Transversality for multiple covers, super-rigidty, and all that

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 J in a symplectic manifold of dimension at least 4, every unbranched cover of a closed somewhere injective J-holomorphic curve is Fredholm regular.[1]

Theorem C. Let {\mathcal M} denote a connected component of the moduli space of closed muliply covered J-holomorphic curves u = v \circ \varphi in a symplectic manifold, where v is a somewhere injective curve with a prescribed number of critical points with prescribed critical orders, and \varphi is a holomorphic branched cover with prescribed numbers of critical values and branch points with prescribed branching orders. Then if J is generic, we have the following alternative:

  1. No curve in {\mathcal M} is Fredholm regular.
  2. Regularity is achieved for all curves in an open and dense subset of {\mathcal M}.

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 J, 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 J. 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:

  1. 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 J-holomorphic curves, not for solutions to abstractly perturbed Cauchy-Riemann type equations.
  2. 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.

Stratification

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 u : (\Sigma,j) \to (M,J) is Fredholm regular if and only if a certain linear Cauchy-Riemann type operator on its normal bundle N_u \to \Sigma, the so called normal Cauchy-Riemann operator

\mathbf{D}_u^N : \Gamma(N_u) \to \Omega^{0,1}(\Sigma,N_u),

is surjective. If you’re only reading this post for an explanation of super-rigidity, then this is all you need to know about \mathbf{D}_u^N 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 u : (\Sigma,j) \to (M,J) is connected and non-constant, then it is necessarily immersed outside of a discrete set \text{Crit}(u) \subset \Sigma, and at each of the critical points z \in \text{Crit}(u), it has a well-defined critical order, which is a positive integer. For instance, the injective holomorphic curve u(z) = (z^{k+1},z^{k+2}) has critical order k at z=0. 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 J-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 T_u \subset u^*TM whose fiber at each immersed point z \in \Sigma \setminus \text{Crit}(u) is simply \text{im}\, du(z). It is therefore natural to choose a complementary complex subbundle N_u \subset u^*TM and call it the generalized normal bundle of u.

Now if \mathbf{D}_u : \Gamma(u^*TM) \to \Omega^{0,1}(\Sigma,u^*TM) denotes the usual linearized Cauchy-Riemann operator at u, the splitting u^*TM = T_u \oplus N_u decomposes it into block form as

\mathbf{D}_u = \begin{pmatrix} \mathbf{D}^T_u & \mathbf{D}^{TN}_u \\ \mathbf{D}_u^{NT} & \mathbf{D}_u^N \end{pmatrix}.

It is easy to check that the diagonal terms \mathbf{D}_u^T and \mathbf{D}_u^N are Cauchy-Riemann type operators on the bundles T_u and N_u respectively, while the off-diagonal terms are zeroth-order terms (in fact \mathbf{D}_u^{NT} vanishes identically). The Fredholm index of \mathbf{D}_u^N is the same as the virtual dimension of the moduli space if u is immersed, though in general they differ because c_1(N_u) differs from c_1(u^*TM) - \chi(\Sigma) by the algebraic count of critical points in u. What you may find more surprising is that even in the non-immersed case, \mathbf{D}_u^N fully characterizes transversality:

Lemma (see Theorem 3 in [W. 2010]). A non-constant J-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 j on \Sigma depend on points z \in \Sigma, or replacing the equation \bar{\partial}_J u = 0 with \bar{\partial}_J u = \nu 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.[2]

Assume u = v \circ \varphi : \widetilde{\Sigma} \to M is a multiply covered J-holomorphic curve, where v : \Sigma \to M is a simple curve and \varphi : \widetilde{\Sigma} \to \Sigma is a d-fold holomorphic branched cover. In this case, the symmetry of u endows its normal Cauchy-Riemann operator with a natural splitting

\mathbf{D}_u^N = \bigoplus_{i=1}^N (\mathbf{D}_u^{\boldsymbol{\theta}_i})^{\oplus k_i}.

Here \boldsymbol{\theta}_1,\ldots,\boldsymbol{\theta}_N are the real irreducible representations of some finite group G, each of which has an associated Fredholm operator \mathbf{D}_u^{\boldsymbol{\theta}_i} that is equivalent to a Cauchy-Riemann type operator on some bundle, and the k_i \ge 0 are integers. To understand what’s going on here, it’s instructive to start with the simplest nontrivial example: assume d=2, so \varphi : \widetilde{\Sigma} \to \Sigma is a branched double cover and therefore has automorphism group \text{Aut}(\varphi) = {\mathbb Z}_2. If \psi \in \text{Aut}(\varphi) denotes the generator of this group, then the space of sections of N_u has a natural splitting as \Gamma_+(N_u) \oplus \Gamma_-(N_u), where

\Gamma_\pm(N_u) := \{ \eta \in \Gamma(N_u)\ |\ \eta \circ \psi = \pm \eta \}.

The target bundle \overline{\text{Hom}}_{\mathbb C}(T\Sigma,N_u) for \mathbf{D}_u^N inherits a similar splitting, and the symmetry of u means that \mathbf{D}_u^N preserves the splitting, thus it decomposes into two pieces

\mathbf{D}_u^N = \mathbf{D}_u^+ \oplus \mathbf{D}_u^-.

The relation between this and the more general splitting above is as follows: G = {\mathbb Z}_2, \boldsymbol{\theta}_1 and \boldsymbol{\theta}_2 are its unique trivial and nontrivial irreducible representations respectively, \mathbf{D}_u^{\boldsymbol{\theta}_1} = \mathbf{D}_u^+, \mathbf{D}_u^{\boldsymbol{\theta}_2} = \mathbf{D}_u^-, and k_1 = k_2 = 1.

Notice that there is a canonical bijection between \Gamma_+(N_u) and \Gamma(N_v) sending any section \eta in the latter to the symmetric section \eta \circ \varphi. Under this identification, \mathbf{D}_u^+ is identified with \mathbf{D}_v^N, the normal Cauchy-Riemann operator for the underlying simple curve. This observation is obviously helpful since we already know how to prove that \mathbf{D}_v^N is surjective for generic J; the transversality problem for u has thus been reduced to a question about the operator \mathbf{D}_u^-.

For covers of arbitrary degree d \ge 2, one can still define a natural splitting \mathbf{D}_u^N = \mathbf{D}_u^+ \oplus \mathbf{D}_u^- in which \mathbf{D}_u^+ is identified with \mathbf{D}_v^N, but the factor \mathbf{D}_u^- can be split further if d \ge 3. 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 G whose real irreducible representations \boldsymbol{\theta}_i give rise to so-called twisted Cauchy-Riemann operators \mathbf{D}_u^{\boldsymbol{\theta}_i}, which are all Fredholm. These should be regarded as the fundamental building blocks of transversality theory for multiple covers. Here we can always assume \boldsymbol{\theta}_1 is the trivial representation of G and k_1 = 1, so the first factor (\mathbf{D}_u^{\boldsymbol{\theta}_1})^{\oplus k_1} in the splitting can be identified with \mathbf{D}_u^+ \cong \mathbf{D}_v^N, and the rest is a splitting of \mathbf{D}_u^-.  The group G always has order at least d, as it is the automorphism group of some normal[3] branched cover of \Sigma that factors through \varphi. Such normal covers always exist, and in fact there is a canonical one up to isomorphism, with degree at most d!, whose automorphism group I refer to as the generalized automorphism group of \varphi. If \varphi is already normal, then the canonical normal cover factoring through it is isomorphic to \varphi itself, so we can take G = \text{Aut}(\varphi).

Moduli spaces of multiple covers

Given a tame almost complex structure J, an integer d \ge 2, a finite group G and some additional combinatorial data to be specified below, we now consider a moduli space

{\mathcal M}^d_G(J) = \{ u = v \circ \varphi \},

where

  • v is a somewhere injective J-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 v is immersed everywhere else.
  • \varphi is a holomorphic map of closed Riemann surfaces with degree d and generalized automorphism group isomorphic to G, 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 {\mathcal M}^d_G(J) as a subset of the usual moduli space of smooth unparametrized J-holomorphic curves (including both simple curves and multiple covers). It should be clear that every d-fold multiply covered J-holomorphic curve u = v \circ \varphi belongs to {\mathcal M}^d_G(J) for appropriate choices of the finite group G and the combinatorial data that prescribes the critical points of v and branching behavior of \varphi. Note that the set of all possible choices of this combinatorial data is countable. It is also easy to see that {\mathcal M}^d_G(J) is a smooth manifold for generic J, 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 v, the moduli space of branched covers \varphi with fixed branching data is a manifold of real dimension twice the number of critical values of \varphi. This is all more or less standard.

The point of prescribing critical and branching behavior in this way is that as v and \varphi move around to produce various multiple covers u = v \circ \varphi \in {\mathcal M}^d_G, the splitting

\mathbf{D}_u^N = \bigoplus_{i=1}^N (\mathbf{D}_u^{\boldsymbol{\theta}_i})^{\oplus k_i}

changes continuously. In particular, prescribing the critical points of v makes the generalized normal bundle N_v into a continuous family of bundles parametrized by v; this would not be true if v were allowed to move arbitrarily through the space of somewhere injective curves, since its critical points could then disappear, changing the topology of N_v. Similarly, prescribing the branching data of \varphi 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 {\mathcal M}^d_G described above, let \boldsymbol{\theta}_i : G \to \text{Aut}_{\mathbb R}(W_i) for i=1,\ldots,N denote the distinct real irreducible representations of G, and denote

{\mathbb K}_i := \text{End}_G(W_i)t_i := \dim {\mathbb K}_i.

By standard results in representation theory, the endomorphism algebra {\mathbb K}_i is always isomorphic to either {\mathbb R}, {\mathbb C} or the quaternions {\mathbb H}, hence t_i \in \{1,2,4\}. This alternative depends on whether the complexification of \boldsymbol{\theta} is also irreducible (as a complex representation) or is the direct sum of a complex irreducible representation \boldsymbol{\lambda}_i with its dual (which is either isomorphic to \boldsymbol{\lambda}_i or not).[4] It is useful to note that the endomorphisms endow W_i and consequently the domain and target spaces of \mathbf{D}_u^{\boldsymbol{\theta}_i} with {\mathbb K}_i-module structures such that \mathbf{D}_u^{\boldsymbol{\theta}_i} is {\mathbb K}_i-linear.

Now for any tuples of nonnegative integers \mathbf{k} = (k_1,\ldots,k_N) and \mathbf{c} = (c_1,\ldots,c_N), we consider the subset

{\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) = \{ u \in {\mathcal M}^d_G(J)\ |\ \dim_{{\mathbb K}_i} \ker \mathbf{D}_u^{\boldsymbol{\theta}_i} = k_i and \dim_{{\mathbb K}_i} \text{coker}\, \mathbf{D}_u^{\boldsymbol{\theta}_i} = c_i for all i \}.

Note that since each of the operators \mathbf{D}_u^{\boldsymbol{\theta}_i} is Fredholm, this subset will be automatically empty unless \mathbf{k} and \mathbf{c} are chosen so that k_i - c_i is the index (with respect to {\mathbb K}_i) of \mathbf{D}_u^{\boldsymbol{\theta}_i} for every i=1,\ldots,N.

Theorem D. For generic J, {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) is a smooth submanifold of {\mathcal M}^d_G(J) with

\text{codim }{\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) = \sum_{i=1}^N t_i k_i c_i.

The theorem can also be stated in a more general form with J replaced by any smooth finite-dimensional family of almost complex structures, so that the definitions of {\mathcal M}^d_G(J) and {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) 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 \mathbf{D}_u^N already tells us that there is no hope for a multiply covered curve u = v \circ \varphi to be Fredholm regular unless

\text{ind}\,\mathbf{D}_u^{\boldsymbol{\theta}_i} \ge 0

for every i=1,\ldots,N such that k_i > 0, as \mathbf{D}_u^N 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 u. But Theorem D now supplements this with the following insight: if the topological obstruction vanishes, then almost every element of {\mathcal M}^d_G(J) will definitely be regular, as the non-regular curves all live in substrata {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) 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 J, all unbranched double covers u = v \circ \varphi of immersed somewhere injective curves v are Fredholm regular. Let \text{ind}(v) and \text{ind}(u) denote the virtual dimensions of the moduli spaces containing v and u 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

\text{ind}(u) = 2\, \text{ind}(v)

Note that \text{ind}(v) \ge 0 without loss of generality since J is generic. Then \text{ind}(v) is also the dimension of the space {\mathcal M}^2_{{\mathbb Z}_2}(J) in which any immersed double cover of v 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 \mathbf{D}_u^N takes the simple form \mathbf{D}_u^+ \oplus \mathbf{D}_u^-, with \mathbf{D}_u^+ \cong \mathbf{D}_v^N, and the relevant representations in this picture are the unique trivial and nontrivial irreducible representations of {\mathbb Z}_2, both of which remain irreducible after complexifying them, so the dimensions of the corresponding {\mathbb Z}_2-equivariant endomorphism algebras are

t_+ = t_- = 1.

The simplicity of the splitting \mathbf{D}_u^N = \mathbf{D}_u^+ \oplus \mathbf{D}_u^- also provides enough information to deduce the index of \mathbf{D}_u^-:

\text{ind}\,\mathbf{D}_u^- = \text{ind}\,\mathbf{D}_u^N - \text{ind}\,\mathbf{D}_u^+ = \text{ind}(u) - \text{ind}(v) = \text{ind}(v).

We can assume \mathbf{D}_u^+ is surjective since v is Fredholm regular, so if u is not regular, it can only be because \dim \text{coker}\, \mathbf{D}_u^- > 0, meaning that u belongs to a space of the form {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) with \mathbf{k} = (k_+,k_-) and \mathbf{c} = (c_+,c_-), where c_+ = 0, c_- > 0, k_+ = c_+ + \text{ind}\,\mathbf{D}_u^+ = \text{ind}(v), and k_- = c_- + \text{ind}\,\mathbf{D}_u^- = c_- + \text{ind}(v). Putting all of this information together, u now lives in a smooth manifold {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) with dimension

\text{ind}(v) - t_+ k_+ c_+ - t_- k_- c_- = \text{ind}(v) - c_- (c_- + \text{ind}(v)) = - c_-^2 - (c_- - 1) \text{ind}(v),

and this is strictly negative since c_- \ge 1 by assumption. This is a contradiction.

In this example we were lucky because the index of \mathbf{D}_u^- could be deduced without any further computation. In more complicated situations, it is often necessary to compute the indices of the twisted operators \mathbf{D}_u^{\boldsymbol{\theta}_i} 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.

Super-rigidity

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

\text{ind}(u) = (n-3) (2-2g) + 2 c_1([u]) = 0

for all holomorphic curves u, 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 u 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

\dim \ker \mathbf{D}_u^N = 0,

and we say that a simple curve v 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 J in any symplectic manifold of dimension 2n \ge 6, all simple curves of index zero are indeed super-rigid. Looking again at our splitting of \mathbf{D}_u^N, you can see that the first step in proving this must be to check that whenever u = v \circ \varphi for an immersed simple curve v of index zero, all the twisted operators satisfy

\text{ind}\,\mathbf{D}_u^{\boldsymbol{\theta}_i} \le 0,

as \mathbf{D}^N clearly could not be injective without this. Actually, the necessary dimension-counting argument requires a stricter upper bound for the case when the representation \boldsymbol{\theta}_i is faithful. To illustrate this, let’s focus again on the case of a degree 2 cover and try to prove that \mathbf{D}_u^N will be injective for generic J if \text{ind}(v) = 0 and n \ge 3. Assume \varphi is a 2-fold branched cover with r \ge 0 branch points, which in this case is the same as the number of critical values. Then assuming our simple curve v is immersed (which is always true for simple index 0 curves if J is generic), the cover u = v \circ \varphi lives in a space of real dimension

\dim {\mathcal M}^2_{{\mathbb Z}_2}(J) = 2r.

Since J is generic, we can assume v is Fredholm regular, hence the index zero operator \mathbf{D}_v^N \cong \mathbf{D}_u^+ is an isomorphism. If \mathbf{D}_u^N is not injective, it therefore means that \mathbf{D}_u^- is not injective, so u belongs to a stratum {\mathcal M}^2_{{\mathbb Z}_2}(J;\mathbf{k},\mathbf{c}) \subset {\mathcal M}^2_{{\mathbb Z}_2}(J) with \mathbf{k} = (k_+,k_-) and \mathbf{c} = (c_+,c_-) where k_+ = c_+ = 0, k_- > 0 and c_- = k_- - \text{ind}\,\mathbf{D}_u^-. To make full use of this, we need more precise information about \text{ind}\,\mathbf{D}_u^-. This is the factor in the splitting that corresponds to a faithful representation of {\mathbb Z}_2; the other factor \mathbf{D}_u^+ corresponds to the trivial (and therefore unfaithful) representation of {\mathbb Z}_2, but we do not need to care about it since \mathbf{D}_u^+ is identified with \mathbf{D}_v^N, so that the standard analysis for simple curves has already told us everything we wanted to know. To compute \text{ind}\,\mathbf{D}_u^-, note that by the Riemann-Hurwitz formula, the count r \ge 0 of branch points of the cover \varphi : \widetilde{\Sigma} \to \Sigma satisfies

r = -\chi(\widetilde{\Sigma}) + d \chi(\Sigma),

with d = 2 in the present situation. Combining this with the Riemann-Roch formula then gives the relation

\text{ind}\,\mathbf{D}_u^N = d \cdot \text{ind}\,\mathbf{D}_v^N - (n-1)r,

thus \text{ind}\,\mathbf{D}_u^N = 2\, \text{ind}(v) - (n-1) r = -(n-1)r. This implies

\text{ind}\,\mathbf{D}_u^- = \text{ind}\,\mathbf{D}_u^N - \text{ind}\,\mathbf{D}_u^+ = -(n-1)r,

and plugging this in to compute c_-, Theorem D now implies that u lives in a substratum {\mathcal M}^2_{{\mathbb Z}_2}(J;\mathbf{k},\mathbf{c}) with dimension

2r - t_+ k_+ c_+ - t_- k_- c_- = 2r - k_- [k_- + (n-1)r] \le -k_-^2 - [k_- (n-1) - 2]r,

and this is strictly negative since we assumed n \ge 3. This is of course a contradiction, and thus proves that the super-rigidity condition is satisfied for branched covers of degree two.[5]

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 \mathbf{D}_u^{\boldsymbol{\theta}_i}, which comes more or less for free when d=2 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:

  1. How to define the splitting of \mathbf{D}_u^N for general multiple covers u = v \circ \varphi, in particular in cases where \varphi is a non-normal branched cover;
  2. 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 {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}), so that some dimensions that used to be real are now dimensions over the endomorphism algebra {\mathbb K}_i = \text{End}_G(W_i). 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.

Footnotes

[1] 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.

[2] 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.

[3] Recall that a covering map p : Y \to X is called normal (or also regular) if for every x \in X and every pair y_1,y_2 \in p^{-1}(x), there exists a deck transformation sending y_1 to y_2. This definition extends in an obvious way to a branched cover of Riemann surfaces \varphi : \widetilde{\Sigma} \to \Sigma, so \varphi is normal if and only if the order of its automorphism group \text{Aut}(\varphi) is the same as its degree.  In general, |\text{Aut}(\varphi)| \le \text{deg}(\varphi). (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”.)

[4] Readers unfamiliar with the real representation theory of finite groups will find these details explained in the classic textbook by Serre.

[5] 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.)

About Chris Wendl

I'm a professor of mathematics at Humboldt University in Berlin. My research deals with symplectic manifolds, contact manifolds, pseudoholomorphic curves and various related things.
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2 Responses to Transversality for multiple covers, super-rigidty, and all that

  1. Paolo says:

    A silly question perhaps: where is Theorem A?

    Like

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