which they typically do. Various people have described various remedies for this over the years, usually involving words like “virtual”, “Kuranishi” or “polyfold”, and there is probably a theorem that for every proposed remedy, you can find at least one well-respected symplectic topologist willing to make denigrating and sarcastic remarks about it after a few drinks.
But that isn’t what this post is about. I want to talk about something more naive and concrete. I want to answer the following questions:
When is it possible for a multiply covered holomorphic curve to be regular? If it cannot be regular, then why not, and what is true instead?
I was surprised to learn sometime last year that these questions actually admit reasonable answers, and I think a lot of people who use holomorphic curves in their research will find the answers interesting and potentially useful. The details are written up in a preprint I put up last year called Transversality and super-rigidity for multiply covered holomorphic curves, which, among other things, answered an open question about the Gromov-Witten invariants of Calabi-Yau 3-folds. In this post I will try to explain some of the main ideas.
First, here are two sample theorems from the paper:
Theorem B. For generic in a symplectic manifold of dimension at least 4, every unbranched cover of a closed somewhere injective -holomorphic curve is Fredholm regular.
Theorem C. Let denote a connected component of the moduli space of closed muliply covered -holomorphic curves in a symplectic manifold, where is a somewhere injective curve with a prescribed number of critical points with prescribed critical orders, and is a holomorphic branched cover with prescribed numbers of critical values and branch points with prescribed branching orders. Then if is generic, we have the following alternative:
- No curve in is Fredholm regular.
- Regularity is achieved for all curves in an open and dense subset of .
The precise nature of the “moduli space of multiple covers” I’m referring to in this second result deserves more explanation, so I’ll come back to this below. The result says in effect that for each branched cover, transversality is either completely impossible for topological reasons or it is achieved with probability 1 for generic , meaning there may be a nonempty subset in the space of multiply covered curves for which transversality fails, but this subset has measure 0. The first result is stronger since it does not have such a probabilistic caveat, but it requires the stronger hypothesis that the cover has no branch points.
Since I’m focusing on transversality in this post, I haven’t yet mentioned Theorem A in the paper, which states that in dimensions six and above, all somewhere injective index 0 curves are “super-rigid” for generic . If you don’t already know what that means, then you don’t know why it was an exciting enough result to get top billing in my paper, but I’ll say a bit more about this below.
Why are we having this conversation?
Let’s clear up one thing before we continue: no matter how cleverly I can prove that certain multiple covers achieve transversality, there is no hope that results like this will ever fully replace virtual cycle or abstract perturbation methods for defining things like Gromov-Witten theory or SFT. That is not the goal. You might therefore ask: if more abstract methods are necessary anyway, what’s the point of struggling to understand transversality issues for actual holomorphic curves?
The answer is that it depends what kind of problem you want to solve. Abstractly perturbing the nonlinear Cauchy-Riemann equation destroys symmetry, which is good if achieving transversality is your only goal—transversality and symmetry are famously incompatible—but it also kills properties that sometimes carry interesting information. I’ll give you two examples:
- Most of the celebrated results about symplectic 4-manifolds or contact 3-manifolds based on holomorphic curve theory rely crucially on positivity of intersections, and so do some of the enumerative invariants in this context such as Taubes’s Gromov invariant and its 3-dimensional cousin, embedded contact homology (ECH). But positivity of intersections holds only for actual -holomorphic curves, not for solutions to abstractly perturbed Cauchy-Riemann type equations.
- The Gopakumar-Vafa formula for Gromov-Witten invariants on Calabi-Yau 3-folds is often interpreted as a relation between the count of embedded curves and their “multiple cover contributions”. This notion ceases to have any meaning as soon as perturbations eliminate the distinction between simple curves and multiple covers.
My own motivation to think about this stuff came mainly from the first example, as I strongly suspect that the phenomena behind Theorems A, B and C will end up filling in a crucial missing piece of the analytical picture necessary for completing the definition of ECH, i.e. for defining cobordism maps and proving invariance without reference to Seiberg-Witten theory. Also, while it’s pretty clear that not all of the important transversality problems of SFT can be solved via these methods, I have a feeling they might at least go far enough to define something akin to the “semipositive case” of SFT, thus giving a rigorous version of the theory that works in some settings and is more concrete than the general case.
The proofs of Theorems B and C are based on a general result about the local structure of the space of multiply covered curves. Here is an outline of the idea.
The normal Cauchy-Riemann operator
You may be familiar with the fact that an immersed holomorphic curve is Fredholm regular if and only if a certain linear Cauchy-Riemann type operator on its normal bundle , the so called normal Cauchy-Riemann operator
is surjective. If you’re only reading this post for an explanation of super-rigidity, then this is all you need to know about and you can now skip to the next subsection. But to discuss Theorems B and C, we also need the generalization of this statement to non-immersed curves.
Recall that if is connected and non-constant, then it is necessarily immersed outside of a discrete set , and at each of the critical points , it has a well-defined critical order, which is a positive integer. For instance, the injective holomorphic curve has critical order at . That this is well defined in general can be regarded as a consequence of the well-known theorem of Micallef and White on the structure of singularities for -holomorphic curves, though there are also other ways to see it. A related fact is that there is a well-defined and smooth complex line bundle whose fiber at each immersed point is simply . It is therefore natural to choose a complementary complex subbundle and call it the generalized normal bundle of .
Now if denotes the usual linearized Cauchy-Riemann operator at , the splitting decomposes it into block form as
It is easy to check that the diagonal terms and are Cauchy-Riemann type operators on the bundles and respectively, while the off-diagonal terms are zeroth-order terms (in fact vanishes identically). The Fredholm index of is the same as the virtual dimension of the moduli space if is immersed, though in general they differ because differs from by the algebraic count of critical points in . What you may find more surprising is that even in the non-immersed case, fully characterizes transversality:
Lemma (see Theorem 3 in [W. 2010]). A non-constant -holomorphic curve is Fredholm regular if and only if its normal Cauchy-Riemann operator is surjective.
Cauchy-Riemann operators with symmetry
Abstract approaches to the transversality problem are typically based on the premise that transversality and symmetry are incompatible, therefore you need a perturbation that breaks the symmetry, e.g. by letting the complex structure on depend on points , or replacing the equation with or various similar ideas. But if you prefer to keep the symmetry instead of breaking it, then you have to make use of it somehow. Here we can use an important observation that I learned from a paper of Taubes.
Assume is a multiply covered -holomorphic curve, where is a simple curve and is a -fold holomorphic branched cover. In this case, the symmetry of endows its normal Cauchy-Riemann operator with a natural splitting
Here are the real irreducible representations of some finite group , each of which has an associated Fredholm operator that is equivalent to a Cauchy-Riemann type operator on some bundle, and the are integers. To understand what’s going on here, it’s instructive to start with the simplest nontrivial example: assume , so is a branched double cover and therefore has automorphism group . If denotes the generator of this group, then the space of sections of has a natural splitting as , where
The target bundle for inherits a similar splitting, and the symmetry of means that preserves the splitting, thus it decomposes into two pieces
The relation between this and the more general splitting above is as follows: , and are its unique trivial and nontrivial irreducible representations respectively, , , and .
Notice that there is a canonical bijection between and sending any section in the latter to the symmetric section . Under this identification, is identified with , the normal Cauchy-Riemann operator for the underlying simple curve. This observation is obviously helpful since we already know how to prove that is surjective for generic ; the transversality problem for has thus been reduced to a question about the operator .
For covers of arbitrary degree , one can still define a natural splitting in which is identified with , but the factor can be split further if . This is a somewhat longer story, so I will make it the subject of a separate post, but the picture in general is as sketched above: there is a finite group whose real irreducible representations give rise to so-called twisted Cauchy-Riemann operators , which are all Fredholm. These should be regarded as the fundamental building blocks of transversality theory for multiple covers. Here we can always assume is the trivial representation of and , so the first factor in the splitting can be identified with , and the rest is a splitting of . The group always has order at least , as it is the automorphism group of some normal branched cover of that factors through . Such normal covers always exist, and in fact there is a canonical one up to isomorphism, with degree at most , whose automorphism group I refer to as the generalized automorphism group of . If is already normal, then the canonical normal cover factoring through it is isomorphic to itself, so we can take .
Moduli spaces of multiple covers
Given a tame almost complex structure , an integer , a finite group and some additional combinatorial data to be specified below, we now consider a moduli space
- is a somewhere injective -holomorphic curve with some prescribed nonnegative number of marked points, each of which is contrained to be a critical point with a prescribed critical order, and is immersed everywhere else.
- is a holomorphic map of closed Riemann surfaces with degree and generalized automorphism group isomorphic to , with some prescribed nonnegative number of marked points such that every branch point is one of the marked points and each has a prescribed branching order. Moreover, we also prescribe whether any given pair of branch points are mapped to the same point, thus determining the total number of critical values.
By forgetting the marked points, we can regard as a subset of the usual moduli space of smooth unparametrized -holomorphic curves (including both simple curves and multiple covers). It should be clear that every -fold multiply covered -holomorphic curve belongs to for appropriate choices of the finite group and the combinatorial data that prescribes the critical points of and branching behavior of . Note that the set of all possible choices of this combinatorial data is countable. It is also easy to see that is a smooth manifold for generic , as the prescribed critical points determine a smooth submanifold of the usual space of simple curves, with codimension depending on the critical orders, while for any fixed , the moduli space of branched covers with fixed branching data is a manifold of real dimension twice the number of critical values of . This is all more or less standard.
The point of prescribing critical and branching behavior in this way is that as and move around to produce various multiple covers , the splitting
changes continuously. In particular, prescribing the critical points of makes the generalized normal bundle into a continuous family of bundles parametrized by ; this would not be true if were allowed to move arbitrarily through the space of somewhere injective curves, since its critical points could then disappear, changing the topology of . Similarly, prescribing the branching data of ensures that every nearby branched cover with the same branching data has the same generalized automorphism group.
A stratification theorem
We can now state the theorem that makes everything else in this story work.
For the moduli space described above, let for denote the distinct real irreducible representations of , and denote
By standard results in representation theory, the endomorphism algebra is always isomorphic to either , or the quaternions , hence . This alternative depends on whether the complexification of is also irreducible (as a complex representation) or is the direct sum of a complex irreducible representation with its dual (which is either isomorphic to or not). It is useful to note that the endomorphisms endow and consequently the domain and target spaces of with -module structures such that is -linear.
Now for any tuples of nonnegative integers and , we consider the subset
and for all
Note that since each of the operators is Fredholm, this subset will be automatically empty unless and are chosen so that is the index (with respect to ) of for every .
Theorem D. For generic , is a smooth submanifold of with
The theorem can also be stated in a more general form with replaced by any smooth finite-dimensional family of almost complex structures, so that the definitions of and become slightly more general, but the formula for the codimension remains the same. This means that in addition to proving transversality results, Theorem D can be used as the starting point of a general bifurcation theory for multiply covered curves under generic deformations of the almost complex structure.
I’ll wait until a followup post before making any attempt to explain why Theorem D is true, but I’d now like to discuss a few of its implications. The first is Theorem C above: the splitting of already tells us that there is no hope for a multiply covered curve to be Fredholm regular unless
for every such that , as cannot be surjective unless every operator in the splitting is surjective. In principle, the indices of these twisted operators can be computed, and they give us a topological obstruction to the regularity of . But Theorem D now supplements this with the following insight: if the topological obstruction vanishes, then almost every element of will definitely be regular, as the non-regular curves all live in substrata that have strictly positive codimension!
Why unbranched double covers are regular
With Theorem D in hand, various generic transversality results can now be proved via dimension-counting arguments. To illustrate this, let’s prove a special case of Theorem B: we claim that for generic , all unbranched double covers of immersed somewhere injective curves are Fredholm regular. Let and denote the virtual dimensions of the moduli spaces containing and respectively, and note that since both are immersed, these dimensions match the Fredholm indices of the respective normal Cauchy-Riemann operators. Combining the Riemann-Hurwitz formula for branched covers with the Riemann-Roch formula for Fredholm indices, we find
Note that without loss of generality since is generic. Then is also the dimension of the space in which any immersed double cover of lives, as there are no critical points to lower the dimension of the space of simple curves, and the space of unbranched covers does not have any moduli of its own. The splitting of takes the simple form , with , and the relevant representations in this picture are the unique trivial and nontrivial irreducible representations of , both of which remain irreducible after complexifying them, so the dimensions of the corresponding -equivariant endomorphism algebras are
The simplicity of the splitting also provides enough information to deduce the index of :
We can assume is surjective since is Fredholm regular, so if is not regular, it can only be because , meaning that belongs to a space of the form with and , where , , , and . Putting all of this information together, now lives in a smooth manifold with dimension
and this is strictly negative since by assumption. This is a contradiction.
In this example we were lucky because the index of could be deduced without any further computation. In more complicated situations, it is often necessary to compute the indices of the twisted operators directly, but this can be done. The values of these indices are typically what determines whether transversality is achievable or not in any given situation.
One famous setting where transversality is clearly impossible is when the symplectic manifold is 6-dimensional with vanishing first Chern class, sometimes called a symplectic Calabi-Yau 3-fold. The usual formula for virtual dimensions then gives
for all holomorphic curves , whether multiply covered or not. For covers with nonempty set of branch points, this is clearly not the desired answer, as the freedom to move branch points around produces nontrivial moduli in the space of branched covers, producing an actual moduli space that is guaranteed not to be 0-dimensional. The best one can hope for in this setting is a kind of “Morse-Bott” or “clean intersection” condition, saying that the linearized deformation operator for a branched cover has kernel of dimension only as large as it manifestly must be, given the dimension of the moduli space of branched covers. This condition turns out to be equivalent to
and we say that a simple curve is super-rigid if it is immersed and all of its multiple covers have injective normal Cauchy-Riemann operators. If this condition holds for all simple curves, then it prevents any sequence of simple curves from converging to any multiple cover, so that for each genus and homology class, only finitely many simple curves exist. Moreover, the moduli spaces of branched covers over these simple curves have well-defined obstruction bundles, whose Euler classes determine the Gromov-Witten invariants (see e.g. this paper by Zinger).
Theorem D gives rise to a proof that for generic in any symplectic manifold of dimension , all simple curves of index zero are indeed super-rigid. Looking again at our splitting of , you can see that the first step in proving this must be to check that whenever for an immersed simple curve of index zero, all the twisted operators satisfy
as clearly could not be injective without this. Actually, the necessary dimension-counting argument requires a stricter upper bound for the case when the representation is faithful. To illustrate this, let’s focus again on the case of a degree 2 cover and try to prove that will be injective for generic if and . Assume is a 2-fold branched cover with branch points, which in this case is the same as the number of critical values. Then assuming our simple curve is immersed (which is always true for simple index 0 curves if is generic), the cover lives in a space of real dimension
Since is generic, we can assume is Fredholm regular, hence the index zero operator is an isomorphism. If is not injective, it therefore means that is not injective, so belongs to a stratum with and where , and . To make full use of this, we need more precise information about . This is the factor in the splitting that corresponds to a faithful representation of ; the other factor corresponds to the trivial (and therefore unfaithful) representation of , but we do not need to care about it since is identified with , so that the standard analysis for simple curves has already told us everything we wanted to know. To compute , note that by the Riemann-Hurwitz formula, the count of branch points of the cover satisfies
with in the present situation. Combining this with the Riemann-Roch formula then gives the relation
thus . This implies
and plugging this in to compute , Theorem D now implies that lives in a substratum with dimension
and this is strictly negative since we assumed . This is of course a contradiction, and thus proves that the super-rigidity condition is satisfied for branched covers of degree two.
I should be careful about attributions here: the partial super-rigidity result I just sketched is not due to me, but was proved first in a paper by Eftekhary, using a quite similar approach based on ideas of Taubes. Theorem D, however, is strong enough to prove super-rigidity for all branched covers of index zero curves. As in the transversality problems discussed above, the main additional piece of input one needs for this is a computation of the indices of the twisted operators , which comes more or less for free when but is more involved in the general case.
This post is more than long enough already, so I’ll conclude it with an I-owe-you. In order to convince skeptics that this whole story isn’t just wishful thinking, there are two important things I need to explain:
- How to define the splitting of for general multiple covers , in particular in cases where is a non-normal branched cover;
- Why Theorem D is true.
I intend to devote a separate post to each of these topics sometime in the near future. Not sure when, but soon.
Update: Both of the followup posts I promised now exist. See “Regular presentations and twisted Cauchy-Riemann operators” and “The transversality machine”.
Update 2 (19.12.2017): I have edited this post (and will shortly also be updating the paper on the arXiv) to correct a minor error in representation theory. The correction necessitated a change in the definition of the space , so that some dimensions that used to be real are now dimensions over the endomorphism algebra . This change (fortunately) has no adverse impact on Theorems A, B and C. Many thanks to Thomas Walpuski and Aleksander Doan for catching the error.
 I use the term Fredholm regular to refer to the condition that is usually meant when we say that a holomorphic curve is “transversely cut out,” i.e. it implies via the implicit function theorem that a neighborhood of the curve in its moduli space is a smooth manifold (or orbifold if it has nontrivial automorphism group). In the present context, the word “regular” also arises with a different meaning, referring to a branched cover whose degree matches the order of its automorphism group. We must be careful to distinguish between these two meanings.
 Taubes’s paper Counting pseudo-holomorphic submanifolds in dimension 4, which explains all of the technical details needed for defining the Gromov invariant, has been something of an anomaly in the symplectic literature for 20 years. I can think of hardly any other papers that address the multiple cover problem by actually proving that transversality generically holds (as indeed it does for the doubly covered tori that the Gromov invariant counts; Theorem B is a generalization of this statement). My own work on this subject began in earnest when I started asking the question, “how did Taubes prove that those tori are regular?”, and it turned out that none of the people I would have expected to know the answer actually did. I suspect most people believed it to be a phenomenon that only happens in dimension four, but Theorem B tells you that that is false. A version of Taubes’s splitting of Cauchy-Riemann operators also appears in Eftekhary’s paper that proved some cases of super-rigidity.
 Recall that a covering map is called normal (or also regular) if for every and every pair , there exists a deck transformation sending to . This definition extends in an obvious way to a branched cover of Riemann surfaces , so is normal if and only if the order of its automorphism group is the same as its degree. In general, . (I personally prefer the term “regular” in place of “normal”, but I’m avoiding using it in this post so as to prevent confusion with the term “Fredholm regular”.)
 Readers unfamiliar with the real representation theory of finite groups will find these details explained in the classic textbook by Serre.
 My dimension-counting argument for super-rigidity conspicuously fails in dimension four, and I do not know whether the result is true in that case. I haven’t thought about it very much, since I don’t immediately know what it would be useful for, but what I do know is the following: one can use uniquely low-dimensional methods to prove that super-rigidity holds in dimension four for index zero curves of genus zero or one. The genus zero case is more or less an example of “automatic transversality,” and can be deduced from the classic paper by Hofer-Lizan-Sikorav on that subject. For genus one curves, the proof uses a method known as the Hutchings magic trick, thanks to the blog post in which Hutchings first described it. (His description was phrased in terms of the ECH index inequality, but for closed curves in symplectic 4-manifolds, it basically reduces to the standard adjunction formula.)