Why Petri’s condition is generic

Part 1: Where it all went wrong

I would like to state a lemma, but it comes with a major caveat: the lemma is false. I guess this means that it’s “not a lemma in the sense of mathematics,” so perhaps I should call it something else, like… an emma? No, let’s call it a lemming.

(Lemmings, as you may have heard, do not generally jump off cliffs. But Lemming 1 did.)

Lemming 1Suppose \mathbf{D} : \Gamma(E) \to \Gamma(F) is a real-linear Cauchy-Riemann type operator over a Riemann surface \Sigma, such that the bundle map E \to F defined by the complex-antilinear part of \mathbf{D} is invertible on some open subset {\mathcal U} \subset \Sigma. Then \mathbf{D} satisfies Petri’s condition on {\mathcal U}.

Recall from the previous post: the words “satisfies Petri’s condition on {\mathcal U}” mean that the natural map

\Pi_{\mathcal U} : \ker \mathbf{D} \otimes \ker\mathbf{D}^* \to \Gamma(E \otimes F|_{\mathcal U})

is injective, where \Pi_{\mathcal U} sends each \eta \otimes \xi \in \ker\mathbf{D} \otimes \ker\mathbf{D}^* to the section \Pi_{\mathcal U}(\eta \otimes \xi)(z) := \eta(z) \otimes \xi(z) of E \otimes F restricted to {\mathcal U}. Put another way, this means that if we fix bases \eta_1,\ldots,\eta_m \in \ker\mathbf{D} and \xi_1,\ldots,\xi_n \in \ker\mathbf{D}^*, then for every nontrivial set of real numbers \Psi_{ij} \in {\mathbb R}, the section

\sum_{i,j} \Psi_{ij} \eta_i \otimes \xi_j \in \Gamma(E \otimes F)

is guaranteed to be nonzero somewhere in {\mathcal U}. I tried to explain in the previous post why this is something you would want to be true if you are studying equivariant transversality problems. And given what we know about unique continuation for Cauchy-Riemann type equations, it certainly looks true on first glance. But as you might gather from the hypothesis in Lemming 1, reality is more complicated. I should emphasize at this point that even through the bundles E and F come with complex structures, the operators \mathbf{D} and \mathbf{D}^* are in general not complex linear, so all tensor products in this discussion must be understood to be real tensor products, even if \ker\mathbf{D} and \ker\mathbf{D}^* happen in some cases to be complex vector spaces. That makes the following an example in which Petri’s condition fails: take E to be the trivial line bundle over {\mathbb D} \subset {\mathbb C}, and \mathbf{D} = \bar{\partial} := \frac{\partial}{\partial \bar{z}} with formal adjoint \mathbf{D}^* = -\partial := -\frac{\partial}{\partial z}. Then

1 \otimes i\bar{z} - i \otimes \bar{z} - z \otimes i + iz \otimes 1 \in \ker\mathbf{D} \otimes \ker\mathbf{D}^*

is a nontrivial element in the kernel of \Pi_{\mathbb D} : \ker\mathbf{D} \otimes \ker\mathbf{D}^* \to \Gamma(E \otimes F). We see of course that this would not be a nontrivial element in the complex tensor product, and in fact: it is not hard to show that for complex-linear Cauchy-Riemann type operators, the complex analogue of Petri’s condition (using complex tensor products) always holds. This is essentially a consequence of unique continuation, together with the fact that, in local coordinates, all \eta \in \ker\mathbf{D} are power series in z while \xi \in \ker\mathbf{D}^* are power series in \bar{z}.

This fact about the complex-linear case was what originally misled me into believing that Lemming 1 should be true. The hypothesis of an invertible antilinear part ensures that all real-linearly independent sets in \ker\mathbf{D} or \ker\mathbf{D}^* are also complex-linearly independent, so that one might realistically hope for properties of \ker \mathbf{D} \otimes_{\mathbb C} \ker\mathbf{D}^* to carry over to \ker\mathbf{D} \otimes_{\mathbb R} \ker\mathbf{D}^*. After developing that intuition and then sitting down to work out the details, I was a bit too quick to believe I had succeeded, leading to the “proof” (or is it a “roof”, or a “proo”?) that appeared in 2016 and had to be withdrawn from the arXiv almost two years later. In reality, I had underestimated the difficulty of this detail, and at least part of my intuition on why it should work out was wrong.

I’ll show you an actual counterexample to Lemming 1 at the end of this post, but first I want to stop talking about things that are false, and say some more about what is actually true.

Part 2: What is right

The point of Lemming 1 was never supposed to be that there is something special about Cauchy-Riemann type operators with invertible antilinear part. The intended point was that Petri’s condition is generic: every Cauchy-Riemann type operator can be perturbed to make its antilinear part somewhere invertible. If Lemming 1 were true, then it would be a short step from there to proving that for generic almost complex structures J, every J-holomorphic curve has a normal Cauchy-Riemann operator that satisfies Petri’s condition… thus making equivariant transversality feasible.

It would therefore suffice if Lemming 1 is replaced with a statement that is less specific about the desirable class of Cauchy-Riemann operators, but still says that they are generic. Something like this:

Lemma 2. For any Cauchy-Riemann type operator \mathbf{D} : \Gamma(E) \to \Gamma(F) on a Riemann surface \Sigma and a fixed open subset {\mathcal U} \subset \Sigma with compact closure, there is a Baire subset {\mathcal A}^{\text{reg}}({\mathcal U}) in the space {\mathcal A}({\mathcal U}) of all smooth linear bundle maps A : E \to F supported in {\mathcal U}, consisting of perturbations A such that \mathbf{D}_A := \mathbf{D} + A satisfies Petri’s condition on {\mathcal U}.

This is essentially what Corollary 5.9 in the new version of my paper on super-rigidity says, and it is again a short step from there to proving that all normal Cauchy-Riemann operators of J-holomorphic curves satisfy Petri’s condition for generic J.

Part of the intuition here is that when you look at examples of operators for which Petri’s condition fails, the counterexamples look very special: the condition \Pi_{\mathcal U}(\sum_i \eta_i \otimes \xi_i) = 0 translates into nontrivial pointwise linear dependence relations among some linearly independent local solutions \eta_i \in \ker\mathbf{D} and \xi_i \in \ker\mathbf{D}^* over an open set, and it would seem surprising somehow for generic operators to admit such relations. Unique continuation also still plays a role, as must be expected since, if there were a nontrivial local solution to \mathbf{D}\eta = 0 that vanishes on some open set {\mathcal U} \subset \Sigma, then one could pair it with any nontrivial \xi \in \ker\mathbf{D}^* and call \eta \otimes \xi an easy counterexample to Petri’s condition. But on balance, I understand Lemma 2 mainly as a genericity result—unique continuation is still an important ingredient in the proof, but the main tool is actually Sard’s theorem.

Part 3: How do you prove a “local genericity result” anyway?

I don’t mind admitting that I was quite puzzled for a while as to how one might go about proving Lemma 2. In the first place, it isn’t immediately clear whether it should be understood analytically as a global or a local result. Calling it “global” in this case would mean that it depends on the global setup of the operator and, in all likelihood, makes use of the fact that \mathbf{D} and \mathbf{D}^* are Fredholm. That sounds good at first, because it seems much more likely for the “Petri map” \Pi_{\mathcal U} : \ker\mathbf{D} \otimes \ker\mathbf{D}^* \to \Gamma(E \otimes F|_{\mathcal U}) to be injective if its domain is finite dimensional. But the problem starts to seem a lot dicier if you imagine what happens to this domain under perturbations: \ker\mathbf{D} and \ker\mathbf{D}^* do not depend continuously on \mathbf{D} in a straightforward way, as their dimensions can jump suddenly downward. One cannot therefore just set up some kind of “universal moduli space”

U := \left\{ (A,t)\ \big|\ A \in {\mathcal A}({\mathcal U}),\ t \in \ker \Pi_{\mathcal U} \subset \ker \mathbf{D}_A \otimes \ker \mathbf{D}_A^* \right\}

and try to apply the Sard-Smale theorem to the obvious projection U \to {\mathcal A}({\mathcal U}) : (A,t) \mapsto A, because U does not closely resemble anything that could reasonably be called a manifold.

The second problem with viewing Lemma 2 globally is that since we want a result that applies to multiply covered holomorphic curves, we would also need a version of the lemma that considers operators \mathbf{D} which are equivariant under the action of some finite symmetry group G, so that the perturbations A are also required to be G-invariant. This makes the problem vulnerable to the same difficulty that this whole endeavor was designed to overcome: transversality and symmetry are not generally compatible with each other. One of the selling points of Lemming 1 had always been that since the condition required on the perturbation was fundamentally local, proving it for linearized Cauchy-Riemann operators along simple curves would immediately imply the same result for all multiple covers of those curves.

All this makes a pretty convincing argument for taking a local approach to Lemma 2: we should not assume any condition (such as compactness) on \Sigma, nor should we assume that \mathbf{D} is Fredholm… whatever can be proven should be provable by considering small zeroth-order perturbations of the standard Cauchy-Riemann operator \bar{\partial} = \partial_s + i\partial_t : C^\infty({\mathbb D},{\mathbb C}^m) \to C^\infty({\mathbb D},{\mathbb C}^m). This idea does not have the two drawbacks mentioned above—in particular, it is a standard result of local elliptic regularity theory that the infinite-dimensional space \ker \mathbf{D} \subset C^\infty({\mathbb D},{\mathbb C}^m) does vary smoothly with the operator \mathbf{D} in suitable functional-analytic settings. But now there is a new problem: nothing in the setup is Fredholm, and there is no Sard’s theorem for non-Fredholm maps between infinite-dimensional manifolds.

There does exist a local approach that doesn’t have this last drawback: one can consider the problem on jet spaces of sections at a point. In this way, everything becomes finite dimensional, and no actual functional analysis is needed.

Part 4: The jet space approach

I will now describe the setup for proving Lemma 2. I’ll focus specifically on Cauchy-Riemann operators, but it’s interesting to note that a large portion of the discussion makes sense for much more general classes of differential operators, for which one might conceivably be interested in studying equivariant transversality (see e.g. the preprint by Doan and Walpuski on this subject).

We are given a Riemann surface \Sigma and complex vector bundle E \to \Sigma, giving rise to the bundle F := \overline{\text{Hom}}_{\mathbb C}(T\Sigma,E) and the affine space of real-linear Cauchy-Riemann type operators {\mathcal CR}_{\mathbb R}(E), which map \Gamma(E) to \Gamma(F). Fix a point p \in \Sigma and let J^k_p E denote the vector space of k-jets of sections of E at p. Each \mathbf{D} \in {\mathcal CR}_{\mathbb R}(E) then descends to a linear map

\mathbf{D} : J^k_p E \to J^{k-1}_p F

for every k \in {\mathbb N}, and usefully, this map is always surjective. The latter can be deduced from standard local existence results for solutions to the equation \bar{\partial} u = f, but in the jet space context, it’s actually much easier than that: first, one can easily just write down a right-inverse for the operator \bar{\partial} : J^k_p E \to J^{k-1}_p F. The general case is then a consequence of the fact that surjectivity is an open condition, using the following observation:

Rescaling principle: Every Cauchy-Riemann type operator \mathbf{D} : J^k_p E \to J^{k-1}_p F is equivalent (via choices of local coordinates and trivializations near p) to an arbitrarily small perturbation of the standard operator \bar{\partial}.

Let {\mathcal CR}^k_{\mathbb R}(E) denote the space of linear maps J^k_p E \to J^{k-1}_p F that are induced by operators in {\mathcal CR}_{\mathbb R}(E). Since the (k-1)-jet of a section \mathbf{D}_A \eta = \mathbf{D}\eta + A\eta depends on the zeroth-order perturbation A only up to its (k-1)-jet, {\mathcal CR}^k_{\mathbb R}(E) is an affine space over the finite-dimensional vector space J^{k-1}_p\text{Hom}(E,F). We can now consider the k-jet Petri map

\Pi^k : J^k_p E \otimes J^k_p F \to J^k_p(E \otimes F),

defined by letting the natural map \Pi : \Gamma(E) \otimes \Gamma(F) \to \Gamma(E \otimes F) descend to quotient spaces. We will be interested particularly in the restriction of \Pi^k to the subspace \ker\mathbf{D} \otimes \ker\mathbf{D}^* \subset J^k_p E \otimes J^k_p F for each \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E). There is a trivial reason why this map will never actually be injective: if \eta \in J^k_p E vanishes to order q-1 \le k and \xi \in J^k_p F vanishes to order r-1 \le k with q+r > k, then their product vanishes to order at least k and is thus trivial in J^k(E \otimes F). The fancy way to say this is that jet spaces carry natural filtrations,

J^k_p E = (J^k_p E)^0 \supset (J^k_p E)^1 \supset \ldots \supset (J^k_p E)^k \supset (J^k_p E)^{k+1} = \{0\},

where we can identify k-jets with Taylor polynomials in coordinates to define (J^k_p E)^\ell as the space of Taylor polynomials that are O(|z|^\ell). Under the natural tensor product filtration that J^k_p E \otimes J^k_p F inherits from the filtrations of J^k_p E and J^k_p F, the Petri map \Pi^k preserves filtrations and thus vanishes on (J^k_p E \otimes J^k_p F)^{k+1}. This observation motivates considering for each \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E) and each k,\ell \in {\mathbb N} with 0 \le \ell \le k+1 the space

{\mathcal M}^k_\ell(\mathbf{D}) := \left\{ t \in \ker\mathbf{D} \otimes \ker\mathbf{D}^*\ \big|\ \Pi^k(t) = 0 \text{ and } t \not\in (J^k_p E \otimes J^k_p F)^\ell \right\}.

Notice how we’ve just quietly reinserted unique continuation into this discussion. If we can find sequences \ell_n,k_n \to \infty such that {\mathcal M}^{k_n}_{\ell_n}(\mathbf{D}) = \emptyset for a given operator \mathbf{D} \in {\mathcal CR}_{\mathbb R}(E), then we’ve proven that the only possible counterexamples to Petri’s condition for \mathbf{D} are nontrivial elements t \in \ker\mathbf{D} \otimes \ker\mathbf{D}^* that vanish to all orders at the point p. One can easily deduce from unique continuation that there are no such elements, so this would imply Petri’s condition.

What I’m about to say will sound like bad news, but it leads to something good. One can easily compute the expected dimension of {\mathcal M}^k_\ell(\mathbf{D}) via a dimension count. My initial naive hope had been that this expected dimension would turn out to be negative, perhaps after choosing k sufficiently large, and one could then argue via Sard’s theorem that {\mathcal M}^k_\ell(\mathbf{D}) is empty for almost every \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E). But the expected dimension isn’t negative. In fact, for all choices k, \ell and \mathbf{D}, {\mathcal M}^k_\ell(\mathbf{D}) turns out to be a nonempty open subset in a nontrivial vector space that depends smoothly on \mathbf{D}. There are good geometric reasons for this, which I can happily explain to anyone who’s curious, but I won’t get into them here—the point for now is just that the naive idea doesn’t work.

You get some interesting insight, however, if you then try to imagine (as I did) how the nonemptiness of {\mathcal M}^k_\ell(\mathbf{D}) might be used to disprove Lemma 2, e.g. to show that Petri’s condition fails for every Cauchy-Riemann operator. The nontriviality of every {\mathcal M}^k_\ell(\mathbf{D}) means that one can associate to every Cauchy-Riemann operator \mathbf{D} and integer \ell \in {\mathbb N} a sequence of tensor products of sections

\displaystyle t_k = \sum_{i=1}^{r_k} \eta_{k,i} \otimes \xi_{k,i} \in \Gamma(E) \otimes \Gamma(F)

such that for each k, t_k does not vanish to order \ell at p, but \mathbf{D}\eta_{k,i} and \mathbf{D}^*\xi_{k,i} vanish to order k and the Petri map takes t_k to a section of E \otimes F that also vanishes to order k at p. It is very far from obvious whether t_k can be made to converge to something as k \to \infty, though if it does, then it would be reasonable to expect that the limit is the infinity-jet of a counterexample to Petri’s condition. One of the big reasons why convergence is unclear is that the numbers r_k may be unbounded. One can rephrase this as follows: given two vector spaces V and W, say that an element t \in V \otimes W has rank r  if one can write t = \sum_{i=1}^r v_i \otimes w_i for two linearly-independent sets v_1,\ldots,v_r \in V and w_1,\ldots,w_r \in W. It is built into the definition of a tensor product of vector spaces that every element in it has finite rank. This is no longer true if one wishes to define a tensor product of infinite-dimensional Hilbert spaces—in that context, one needs to enlarge the algebraic tensor product to an analytical completion that includes elements of infinite rank. I find it conceivable that Petri’s condition really will fail at the local level for all Cauchy-Riemann operators if one replaces \Gamma(E) \otimes \Gamma(F) with a Hilbert space tensor product of local sections. But that is not what we are doing; the sequence t_k described above only has any chance of converging to a counterexample if the rank of t_k stays bounded.

With this in mind, let’s modify our definition of {\mathcal M^k_\ell}(\mathbf{D}): define for each r  \in {\mathbb N} the space

{\mathcal M}^k_{r,\ell}(\mathbf{D}) := \left\{ t \in \ker\mathbf{D} \otimes \ker\mathbf{D}^*\ \big|\ \Pi^k(t) = 0,\ \text{rank}(t) = r \text{ and } t \not\in (J^k_p E \otimes J^k_p F)^\ell \right\}

One should view this as a subset of

{\mathcal V}^k_{r,\ell}(\mathbf{D}) := \left\{ t \in \ker\mathbf{D} \otimes \ker\mathbf{D}^*\ \big|\ \text{rank}(t) = r \text{ and } t \not\in (J^k_p E \otimes J^k_p F)^\ell \right\},

which is a smooth submanifold of the vector space \ker\mathbf{D} \otimes \ker\mathbf{D}^* \subset J^k_p E \otimes J^k_p F, for the same reason that the space of matrices of a fixed rank is a submanifold in the space of all matrices. Its codimension depends on r and produces a general formula for the dimension of {\mathcal V}^k_{r,\ell}(\mathbf{D}) that grows linearly with k. On the other hand, the extra condition \Pi^k(t) = 0 \in J^k_p(E \otimes F) cuts out a subset whose expected codimension is the dimension of J^k_p(E \otimes F); that is the number of distinct Taylor polynomials up to degree k in z and \bar{z} with values in a fiber of E \otimes F, and it grows quadratically with k. As a result, the expected dimension of {\mathcal M}^k_{r,\ell}(\mathbf{D}) becomes negative as soon as k is sufficiently large. This is, in my opinion, the main reason why you should believe that Lemma 2 is true. It now becomes a consequence of the following more technical statement:

Lemma 3. For every r,\ell \in {\mathbb N}, there exists k_0 \in {\mathbb N} such that {\mathcal M}^k_{r,\ell}(\mathbf{D}) = \emptyset for all k \ge k_0 and almost every \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E).

I’ll add just a few comments about the proof of this lemma. To set it up for Sard’s theorem, one needs to consider a “universal” version of the space {\mathcal M}^k_{r,\ell}(\mathbf{D}), namely

{\mathcal M}^k_{r,\ell} := \left\{ (\mathbf{D},t)\ \big|\ \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E) \text{ and } t \in {\mathcal M}^k_{r,\ell}(\mathbf{D}) \right\},

which one might hope should be a smooth submanifold of the manifold

{\mathcal V}^k_{r,\ell} := \left\{ (\mathbf{D},t)\ \big|\ \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E) \text{ and } t \in {\mathcal V}^k_{r,\ell}(\mathbf{D}) \right\},

e.g. because the smooth map {\mathcal V}^k_{r,\ell} \to J^k_p(E \otimes F) : (\mathbf{D},t) \mapsto \Pi^k(t) is transverse to zero. That seems to be not quite true in general, but what can be proved is close enough to that statement that it gives the desired result: one can show namely that the lineaization of (\mathbf{D},t) \mapsto \Pi^k(t) with respect to changes in \mathbf{D} has its rank bounded below by some quadratic function of k. As a measure of plausibility for this claim, notice that since the space of perturbations {\mathcal CR}^k_{\mathbb R}(E) is an affine space over J^{k-1}_p\text{Hom}(E,F), its dimension is also a quadratic function of k. The bound on the rank does not prove that {\mathcal M}^k_{r,\ell} is a submanifold, but it does prove that it’s something I like to call a C^\infty-subvariety, which has the property that it locally is contained in (locally defined) submanifolds whose codimension is given by the lower bound on the rank. That is enough structure to apply Sard’s theorem and prove, given that the codimension will exceed the dimension of {\mathcal V}^k_{r,\ell}(\mathbf{D}) when k is large enough, that for almost every \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E), {\mathcal M}^k_{r,\ell}(\mathbf{D}) itself is locally contained in submanifolds that have negative dimension, meaning {\mathcal M}^k_{r,\ell}(\mathbf{D}) is empty.

This general picture reduces the proof of Lemma 2 to a linear algebra problem: in principle, one needs to write down the linearization of the map {\mathcal V}^k_{r,\ell} \to J^k_p(E \otimes F) : (\mathbf{D},t) \mapsto \Pi^k(t) with respect to variations in \mathbf{D} \in {\mathcal CR}^k_{\mathbb R}(E) at an arbitrary point (\mathbf{D},t) \in {\mathcal M}^k_{r,\ell}, and find a good lower bound on the rank of this linear map. My final remark about this is that due to the rescaling principle mentioned above, one does not really need to consider arbitrary (\mathbf{D},t) \in {\mathcal M}^k_{r,\ell}; it suffices instead to establish this bound only for the special case \mathbf{D} = \bar{\partial}, for which it is a bit tedious but not very hard in principle to write down \ker\Pi^k \subset \ker\mathbf{D} \otimes \ker\mathbf{D}^* and the linearized map explicitly. Once you’ve done that, the rank bound carries over to an open neighborhood of such pairs in {\mathcal M}^k_{r,\ell}, and since every Cauchy-Riemann operator is (up to choices of coordinates and trivializations) an arbitrarily small perturbation of \bar{\partial}, the result applies to all operators.

Epilogue: The fall of Lemming 1

By this point, no one is reading this post anymore except for the referees of my paper and possibly one or two stalkers, so just for amusement, I might as well tell you how to find a concrete counterexample to Lemming 1. Take E and F to be the trivial line bundle over {\mathbb D} \subset {\mathbb C} and consider the operators

\mathbf{D} := \bar{\partial} + \kappa,    \mathbf{D}^* := -\partial + \kappa^*,

where \kappa : E \to F and \kappa^* : F \to E both denote the real-linear bundle map defined by complex conjugation. This is the simplest Cauchy-Riemann operator with invertible antilinear part that one can possibly write down, but I was stuck for an embarrassingly long time on how to write down precise local solutions to \mathbf{D}\eta = 0 and \mathbf{D}^*\xi = 0. There’s an easy trick for this that will be familiar to anyone who knows about asymptotic formulas for punctured holomorphic curves. In that context, we often have occasion to consider operators of the form \partial_s + i\partial_t + S(t) for functions on a half-cylinder [0,\infty) \times S^1, with S(t) an S^1-family of real-linear transformations on {\mathbb C}^m, and the equation (\partial_s + i\partial_t + S)\eta = 0 then has a special solution of the form

\eta(s,t) = e^{\lambda s} f_\lambda(t)

whenever f_\lambda : S^1 \to {\mathbb C}^m is an eigenfunction of the operator -i\partial_t - S with eigenvalue \lambda. Now, \mathbf{D} = \partial_s + i\partial_t + \kappa can be viewed as such an operator on a half-cylinder, but if we are truly only interested in local solutions, then we can ignore the requirement for the eigenfunction f_\lambda(t) to be periodic in t, which makes arbitrary real numbers possible for the eigenvalue \lambda. Once you’ve thought of this, you can do some calculations and are led sooner or later to write down an example like the following: define local sections \eta_\lambda \in \ker\mathbf{D} and \xi_\lambda \in \ker\mathbf{D}^* for \lambda \in (-1,1) by

\eta_\lambda(s,t) := e^{\lambda s + \sqrt{1 - \lambda^2} t}\left( \sqrt{1-\lambda} + i \sqrt{1 + \lambda} \right),

\xi_\lambda(s,t) := e^{-\lambda s - \sqrt{1 - \lambda^2} t} \left( \sqrt{1 - \lambda} - i \sqrt{1 + \lambda} \right),

If we identify the fibers of E and F with {\mathbb R}^2 so that the fibers of E \otimes F become the space of real 2-by-2 matrices, then feeding \eta_\lambda \otimes \xi_\lambda into the Petri map \Pi : \Gamma(E) \otimes \Gamma(F) \to \Gamma(E \otimes F) gives constant sections,

\Pi(\eta_\lambda \otimes \xi_\lambda)(s,t) = \begin{pmatrix} 1 - \lambda & -\sqrt{1 - \lambda^2} \\ \sqrt{1 - \lambda^2} & -1 - \lambda \end{pmatrix}.

These all take values in the 3-dimensional vector space of matrices of the form \begin{pmatrix} a & b \\ -b & c \end{pmatrix}, thus any four such products must be linearly dependent, and the dependence relation yields counterexamples to Petri’s condition if you choose four distinct numbers for the eigenvalue \lambda \in (-1,1).

Shit happens.

Acknowledgement: A substantial proportion of what I understand about the subject of this post emerged from conversations with Aleksander Doan and Thomas Walpuski.

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Super-rigidity is fixed

A new version of the paper Transversality and super-rigidity for multiply covered holomorphic curves has just been uploaded to my homepage, and will be replacing the previous (withdrawn) version on the arXiv within the next couple of days. Here’s the quick update for those who are keeping score but don’t have time for the details: the main theorems remain unchanged, and all gaps in their proofs have been filled.

For those who do have time for the details, my intention in this post is to review what the problem was and clarify why it was essential to fix it—I’ve come to view it as something more interesting and possibly more important than a mere technical difficulty, and I want to explain why. In the sequel I will then explain how the problem has been solved.

I’m not going to assume that everyone has read my series of previous posts on the super-rigidity paper and what went wrong in the proof. The main thing you need to know is this: the goal is to understand, in precise terms, when it is possible or impossible to establish transversality (or related conditions) for multiply covered J-holomorphic curves via the standard method of perturbing the almost complex structure generically. At the linearized level, this becomes an equivariant transversality problem: given a linear Cauchy-Riemann type operator that is invariant under a group action, when can you add generic zeroth-order perturbations to make the operator surjective/injective without breaking the symmetry?

Since I wasn’t the first person to have thought about such issues, I’ve been asked by several colleagues how my approach differs from earlier work by other authors… in particular the three or four previous attempted proofs (later withdrawn) that super-rigidity holds for generic J in Calabi-Yau 3-folds. Many elements in my approach have indeed appeared before: the twisted bundle decomposition for Cauchy-Riemann operators originated in work of Taubes, as did the idea (to be discussed below) of stratifying a moduli space via conditions on kernels and cokernels of Cauchy-Riemann operators. These two ideas later served as the basis for Eftekhary’s partial result on super-rigidity, and I’ve also seen the stratification idea appear in the wall-crossing argument in Ionel and Parker’s paper on the Gopakumar-Vafa formula. The main element in my approach that was not present in any of the others is a result that I used to call quadratic unique continuation, though for reasons that I’ll get into in the next post, I now find that to be a bad choice of words and am instead calling it Petri’s condition (thanks to Aleksander Doan and Thomas Walpuski for the terminology). The technical foundation of my paper is based on a result saying that Petri’s condition can be achieved locally under generic local perturbations of any Cauchy-Riemann type operator. That is the lemma that was wrong in the previous version, and has now been corrected.

Stratification and Petri’s condition

I want to explain a bit why Petri’s condition arises as an essential obstacle to overcome in equivariant transversality problems. This issue is quite general—as demonstrated in recent work by Doan and Walpuski, it pertains to more than just Cauchy-Riemann type operators or holomorphic curves, thus I will try to frame it in the generality that it deserves.

Consider a linear first-order partial differential operator \mathbf{D} : \Gamma(E) \to \Gamma(F) between two vector bundles over a smooth manifold M. We will assume that \mathbf{D} satisfies some nice condition such as ellipticity, so that it will be Fredholm when extended to suitable Banach space settings (which I won’t talk about here) and all local solutions to \mathbf{D}\eta = 0 are smooth. Fix also an open subset {\mathcal U} \subset M with compact closure and let

{\mathcal A}({\mathcal U}) \subset \Gamma(\text{Hom}(E,F))

denote the space of all smooth bundle maps E \to F with support in \mathcal{U}. These define compact perturbations of \mathbf{D} in the relevant Banach space setting, so that the perturbed operator

\mathbf{D}_A := \mathbf{D} + A : \Gamma(E) \to \Gamma(F)

for each A \in {\mathcal A}({\mathcal U}) is also Fredholm. The main idea of the stratification approach is now to consider subsets of the form

{\mathcal A}_{k,\ell}({\mathcal U}) := \left\{ A \in {\mathcal A}({\mathcal U})\ \big|\ \dim\ker(\mathbf{D}_A) = k \text{ and } \dim\text{coker}(\mathbf{D}_A) = \ell \right\} \subset {\mathcal A}({\mathcal U}).

If we are lucky, then this space will be a smooth finite-codimensional submanifold of {\mathcal A}({\mathcal U}), and its codimension in this particular setting should be k\ell. This is analogous to the fact that the space of all linear transformations {\mathbb R}^m \to {\mathbb R}^n of a fixed rank forms a smooth submanifold, and it can be proved in much the same way: one can associate to each A \in {\mathcal A}_{k,\ell}({\mathcal U}) a neighborhood {\mathcal O} \subset {\mathcal A}({\mathcal U}) and a smooth map

\Phi : {\mathcal O} \to \text{Hom}(\ker\mathbf{D}_{A} , \text{coker} \mathbf{D}_{A})

whose zero set is a neighborhood of A in {\mathcal A}_{k,\ell}({\mathcal U}), hence {\mathcal A}_{k,\ell}({\mathcal U}) is indeed a submanifold with the aforementioned codimension if we can arrange for the linearization of \Phi at A to be surjective. (For details on how to define \Phi, see the discussion of walls in the space of Fredholm operators in an earlier post.)

Surjectivity is the subtle part. The linearization in question takes the form

\mathbf{L} := d\Phi(A) : {\mathcal A}({\mathcal U}) \to \text{Hom}(\ker \mathbf{D}_{A} , \text{coker} \mathbf{D}_{A}),

\mathbf{L}(B) \eta := \pi(B\eta),

where \pi denotes the projection from the relevant Banach space of sections of F to the quotient \text{coker} \mathbf{D}_{A}, or equivalently, to the kernel of the formal adjoint operator \mathbf{D}_{A}^* with respect to some fixed choices of geometric data (i.e. bundle metrics and volume forms) on E, F and M. Let us fix such geometric data and denote the resulting L^2-inner product for sections of E or F by \langle\ ,\ \rangle_{L^2}. Choosing bases \eta_1,\ldots,\eta_m \in \ker\mathbf{D}_{A} and \xi_1,\ldots,\xi_n \in \ker\mathbf{D}_{A}^*, the difference between \mathbf{L}(B)\eta_i and B\eta_i is L^2-orthogonal to each \xi_j, thus the matrix elements that determine the linear map \mathbf{L}(B) : \ker \mathbf{D}_A \to \ker \mathbf{D}_A^* for each B \in {\mathcal A}({\mathcal U}) are

\langle \mathbf{L}(B) \eta_i , \xi_j \rangle_{L^2} = \langle B\eta_i , \xi_j \rangle_{L^2}.

The map \mathbf{L} then fails to be surjective onto \text{Hom}(\ker \mathbf{D}_A , \ker \mathbf{D}_A^*) if and only if there exists a nontrivial set of constants \Psi_{ij} \in {\mathbb R} that are “orthogonal” to the image of \mathbf{L} in the sense that for all B \in {\mathcal A}({\mathcal U}),

\sum_{i,j} \Psi_{ij} \langle B\eta_i , \xi_j \rangle_{L^2} = \int_{\mathcal U} \langle\cdot,\cdot\rangle_F \circ (B \otimes \text{Id}) \left( \sum_{i,j} \Psi_{ij} \eta_i \otimes \xi_j\right) \, d\text{vol} = 0.

The interesting term in this expression is the summation in parentheses: \sum_{i,j} \Psi_{ij} \eta_i \otimes \xi_j is a section of the tensor product bundle E \otimes F, which we are free to restrict to the subset {\mathcal U} \subset \Sigma since the support of B is contained there. In particular, \sum_{i,j} \Psi_{ij} \eta_i \otimes \xi_j is an element in the image of the natural linear map

\Pi_{\mathcal U} : \ker\mathbf{D}_A \otimes \ker\mathbf{D}_A^* \longrightarrow \Gamma(E \otimes F|_{\mathcal U})

which sends each product \eta \otimes \xi to the section \Pi(\eta \otimes \xi)(z) := \eta(z) \otimes \xi(z) restricted to {\mathcal U}. It is an easy linear algebra exercise to show that if \sum_{i,j} \Psi_{ij} \eta_i \otimes \xi_j \in \Gamma(E \otimes F) is nonzero on some open set in {\mathcal U}, then one can find some B \in {\mathcal A}({\mathcal U}) to make sure that the integral above does not vanish. In other words, \mathbf{L} is guaranteed to be surjective if the following condition is achieved:

Definition. The operator \mathbf{D}_A : \Gamma(E) \to \Gamma(F) satisfies Petri’s condition on the subset {\mathcal U} \subset M if the natural map \Pi_{\mathcal U} : \ker\mathbf{D}_A \otimes \ker\mathbf{D}_A^* \to \Gamma(E \otimes F|_{\mathcal U}) is injective.

Equivariance

One of the beautiful things about this approach to transversality issues is that if the program I’ve just sketched can be carried out at all, then it can also be carried out equivariantly. In particular, if the operators \mathbf{D}_A arise as linearized operators for something like a multiply covered holomorphic curve, then they come with symmetry, e.g. there may be a finite group G acting on M and the two bundles such that \mathbf{D} is G-equivariant and we are only allowed to perturb within the space {\mathcal A}_G({\mathcal U}) \subset {\mathcal A}({\mathcal U}) of G-invariant zeroth-order perturbations. In this case, the map \Phi automatically takes values in the space of G-equivariant linear maps \ker\mathbf{D}_A \to \ker\mathbf{D}_A^*, so that the linearized problem becomes to show that the map

\mathbf{L} : {\mathcal A}_G({\mathcal U}) \to \text{Hom}_G(\ker\mathbf{D}_A , \ker\mathbf{D}_A^*)

given by the same formula as before is surjective. If we have Petri’s condition, then this is easy: given \Psi \in \text{Hom}_G(\ker\mathbf{D}_A,\ker\mathbf{D}_A^*), we can use the non-equivariant case to find a (not necessarily G-invariant) solution \widetilde{B} \in {\mathcal A}({\mathcal U}) to \mathbf{L}(\widetilde{B}) = \Psi, but then symmetrize it to produce a solution

B := \frac{1}{|G|} \sum_{g \in G} g^*\widetilde{B} \in {\mathcal A}_G({\mathcal U}), satisfying \mathbf{L}(B) = \Psi.

Here’s the punchline. In certain settings, depending on the overall goal, it may well be that you can get away with proving less than the statement that {\mathcal A}_{k,\ell}({\mathcal U}) is a smooth submanifold of the right codimension, in which case you might not need to know whether Petri’s condition holds. But for almost any such work-around you might choose, the equivariant case will not work—at least, not in as much generality as one would like. Let me expand on that a bit. The papers I mentioned above by Taubes, Eftekhary and Ionel-Parker all make use of this stratification idea, so some form of the operator that I’m calling \mathbf{L} appears in all of them. But in all three papers, it turns out that the main results do not really require {\mathcal A}_{k,\ell}({\mathcal U}) to be a submanifold of the predicted codimension—it suffices to prove that it’s some kind of “subvariety” that resembles a manifold and whose codimension can be bounded from below, which means not necessarily proving that \mathbf{L} is surjective, but establishing a good lower bound on its rank. Taubes, for instance, uses the following cute trick: if we fix a nontrivial element \eta_0 \in \ker\mathbf{D}_A, then we can associate to every \xi \in \ker\mathbf{D}_A^* a zeroth-order perturbation of the form

B_\xi := \langle \eta_0,\cdot \rangle_E \, \xi \in \Gamma(\text{Hom}(E,F)),

which then satisfies

\langle \mathbf{L}(B_\xi) \eta_0 , \xi \rangle_{L^2} = \langle B_\xi \eta_0 , \xi \rangle_{L^2} = \int_M \langle \eta_0,\eta_0 \rangle_E \cdot \langle \xi,\xi \rangle_F, d\text{vol} > 0

due to unique continuation. One therefore obtains an injective linear map \ker\mathbf{D}_A^* \to \text{Hom}(\ker\mathbf{D}_A,\ker\mathbf{D}_A^*) : \xi \mapsto \mathbf{L}(B_\xi), which proves \text{rank} \mathbf{L} \ge \dim \text{coker} \mathbf{D}_A.

This argument suffices for certain applications, but outside of a very restrictive range of special cases (such as the regular double covers of tori in Taubes’s paper), it doesn’t give anything for the equivariant case: one can symmetrize the perturbations B_\xi constructed above, but there’s no guarantee that they won’t all become zero.

This is just one example; there are a few other tricks that I found in various other papers and attempted to implement as work-arounds when I wanted to prove Petri’s condition but didn’t know how to do it. None of them seemed sufficient to produce equivariant results in full generality. The conclusion I came to was that if you want to understand equivariant transversality for nonlinear PDEs, then Petri’s condition is one of the main necessary ingredients, and it is absolutely necessary.

As you can imagine, I was therefore fairly distraught when my original proof of Petri’s condition for Cauchy-Riemann type operators broke down. I still believed that it was very likely to be a generic property, and I also suspected that someone in either geometric analysis or algebraic geometry must have thought about this before and could simply give me the solution, if I only knew whom to ask. But having now asked around quite a bit more, I’m left with the impression that, in fact, hardly anyone has thought very much about this before. Thus I decided to write this post, telling you why Petri’s condition is something worth thinking about. In the next one, I’ll tell you what I’ve learned in the effort to prove it.

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One postdoc and two Ph.D. studentships in symplectic topology available in Berlin

The following jobs in my symplectic research group at the HU Berlin have just been advertised with application deadline of January 1, 2019:

Starting dates are planned for Autumn 2019 but can be moved a bit if desired by the candidate. They are all non-teaching positions, so no knowledge of German is required, though some voluntary teaching is possible, even in English (for upper-level courses).

A note about the Ph.D. studentships: these positions are conceived so that the salary should be comparable to a standard Ph.D. fellowship as offered e.g. by the Berlin Mathematical School (BMS).* If you are interested in working with me as a Ph.D. student, then I recommend applying both for BMS Phase 2 admission and for one of these positions, as I might not be able to consider you for this funding unless you have specifically applied for it. (There are advantages to being a member of the BMS regardless.)

* My standard practical note for those unfamiliar with the vagaries of public employment in Germany: the precise salaries for such positions are determined by a complicated formula that depends on various details, including your family circumstances, but in theory you can compute them more exactly (and also the after-tax version) using this salary calculator. The main thing you need to know is that these positions are in Entgeltgruppe E13 of the TV-L. For the PhD studentships you need to input “50%” for Arbeitszeit, as the positions are officially half-time (because you spend the other half of the time learning?). Please don’t ask me about Zusatzversorgung or Lohnsteuerklassen… if you want to know what these things mean, your best bet is to find an actual German, which, as you probably know, I am not.

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Super-rigidity is open again

Update 28.05.2019: The error that this post talks about has now been fixed. The details are explained in a pair of more recent posts.


I’ll start with the bad news: as of today, I am withdrawing the 2016 preprint Transversality and super-rigidity for multiply covered holomorphic curves. There is a mistake in that paper which makes the proofs of all four of its main results incomplete, and I cannot currently say with confidence that I know how to fix the mistake. As some readers will be aware, this is not the first time in history that an attempted proof of the super-rigidity conjecture for holomorphic curves has been withdrawn—in fact, it is not even the first time in history that such a proof has been withdrawn (or at least drastically downgraded) by me. The last time I did that, it was because I had come to believe the whole idea of my approach was unsuitable for the problem at hand.

The good news is that that is not the case this time: I strongly believe that the current situation is temporary, that the approach attempted in my 2016 paper is fundamentally correct, that 90% of the contents of the paper are correct and the remaining 10% is correctable. Moreover, I do currently know how to fill the gap to prove transversality results for multiple covers in certain cases, and there is good reason to believe that those cases are general enough for the applications to Embedded Contact Homology that originally motivated this project.

Nonetheless, since super-rigidity (Theorem A in my paper) is a well-known open problem that has been studied by several people in the Gromov-Witten community in the past, it is important to acknowledge publicly at this point that the problem is still open.

I would now like to explain a bit what goes wrong in my proof and what would be needed in order to fix it. I have explored a few ideas for a fix without making much progress, but it still seems quite plausible to me that a relatively easy fix may be possible, perhaps using ideas that are well-known among experts in either elliptic PDE theory or algebraic geometry (I would not call myself an expert in either). As I have recently learned from Thomas Walpuski and Aleksander Doan, the required lemma is not necessarily unique to the context of holomorphic curves, but can be formulated and has applications to a much wider class of equivariant transversality problems for elliptic PDEs (as discussed e.g. in Section 4 of their paper on associative submanifolds and monopoles). So, if you can prove it, you should write it up—it would be publishable as a stand-alone result!

(Acknowledgements: the error in my proof was found by Thomas Walpuski and Aleksander Doan, and my current understanding of the problem has been greatly influenced by discussions with them.)

What went wrong: quadratic unique continuation (AKA “Petri’s condition”)

I explained the main ideas behind my transversality approach in a series of three posts last December, and the vast majority of what it says in those posts is still correct as far as I’m aware. The detail that goes wrong is discussed in the penultimate section of the third post: it is a result I referred to at the time as a quadratic unique continuation lemma. Recent discussions with Doan and Walpuski have taught me some new terminology for it, which is borrowed from algebraic geometry.

Definition: Assume E and F are real Euclidean vector bundles over a manifold \Sigma with a fixed volume form, \mathbf{D} : \Gamma(E) \to \Gamma(F) is a real-linear partial differential operator and \mathbf{D}^* : \Gamma(F) \to \Gamma(E) denotes its formal adjoint. We say that \mathbf{D} satisfies Petri’s condition on some region {\mathcal U} \subset \Sigma if the natural map

\ker\mathbf{D} \otimes_{\mathbb R} \ker\mathbf{D}^* \stackrel{\iota}{\longrightarrow} \Gamma(E \otimes_{\mathbb R} F|_{\mathcal U})

defined by \iota(\eta \otimes \xi)(z) = \eta(z) \otimes \xi(z) is injective.

For reasons that I tried to explain in one of those posts last December, the main theorems on transversality and super-rigidity in my paper all follow if one restricts attention to J-holomorphic curves whose normal Cauchy-Riemann operators all satisfy Petri’s condition on some open subset where J can be perturbed. As I also explained in that post, it is not true that all Cauchy-Riemann type operators satisfy Petri’s condition, at least not locally:  for any complex-linear Cauchy-Riemann operator \mathbf{D} on a trivial line bundle over the disk, one can find complex-linearly independent sections \eta_1,\eta_2 \in \ker\mathbf{D} and \xi_1,\xi_2 \in \ker\mathbf{D}^* such that the section

\eta_1 \otimes_{\mathbb R} i\xi_2 - i\eta_1 \otimes_{\mathbb R} \xi_2 - \eta_2 \otimes_{\mathbb R} i\xi_1 + i\eta_2 \otimes_{\mathbb R} \xi_1 \in \Gamma(E \otimes_{\mathbb R} F)

is identically zero. Evidently, Petri’s condition can only hold for real Cauchy-Riemann type operators, and only under some condition preventing \mathbf{D} and \mathbf{D}^* from restricting to complex-linear operators on some subbundle. There are many generic conditions that prevent the latter: the simplest one I know is requiring the bundle map E \to F defined as the complex-antilinear part of \mathbf{D} to be invertible at some point z_0 \in \Sigma, a condition that holds for all normal Cauchy-Riemann operators of J-holomorphic curves if J is generic. When that condition holds, it forces \ker \mathbf{D} and \ker \mathbf{D}^* to be totally real subspaces of \Gamma(E) and \Gamma(F) respectively, and my claim in the paper was that whenever the latter is true, Petri’s condition holds.

That might be true, but my proof of it had a careless mistake that, when discovered, called the whole idea behind my proof into question. It’s probably not worth telling you what the precise mistake was—suffice it to say it was a problem of linear algebra (I claimed some set of vectors was complex-linearly independent when I actually only knew they were real-linearly independent). In any case, the idea had been to carry out a mild generalization of the proof that complex-linear Cauchy-Riemann operators satisfy the complex analogue of Petri’s condition involving complex tensor products. The latter is easy to prove using Taylor series, due to the fact that sections in \ker\mathbf{D} are power series in z while sections in \ker\mathbf{D}^* are power series in \bar{z}. The idea was then to use the totally real condition as a bridge between the complex- and real-linear worlds, i.e. if we know that real-linearly independent sets in \ker\mathbf{D} and \ker\mathbf{D}^* are also complex-linearly independent, then a result on the complex Petri condition should imply a corresponding real result.

On closer inspection, it is not as easy as I thought at first to make a rigorous argument out of this intuition involving the totally real condition, and I would not bet anyone’s life on the original lemma being true the way I stated it. Nonetheless, it seems to me almost inconceivable for something like the following statement not to hold:

Conjecture: Suppose \mathbf{D} : \Gamma(E) \to \Omega^{0,1}(\Sigma,E) is a complex-linear Cauchy-Riemann type operator on a vector bundle E over a Riemann surface \Sigma, and z_0 \in \Sigma is a point. Then there exists a generic condition on the \infty-jet of a complex-antilinear bundle map A : E \to \Lambda^{0,1}T^*\Sigma \otimes_{\mathbb C} E at z_0 such that for all A satisfying this condition, the operator \mathbf{D} + A satisfies Petri’s condition on neighborhoods of z_0.

What we know

Petri’s condition makes sense for arbitrary linear PDEs on vector bundles, and it tells us something about the relationship between global and pointwise linear independence for solutions to those PDEs. Here are some situations in which we can easily say that it holds:

  1. It holds for any first-order operator over a 1-dimensional domain, e.g. for the asymptotic operators that arise by linearizing Reeb vector fields in contact manifolds along periodic orbits. The condition holds in this case just because the PDE is an ODE, so by local existence and uniqueness, global linear independence implies pointwise linear independence. This observation is actually useful, e.g. one can adapt the methods described in my super-rigidity paper to classify the bifurcations of closed Reeb orbits under generic deformations of a contact form.
  2. It also holds for any Cauchy-Riemann type operator that splits over a direct sum of trivial line bundles on a closed surface, or more generally, for any operator whose solutions are constrained (e.g. via the similarity principle and asymptotic winding estimates) to be nowhere zero. This is again because global linear independence in this case implies pointwise linear independence. In higher dimensions this condition is rather special and difficult to observe in nature, but it is quite a familiar phenomenon in dimension four, where normal Cauchy-Riemann operators are defined on line bundles: for instance, this includes the case of the multiply covered holomorphic tori that arose in Taubes’s work on the Gromov invariant, and it similarly appears to include the situations that are important for applications to Embedded Contact Homology.
  3. For any differential operator that is Fredholm, Petri’s condition is clearly open. This makes the scenario in the previous bullet point seem slightly less special and more applicable.
  4. As mentioned above, the complex analogue of the Petri condition holds for all complex-linear Cauchy-Riemann operators. I do not currently know whether this observation is useful, since my original idea to produce a proof of the real Petri condition out of it has not panned out. I can still imagine clever ways that such an argument might work, though to be honest, I’d rather see a different proof—what makes me uneasy about appealing to complex analysis in this way is that it would be hard to imagine how such an argument might generalize to other elliptic operators beyond the Cauchy-Riemann case.
  5. There is also a weaker condition that does hold for all Cauchy-Riemann type operators: say that \mathbf{D} satisfies Petri’s condition “up to rank r” on a region {\mathcal U} \subset \Sigma if for all subspaces K \subset \mathbf{D} and C \subset \mathbf{D}^* of dimensions at most r, the natural map K \otimes_{\mathbb R} C \to \Gamma(E \otimes_{\mathbb R} F|_{\mathcal U}) is injective. The rank 1 Petri condition is an immediate consequence of the usual unique continuation results for Cauchy-Riemann operators, and it is not hard to extend this to ranks 2 and 3 using the fact that two real-linearly independent solutions to a Cauchy-Riemann type equation are always also pointwise real-linearly independent on an open and dense subset. (The latter follows from the Leibniz rule for Cauchy-Riemann operators together with the fact that real-valued holomorphic functions are all constant.) These partial results toward Petri’s condition appear to be the reason why Eftekhary’s partial proof of super-rigidity works. The counterexample mentioned above shows, however, that from rank 4 upward, Petri’s condition does not come for free.

That’s all for the moment. My paper will reappear in some revised form in the future, though it remains to be seen whether that means fixing the original proof or downgrading the generality of the stated results. I’ll surely write more on this topic when I know more.

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Postdoc and PhD studentship for symplectic topology in Berlin

I am happy to announce that my research group in symplectic and contact topology at the Humboldt-Universität in Berlin is hiring for Autumn 2018:

  • One 2-year postdoc position, salary approx. 44,000€/year (before taxes).* Here is the official advert in English and in German.
  • One 3-year PhD studentship, salary approx. 22,000€/year (before taxes).* Here is the official advert in English and in German.

Both positions are research-only; no teaching is required, so applicants need not know any German. (Opportunities to teach will nonetheless be available if desired.) They are both part of an ERC-funded project involving pseudoholomorphic curves in symplectic and contact topology, so postdoc applicants should ideally have some background in that subject, and PhD applicants should at least have a solid background in differential geometry and analysis. Applications should be e-mailed directly to me (including recommendation letters sent separately) by March 15, and we aim to make decisions as soon as possible after that date. The adverts specify what documents should be sent in the application: if you’ve already been applying for other postdoc or PhD positions, it is mostly the same stuff as usual. (Applicants for the PhD studentship: please be sure to include your university transcripts in addition to your CV and a statement of research experience/interests).

If there’s anything else I can clarify, please feel free to contact me by e-mail!

* For those unfamiliar with the vagaries of public employment in Germany: the precise salaries are determined by a complicated formula that depends on various details, including your family circumstances, but in theory you can compute them more exactly (and also the after-tax version) using this salary calculator. The main thing you need to know is that both positions are in Entgeltgruppe E13 of the TV-L. For the PhD studentship you need to input “50%” for Arbeitszeit, as the position is officially half-time (because you spend the other half of the time learning?). Please don’t ask me about Zusatzversorgung or Lohnsteuerklassen… if you want to know what these things mean, your best bet is to find an actual German, which, as you probably know, I am not.

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The transversality machine

UPDATE added 28.05.2019: An error (detailed in another post) in the paper that this post is about was discovered in Summer 2018, and has since been fixed, but it renders some portion of what is explained in this post either irrelevant or misleading. In particular, everything this post says about the “quadratic unique continuation” lemma should be taken with a grain of salt. I’ve written a pair of more recent posts to try and explain these matters properly.


In the last two posts, I have been describing a machine…

…well, no, not quite like that machine. I was speaking figuratively.

The machine is a stratification theorem, whose purpose is to demystify the transversality properties of multiply covered holomorphic curves. Its user interface consists mainly of a set of “twisted” Cauchy-Riemann operators associated to the normal operator \mathbf{D}_u^N of any multiply covered curve u = v \circ \varphi, producing a splitting

\mathbf{D}_u^N \cong (\mathbf{D}_u^{\boldsymbol{\theta}_1})^{\oplus k_1} \oplus \ldots \oplus (\mathbf{D}_u^{\boldsymbol{\theta}_N})^{\oplus k_N}

as described in the previous post. Operating the machine requires no specialized training beyond the ability to compute the indices of these operators—which are Cauchy-Riemann type operators defined on Sobolev spaces with exponential weight conditions over a punctured surface—plus a certain amount of patience with dimension-counting arguments. If you have this, then the machine gives you a stratification of the moduli space of multiple covers, with chambers in which transversality (or possibly the next best thing) is achieved, separated by smooth walls and further strata whose dimensions can all be computed.

Anyway, that’s what the instruction manual says it does. In this post I want to open up the machine and try to explain why it works.

The statement of the theorem

I am assuming you’ve read the original post in which all of the following notation was explained, but recall in particular that {\mathcal M}^d_G(J) is a moduli space of d-fold covered J-holomorphic curves u = v \circ \varphi with prescribed critical and branching behavior for v and \varphi respectively, and for \mathbf{k} = (k_1,\ldots,k_N) and \mathbf{c} = (c_1,\ldots,c_N) we consider the subset

\displaystyle {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) = \big\{ u \in {\mathcal M}^d_G(J)\ \big|\ \dim_{{\mathbb K}_i} \ker\mathbf{D}_u^{\boldsymbol{\theta}_i} = k_i \text{ and } \dim_{{\mathbb K}_i} \text{coker}\,\mathbf{D}_u^{\boldsymbol{\theta}_i} = c_i for i=1,\ldots,N \big\}.

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

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

where for each of the irreducible representations \boldsymbol{\theta}_i : G \to \text{Aut}_{\mathbb R}(W_i), the number t_i \in \{1,2,4\} is defined as the real dimension of {\mathbb K}_i := \text{End}_G(W_i).

The proof of this theorem follows the same general outline as most other theorems you’ve seen before that begin with the words “For generic J…”. The relevant generic set comes from applying the Sard-Smale theorem to a projection map

{\mathcal M}^d_G(\mathbf{k},\mathbf{c}) \to {\mathcal J} : (J,u) \mapsto J,

where {\mathcal J} is a suitable Banach manifold of perturbed almost complex structures and {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) is the resulting universal moduli space

{\mathcal M}^d_G(\mathbf{k},\mathbf{c}) = \big\{ (J,u) \ \big|\ J \in {\mathcal J} and u \in {\mathcal M}^d_G(J;\mathbf{k},\mathbf{c}) \big\}.

The main step is thus to prove that this universal moduli space is a smooth Banach manifold, which will follow from the implicit function theorem after proving that a certain linearized operator is surjective. The latter is, as usual, the hard part, and it resembles more familiar arguments in that it requires a unique continuation result for linear Cauchy-Riemann type equations, but the precise lemma we need is probably different than what you are used to. This so-called “quadratic” unique continuation lemma is the heart of the machine, and in itself it is not very hard to prove, but it is somewhat of a challenge to understand what it means and what it is good for, so one of my main goals for this post will be to explain that.

It should go without saying that when the kernel and cokernel conditions in the definition of {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) are dropped, the resulting universal moduli space

{\mathcal M}^d_G := \big\{ (J,u)\ \big|\ J \in {\mathcal J} and u \in {\mathcal M}^d_G(J) \big\}

is indeed a smooth Banach manifold; this follows from standard arguments. Thus assuming you’re on board with what I said above about the Sard-Smale theorem, we can agree that the goal is to prove the following:

Main lemma. {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) \subset {\mathcal M}^d_G is a smooth finite-codimensional submanifold with codimension \sum_i t_i k_i c_i.

I cheated a bit in this statement by replacing {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) with a new object {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) that I have not yet defined. Its precise definition will come in a bit, but for now, suffice it to say that it is an open subset of {\mathcal M}^d_G(\mathbf{k},\mathbf{c}), defined via an extra condition without which I wouldn’t know how to prove the lemma—but this extra condition will have no impact on the rest of the discussion.

What follows is an extended sketch of the proof, with occasional references to specific results in the paper where the full details are carried out. This may turn out to be somewhat weightier material than you are used to reading on a blog, so maybe it will help to have some appropriately weighty music as accompaniment; I personally recommend Beethoven’s Große Fuge.

Reduction to the case of regular covers

For reasons that were hinted at near the end of the previous post, we can impose the following simplifying assumption without loss of generality:

Assumption (cf. the beginning of Section 3.5). The branched covers \varphi : (\Sigma',j') \to (\Sigma,j) for elements (J,u = v \circ \varphi) \in {\mathcal M}^d_G are regular (i.e. normal).

To see why this is not a loss of generality, you need to recall how the splitting of \mathbf{D}_u^N is defined in terms of a regular presentation of \varphi (see the previous post). Each branched cover \varphi : (\Sigma',j') \to (\Sigma,j) of degree d with generalized automorphism group G has a (canonical up to isomorphism) regular branched cover \pi : (\Sigma'',j'') \to (\Sigma,j) that factors through it, with degree \tilde{d} = |G| \ge d and \text{Aut}(\pi) = G. The holomorphic curve \tilde{u} := v \circ \pi then has a splitting

\mathbf{D}_{\tilde{u}}^N = (\mathbf{D}_{\tilde{u}}^{\boldsymbol{\theta}_1})^{\oplus \tilde{k}_1} \oplus \ldots \oplus (\mathbf{D}_{\tilde{u}}^{\boldsymbol{\theta}_N})^{\oplus \tilde{k}_N}

whose summands are the same twisted Cauchy-Riemann operators that appear in the splitting of \mathbf{D}_u^N, but with different multiplicities \tilde{k}_i \ge k_i; in particular, the regularity of \pi implies via a standard theorem in representation theory that all of the \tilde{k}_i must be nonzero. Since \pi and \varphi have exactly the same critical values, their respective neighborhoods in the space of branched covers with fixed branching data can be identified naturally, so that there are canonical identifications {\mathcal M}^{\tilde{d}}_G = {\mathcal M}^d_G and {\mathcal M}^{\tilde{d}}_G(\mathbf{k},\mathbf{c}) = {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) in the neighborhoods of these two curves.

With the preceding understood, the regularity assumption will be in effect from now on. It has the following important consequences:

  1. G = \text{Aut}(\varphi) acts by biholomorphic maps on the domain (\Sigma',j'), and therefore also on the kernel and cokernel of the normal operator \mathbf{D}_u^N, by reparametrization.
  2. The multiplicities k_i in the splitting of \mathbf{D}_u^N are all positive.

Notice now that if (J_0,u_0) \in {\mathcal M}^d_G(\mathbf{k},\mathbf{c}), then sufficiently nearby elements (J,u) \in {\mathcal M}^d_G will belong to {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) if and only if

\dim \ker \mathbf{D}_u^N = \dim \ker \mathbf{D}_{u_0}^N.

Indeed, the splittings of these operators vary continuously as u_0 moves toward u, and each summand is a Fredholm operator, so the dimension of its kernel can jump downward under small perturbations, but not upward. (If you are unfamiliar with this fact, it will follow easily from the discussion of the space of Fredholm operators below.) Since every twisted operator \mathbf{D}_u^{\boldsymbol{\theta}_i} appears with positive multiplicity in the splitting of \mathbf{D}_u^N, any downward jump in \dim \ker \mathbf{D}_u^{\boldsymbol{\theta}_i} will necessarily cause a downward jump in \dim \ker\mathbf{D}_u^N.

How to perturb a normal Cauchy-Riemann operator

To understand the local structure of {\mathcal M}^d_G(\mathbf{k},\mathbf{c}), we need to understand the effect that moving (J,u) through {\mathcal M}^d_G has on the normal operators \mathbf{D}_u^N.  This is fairly straightforward if we consider only variations in (J,u) that leave u fixed, which will suffice for our purposes. Fix a closed J-holomorphic curve u = v \circ \varphi, where v : \Sigma \to M is somewhere injective, and consider a 1-parameter family \{J_s\in {\mathcal J}\}_{s \in (-\epsilon,\epsilon)} with J_0 \equiv J such that J_s = J along the image of v, so u remains J_s-holomorphic for all s. Its normal bundle N_u also remains unchanged as the parameter moves, but the normal Cauchy-Riemann operator depends on J_s, so let us denote the resulting 1-parameter family of operators by

\mathbf{D}^N_{u,s} : \Gamma(N_u) \to \Omega^{0,1}(\Sigma',N_u).

Now Y := \partial_s J_s\big|_{s=0} vanishes along the image of v, but if z \in \Sigma is any injective point of v, then there is considerable freedom to choose the derivative of Y near v(z) in directions normal to v. Choose \eta \in (N_v)_z and write the normal derivative \nabla_\eta Y in terms of the tangent-normal splitting T_{v(z)} M = (T_v)_z \oplus (N_v)_z as

\nabla_\eta Y = \begin{pmatrix} \nabla_\eta^T Y & \nabla_\eta^{TN} Y \\ \nabla_\eta^{NT} Y & \nabla_\eta^N Y \end{pmatrix}.

One can now compute that if \nabla Y is chosen to vanish near all non-injective points of v, then the change in the normal Cauchy-Riemann operator is precisely

\partial_s\mathbf{D}_{u,s}^N \eta\big|_{s=0} = \nabla^{NT}_\eta Y \circ Tu \circ j'.

The point is that we can choose families J_s to make this perturbation more or less anything we want. If we’re working in a symplectic manifold (M,\omega), then of course we’d like to require all the J_s to be compatible with \omega, which gives a nontrivial relation between \nabla^{NT}_\eta Y and \nabla^{TN}_\eta Y, but this does not constrain \nabla^{NT}_\eta Y at all if we are willing to let it determine \nabla^{TN}_\eta Y. The one caveat is that if u = v \circ \varphi has \text{Aut}(\varphi) = G, then every perturbation of \mathbf{D}_u^N produced in this way will automatically be G-invariant. The important result can thus be stated as follows:

Lemma 1 (see Lemma 6.1). Let {\mathcal U} \subset \Sigma denote the open and dense set of injective points of v. Then given any smooth G-invariant zeroth-order perturbation A : N_u \to \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u) with support in \varphi^{-1}({\mathcal U}), there exists a smooth 1-parameter family of compatible almost complex structures J_s, satisfying J_s = J_0 and matching J_0 along the image of u for all s, such that \partial_s \mathbf{D}_{u,s}^N\big|_{s=0} = A.

Generic J are nowhere integrable

It’s time to fill in the missing detail about the definition of the constrained universal moduli space {\mathcal M}^d_G(\mathbf{k},\mathbf{c}). For technical reasons that will become clear when we discuss unique continuation at the end of this post, I need to impose an extra open condition on pairs (J,u). It amounts to the requirement that J must not be integrable on any neighborhood of the image of u, though non-integrability as such is not really the point, but is more of a side-effect.

Recall that for any complex vector space (V,J), a real subspace W \subset V is called totally real if W \cap J(W) = \{0\}. (Sometimes one also requires W to have half the dimension of V, but I am not requiring that here.) Since \mathbf{D}_u^N is a real-linear operator between two complex Banach spaces, one can ask in particular whether its kernel and cokernel (meaning the kernel of its formal adjoint) are totally real. There is an easy criterion for this: recall that if we break up \mathbf{D}_u^N into its complex-linear part \mathbf{D}_u^{\mathbb C} and antilinear part \mathbf{D}_u^{\bar{\mathbb C}}, then the latter is a zeroth-order term, meaning a smooth bundle map N_u \to \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u). Using the standard unique continuation results for linear Cauchy-Riemann operators, one can easily show (cf. Lemma 3.11) that \ker \mathbf{D}_u^N and \text{coker}\, \mathbf{D}_u^N are guaranteed to be totally real whenever the antilinear bundle map \mathbf{D}_u^{\bar{\mathbb C}} is invertible on some fiber. This is manifestly an open condition, and we shall say that \mathbf{D}_u^N itself is totally real whenever it holds. Notice that if this holds for a given curve v, then it automatically also holds for all of its multiple covers u = v \circ \varphi.

Definition. Let {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) \subset {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) denote the open subset consisting of pairs (J,u) for which \mathbf{D}_u^N is totally real.

If J is integrable, then \mathbf{D}_u^N is always complex linear and the totally real condition can never be satisfied. Of course, this should not worry us very much, because we are trying to prove a theorem about generic J, which cannot be expected to be integrable. It turns out in fact that genericity is enough to guarantee that the totally real condition is always satisfied, and this is why establishing the smoothness of {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) will suffice for proving Theorem D.

Lemma 2 (cf. Lemma 6.2). For generic J, every closed J-holomorphic curve has the property that \mathbf{D}_u^N is totally real.

The proof of this is not hard once you’ve absorbed the implications of Lemma 1. The point is to show that the universal moduli space of somewhere injective curves that fail to satisfy the totally real condition lives inside a submanifold of arbitrarily large codimension, namely the set of curves v : \Sigma \to M such that \mathbf{D}_v^{\bar{\mathbb C}} satisfies an incidence relation with the subvariety of noninvertible linear maps (N_v)_z \to \overline{\text{Hom}}_{\mathbb C}(T_z\Sigma,(N_v)_z) at some chosen point z \in \Sigma. Having the invertibility condition fail on an open neighborhood of z means that the incidence relation can be taken to involve jets of arbitrarily high order, cutting out submanifolds of arbitrarily large codimension.

Walls in the space of Fredholm operators


I would now like to tell you a lovely fact about the space of Fredholm operators that everyone ought to know: for any pair of real Banach spaces X and Y and any integers k,\ell \ge 0, the subset

V_{k,\ell} := \big\{ \mathbf{T} : X \to Y\ \big|\ \dim \ker \mathbf{T} = k and \dim \text{coker}\, \mathbf{T} = \ell \big\}

is a smooth finite-codimensional submanifold in the space of bounded linear maps X \to Y, with

\text{codim}\, V_{k,\ell} = k\ell.

One can see it as follows. Given an operator \mathbf{T}_0 \in V_{k,\ell}, \mathbf{T}_0 is evidently Fredholm, so there exist splittings X = V \oplus K, Y = W \oplus C into closed linear subspaces, with K = \ker\mathbf{T}_0 and W = \text{im}\, \mathbf{T}_0, thus C \cong \text{coker}\, \mathbf{T}_0, and \mathbf{T}_0 restricts to V as a Banach space isomorphism V \to W. These produce a block decomposition for any bounded linear operator \mathbf{T} : X \to Y in the form

\mathbf{T} = \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ \mathbf{C} & \mathbf{D} \end{pmatrix},

such that \mathbf{A} : V \to W will necessarily be invertible whenever \mathbf{T} lies in a sufficiently small neighborhood {\mathcal O} of \mathbf{T}_0. We can therefore define a smooth map

\boldsymbol{\Phi} : {\mathcal O} \to \text{Hom}(\ker \mathbf{T}_0,\text{coker}\, \mathbf{T}_0) : \mathbf{T} \mapsto \mathbf{D} - \mathbf{C}\mathbf{A}^{-1}\mathbf{B},

whose derivative at \mathbf{T}_0 is

d\boldsymbol{\Phi}(\mathbf{T}_0) \begin{pmatrix} \mathbf{a} & \mathbf{b} \\ \mathbf{c} & \mathbf{d} \end{pmatrix} = \mathbf{d}

and is thus manifestly surjective. Now if we also associate to each \mathbf{T} \in {\mathcal O} the linear “coordinate change” on X defined in terms of the splitting X = V \oplus K by

\boldsymbol{\Psi} = \begin{pmatrix} \text{Id} & -\mathbf{A}^{-1}\mathbf{B} \\ 0 & \text{Id} \end{pmatrix},

we have \mathbf{T} \boldsymbol{\Psi} = \begin{pmatrix} \mathbf{A} & 0 \\ \mathbf{C} & \boldsymbol{\Phi}(\mathbf{T}) \end{pmatrix}, implying

\ker \mathbf{T} \cong \ker \boldsymbol{\Phi}(\mathbf{T}).

Since \boldsymbol{\Phi}(\mathbf{T}) is defined on the space K with dimension \dim \ker\mathbf{T}_0, this implies that \dim \ker\mathbf{T} \le \dim \ker\mathbf{T}_0 for all \mathbf{T} sufficiently close to \mathbf{T}_0, and equality is satisfied if and only if \boldsymbol{\Phi}(\mathbf{T}) = 0. As a consequence,

V_{k,\ell} \cap {\mathcal O} = \boldsymbol{\Phi}^{-1}(0),

and since d\boldsymbol{\Phi}(\mathbf{T}_0) is surjective, the implicit function theorem gives this zero set the structure of a smooth submanifold with codimension equal to \dim \text{Hom}(K,C) = k\ell.

Walls in the universal moduli space of multiple covers

We now proceed toward the proof of the main lemma. Given (J_0,u_0) \in {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}), we observed already that a sufficiently close element (J,u) \in {\mathcal M}^d_G will also belong to {\mathcal M}^d_G(\mathbf{k},\mathbf{c}) if and only if \dim \ker \mathbf{D}_u^N = \dim \ker \mathbf{D}_{u_0}^N, or equivalently if the two cokernels have matching dimensions. Plugging this into the above discussion about Fredholm operators in general, there is a neighborhood {\mathcal O} \subset {\mathcal M}^d_G of (J_0,u_0) and a smooth map

\boldsymbol{\Phi} : {\mathcal O} \to \text{Hom}(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N)

whose zero set is a neighborhood of (J_0,u_0) in {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}). It is important to notice moreover that in light of the natural action of G = \text{Aut}(\varphi), \boldsymbol{\Phi} can be arranged to have its image in the space of Gequivariant linear maps \ker\mathbf{D}_{u_0}^N \to \text{coker}\, \mathbf{D}_{u_0}^N, i.e.

\boldsymbol{\Phi} : {\mathcal O} \to \text{Hom}_G(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N).

Computing the dimension of the space on the right hand side requires some representation theory: each of \ker\mathbf{D}_{u_0}^N and \text{coker}\, \mathbf{D}_{u_0}^N are now representations of G, and their decompositions into irreducible representations can be deduced from our splitting of \mathbf{D}_{u_0}^N. Schur’s lemma then breaks up the elements of \text{Hom}_G(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N) into blocks that always must vanish when they correspond to two non-isomorphic representations, and the result (cf. Equation (3.22)) is

\displaystyle \dim \text{Hom}_G(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N) = \sum_{i=1}^N \dim \text{End}_G(W_i) \cdot \dim_{{\mathbb K}_i} \text{Hom}(\ker \mathbf{D}_{u_0}^{\boldsymbol{\theta}_i} , \text{coker}\, \mathbf{D}_{u_0}^{\boldsymbol{\theta}_i})

= \sum_{i=1}^N t_i k_i c_i.

The implicit function theorem will now complete the proof of the main lemma if we can prove that the linearization of \boldsymbol{\Phi} at (J_0,u_0) is surjective. Let us write down the derivative of this map in directions of the form (Y,0) \in T_{(J_0,u_0)} {\mathcal M}^d_G that we considered in Lemma 1. Denoting by A_Y the zeroth-order perturbation of \mathbf{D}_{u_0}^N that results from varying (J_0,u_0) in the (Y,0) direction, the derivative in question gives a linear map

T_{J_0}{\mathcal J} \to \text{Hom}_G(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N) : Y \mapsto \mathbf{L}_Y

of the form

\mathbf{L}_Y \eta = \pi_C (A_Y \eta),

where \pi_C denotes the natural linear projection map from the relevant Sobolev space of sections of \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u) to \text{coker}\, \mathbf{D}_u^N.

Lemma 3 (cf. Lemmas 5.4 and 6.4). The operator Y \mapsto \mathbf{L}_Y described above is surjective.

This lemma is the main technical step. To prove it, one can restate the problem in light of Lemma 1 as follows. Let us choose suitable bundle metrics and area forms so that there are well-defined L^2-pairings on spaces of sections, and we can thus identify \text{coker}\,\mathbf{D}_u^N with the kernel of the formal adjoint of \mathbf{D}_u^N. This makes \text{coker}\, \mathbf{D}_{u_0}^N the L^2-orthogonal complement of \text{im}\, \mathbf{D}_{u_0}^N, so that the matrix elements of the linear transformation \mathbf{L}_Y for any given Y \in T_{J_0}{\mathcal J} take the form

\langle \xi , \mathbf{L}_Y \eta \rangle_{L^2} = \langle \xi , A_Y \eta \rangle_{L^2} for \eta \in \ker\mathbf{D}_{u_0}^N and \xi \in \text{coker}\,\mathbf{D}_{u_0}^N.

So we need to know that for any given G-equivariant linear map \Psi : \ker\mathbf{D}_{u_0}^N \to \text{coker}\, \mathbf{D}_{u_0}^N, we can find a G-invariant zeroth-order perturbation A : N_u \to \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u), with support away from the non-injective points of the underlying simple curve v, such that

\langle \xi, A \eta \rangle_{L^2} = \langle \xi , \Psi \eta \rangle_{L^2} for all \eta \in \ker\mathbf{D}_{u_0}^N and \xi \in \text{coker}\, \mathbf{D}_{u_0}^N.

Notice that if we can find a solution A to this problem that is not G-invariant, then we can always symmetrize it to produce one that is; this is possible due to the G-equivariance of \Psi. We are therefore free to ignore the G-symmetry from now on, and simply look for any zeroth-order perturbation A that is supported in a given open set and satisfies the above relation for a given linear map \Psi \in \text{Hom}(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N).

This problem does not sound unsolvable when you consider that \Psi is required to live in a finite-dimensional vector space, while we are free to choose A from a space that is infinite-dimensional. Arguing by contradiction, suppose there is no solution, or equivalently, that \text{Hom}(\ker \mathbf{D}_{u_0}^N,\text{coker}\, \mathbf{D}_{u_0}^N) contains a nontrivial element orthogonal to every element that can be produced by choices of zeroth-order perturbations A. This can be expressed more concretely in terms of the matrix elements with respect to orthonormal bases (\eta_i) of \ker\mathbf{D}_{u_0}^N and (\xi_j) of \text{coker}\, \mathbf{D}_{u_0}^N: if the solution we’re looking for does not exist, then there is a set of real numbers \Psi^{ij}, not all equal to zero, such that

\displaystyle \sum_{i,j} \Psi^{ij} \langle \xi_j , A \eta_i \rangle_{L^2} = 0

for every A in the space of allowed perturbations. This doesn’t sound very likely, but there is one conceivable situation where we would now be in big trouble: to see this, let us write the L^2-product more explicitly as an integral of a real bundle metric \langle\cdot,\cdot \rangle, which we can view as a fiberwise linear form on the tensor product of the bundle \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u) with itself. The above expression then becomes

\displaystyle \int_{\Sigma'} \langle\cdot,\cdot\rangle \circ (\text{Id} \otimes A) \left( \sum_{i,j} \Psi^{i,j} \xi_j(z) \otimes \eta(z) \right) \, d\text{vol} = 0.

The summation in parentheses in this integral is a section of the tensor bundle \overline{\text{Hom}}_{\mathbb C}(T\Sigma',N_u) \otimes_{\mathbb R} N_u, defined as a linear combination of products \xi_j \otimes \eta_i where \eta_i satisfies a linear Cauchy-Riemann type equation and \xi_j satisfies its formal adjoint equation. It is not difficult to show (cf. Lemma 5.5) that if this linear combination is nonzero on some open set, then A can be chosen with support in that set to ensure that the integral is nonzero, thus giving a contradiction. We know that the \eta_i and \xi_j all satisfy unique continuation results, so we can easily find an open set on which all of them are nonvanishing. Is it really possible that a linear combination of this form could nonetheless vanish?

Quadratic unique continuation


Actually yes: if we’re not careful, such a linear combination certainly could vanish. Let’s put the problem in slightly more general terms: assume \mathbf{D} : \Gamma(E) \to \Gamma(F) is a real-linear Cauchy-Riemann type operator on some complex vector bundle E over a Riemann surface \Sigma, with F := \overline{\text{Hom}}_{\mathbb C}(T\Sigma,E), and \mathbf{D}^* : \Gamma(F) \to \Gamma(E) denotes its formal adjoint with respect to some chosen L^2-pairing. The question we need to consider is local, so we place no assumptions on the base \Sigma, i.e. it could be simply a disk.

Question: If K \subset \ker \mathbf{D} and C \subset \ker \mathbf{D}^* are finite-dimensional subspaces, must the natural map

C \otimes_{\mathbb R} K \stackrel{\iota}{\longrightarrow} \Gamma(F \otimes_{\mathbb R} E)

defined by \iota(\xi \otimes \eta)(z) = \xi(z) \otimes \eta(z) be injective?

Answer: In general, no. For example take E and F to be the trivial line bundle over \Sigma = {\mathbb C}, with \mathbf{D} = \bar{\partial} and \mathbf{D}^* = -\partial, so their kernels are the spaces of holomorphic and antiholomorphic functions respectively. Now define K \subset \ker \bar{\partial} to be the complex span of the functions \eta_1(z) = 1 and \eta_2(z) = z, while C \subset \ker \partial is the complex span of \xi_1(z) = 1 and \xi_2(z) = \bar{z}. Then

\xi_1 \otimes i \eta_2 - i \xi_1 \otimes \eta_2 - \xi_2 \otimes i \eta_1 + i \xi_2 \otimes \eta_1 \in C \otimes_{\mathbb R} K

defines a nontrivial element in the real tensor product of the vector spaces C and K, but it also defines the zero-section of the bundle F \otimes_{\mathbb R} E.

It turns out that the reason for the horror scenario in the above example is that we allowed the spaces K and C to be complex. This is where the extra condition in our definition of {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) comes into play, as we will see that the problem goes away if we require K and C to be totally real.

UPDATE 28.05.2019: The following lemma was the error that is mentioned in the caveat at the top of this post. It is now known to be false, but has been replaced with a similar statement that is true. See this more recent post for details.

Lemma 4 (quadratic unique continuation, cf. Proposition 5.1). For any sets of complex-linearly independent sections \eta_1,\ldots,\eta_p \in \ker \mathbf{D} and \xi_1,\ldots,\xi_q \in \ker \mathbf{D}^* in the above setting, no linear combination

\displaystyle \sum_{i,j} c^{ij} \xi_j \otimes \eta_i \in \Gamma(F \otimes_{\mathbb C} E)

with coefficients c^{ij} \in {\mathbb C} satisfying c^{ij} \ne 0 for some i,j vanishes to infinite order at any point.

This is a statement about complex bases and a complex tensor product, but in light of the canonical surjective bundle map F \otimes_{\mathbb R} E \to F \otimes_{\mathbb C} E and the fact that any real basis of a totally real subspace is also complex-linearly independent, it has the following consequence:

Corollary. If the finite-dimensional subspaces K \subset \ker\mathbf{D} and C \subset \ker \mathbf{D}^* are both totally real, then the natural map

C \otimes_{\mathbb R} K \to \Gamma(F \otimes_{\mathbb R} E)

is injective, and the nontrivial sections in its image do not vanish to infinite order at any point.

You’ll gain some valuable intuition about Lemma 4 if you work out the case where \mathbf{D} = \bar{\partial} on a trivial vector bundle, so the \eta_i are all holomorphic functions and the \xi_j are antiholomorphic functions. The proof in this situation is not very hard: it hinges on the fact that all terms in the Taylor series of \eta_i are powers of z, while those in the Taylor series of \xi_j are powers of \bar{z}. Putting them together in a complex tensor product thus leaves them “decoupled” so that no nontrivial linear combination can kill them, so long as the original sets of Taylor coefficients are complex-linearly independent. For the general case, one cannot use Taylor series, but it is still useful to note that—as a corollary of the usual unique continuation results (based on the similarity principle or whatever else you prefer)—the first nonvanishing term in the Taylor expansion of each \eta_i is a power of z, and similarly for \xi_j with powers of \bar{z}. The proof thus proceeds following the same idea as in the (anti-)holomorphic case, but paying specific attention to the first nontrivial term in the Taylor expansion at each step.

A concluding remark on integrability

I’m sure no one will argue with me when I say that integrable complex structures are not “generic,” and indeed, some of the corollaries of Theorem D are known to be false in the integrable case. For instance, Bryan and Pandharipande have found examples in algebraic Calabi-Yau 3-folds for which super-rigidity fails. While I honestly don’t know whether the totally real condition in my definition of {\mathcal M}^{d,*}_G(\mathbf{k},\mathbf{c}) is essential, I find it amusing and slightly poetic that the generic almost complex structures provided by Theorem D are guaranteed to be non-integrable. That does not mean of course that results like super-rigidity cannot possibly hold in integrable settings: there are also known cases in which they do, but when it happens, it is for reasons completely unrelated to any of what I’ve been talking about here.

Is the Große Fuge over yet?

Update (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 the main applications concerning super-rigidity and transversality for multiple covers. Many thanks to Thomas Walpuski and Aleksander Doan for catching the error.

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Regular presentations and twisted Cauchy-Riemann operators

 

This is the first of two followup posts that I promised at the end of “Transversality for multiple covers, super-rigidty, and all that”. I want to fill in some details about the natural splitting

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

that exists for the normal Cauchy-Riemann operator \mathbf{D}_u^N : \Gamma(N_u) \to \Omega^{0,1}(\Sigma,N_u) of any multiply covered J-holomorphic curve u = v \circ \varphi. As I mentioned in the other post, the main thing you need to know before solving any given transversality problem for multiple covers is the Fredholm indices of these operators. So let’s start by understanding what the operators are.

The idea behind this splitting is presumably standard in some circles, but it was new to me when I read about it in Taubes’s 1996 paper Counting pseudo-holomorphic submanifolds in dimension 4. The starting point is the observation that if \varphi is a regular covering map, then sections of the pullback bundle N_u = \varphi^*N_v can be identified naturally with sections of some tensor product bundle N_v \otimes V determined by a permutation representation of the automorphism group of \varphi. The irreducible subrepresentations then give rise to subbundles of N_v \otimes V, which correspond to subspaces of \Gamma(N_u) and \Omega^{0,1}(\Sigma,N_u) that are respected by \mathbf{D}_u^N.

Since Taubes was mainly interested in multiply covered tori covering embedded tori, he did not need to consider cases where \varphi has branch points or fails to be regular (meaning |\text{Aut}(\varphi)| < \text{deg}(\varphi)), but we do. In the following, we will deal with curves u = v \circ \varphi where v : (\Sigma,j) \to (M,J) is any closed somewhere injective J-holomorphic curve and

\varphi : (\Sigma',j') \to (\Sigma,j)

is an arbitrary holomorphic branched cover of closed Riemann surfaces, with degree d, which we assume to be at least 2. Everything I say here can be generalized to punctured surfaces—which I expect to have some interesting applications in SFT and ECH—but I will not discuss that in this post. I should mention that a slightly different variation on Taubes’s splitting idea has been used in Eftekhary’s approach to the super-rigidity problem, though it appears to be limited to the case of regular branched covers.

To simplify notation, let us denote the Cauchy-Riemann type operator \mathbf{D}_v^N simply by \mathbf{D}, with domain and target bundles

E := N_v, F := \overline{\text{Hom}}_{\mathbb C}(T\Sigma,N_v).

We can regard \mathbf{D}_u^N as a pullback of \mathbf{D} and will thus denote it by \varphi^*\mathbf{D}. Its domain and target bundles are

E^\varphi := N_u = \varphi^*N_v and F^\varphi := \overline{\text{Hom}}_{\mathbb{C}}(T\Sigma',N_u),

and the two operators are related to each other by

(\varphi^*\mathbf{D})(\eta \circ \varphi) = \varphi^*(\mathbf{D}\eta) for all \eta \in \Gamma(E).

Branch points as punctures

Now is the moment to reveal a very small lie that I’ve already told you a few times: the splitting I’m going to define is not actually a decomposition of the operator \varphi^*\mathbf{D} : \Gamma(E^\varphi) \to \Gamma(F^\varphi), but is instead a decomposition of a different operator that is, for purposes of transversality questions, equivalent to it. The reason for this extra complication is that I would like to apply a certain amount of standard covering space theory to the branched cover \varphi : \Sigma' \to \Sigma, but before I can do that, I need to remove its branch points so that it becomes an honest covering map. Let

\Theta \subset \Sigma

denote a finite set that contains all the critical values of \varphi; in most situations we will simply take \Theta to be the set of critical values, but in some cases it is useful to have a bit more freedom. With this choice in place, define

\dot{\Sigma} := \Sigma \setminus \Theta, \Theta' := \varphi^{-1}(\Theta), \dot{\Sigma}' := \Sigma' \setminus \Theta',

so that \varphi now restricts to a degree d covering map of punctured surfaces

\dot{\Sigma}' \stackrel{\varphi}{\longrightarrow} \dot{\Sigma}.

We will denote the restrictions of the bundles E and F to the punctured domain \dot{\Sigma} by

\dot{E} := E|_{\dot{\Sigma}} and \dot{F} := F|_{\dot{\Sigma}},

and denote by \dot{\mathbf{D}} : \Gamma(\dot{E}) \to \Gamma(\dot{F}) the corresponding restriction of the operator \mathbf{D} : \Gamma(E) \to \Gamma(F). In order to get useful analytical results from such a restriction, we need to choose suitable Banach space completions so that \mathbf{D} and \dot{\mathbf{D}} end up having kernels and cokernels of the same dimension. The way to do this is straightforward, but it may strike some readers as odd at first glance.

Choose any k \in \mathbb{N} and p \in (1,\infty), so that \mathbf{D} becomes a bounded linear operator between Sobolev spaces

\mathbf{D} : W^{k,p}(E) \to W^{k-1,p}(F),

and elliptic regularity implies that \ker\mathbf{D} is a fixed finite-dimensional space of smooth sections independent of the choice of k and p; for similar reasons, the Fredholm index of \mathbf{D} also does not depend on these choices, hence neither does the dimension of its cokernel. The corresponding choice of Sobolev spaces for \dot{\mathbf{D}} requires exponential weights at the punctures. To define this, choose a trivialization of E over a neighborhood of each point z_0 \in \Theta, together with holomorphic cylindrical coordinates (s,t) identifying a punctured neighborhood of z_0 with the cylindrical end [0,\infty) \times S^1. Then for any \delta > 0, we define the Banach space

W^{k,p,-\delta}(\dot{E}) = \{ \eta \in W^{k,p}_{\text{loc}}(\dot{E})\ |\ e^{-\delta s} \eta \in W^{k,p}([0,\infty) \times S^1) on each cylindrical end near \Theta \}.

Pay careful attention to the signs here: sections in W^{k,p,-\delta}(\dot{E}) need not be bounded near the punctures, as they are allowed to grow exponentially at a rate bounded by \delta. A Banach space of this type will never arise as a tangent space to any reasonable Banach manifold of nonlinear maps, but it is the right space for our present purposes, due to the following result:

Lemma. For any \delta > 0 sufficiently small, the operators \mathbf{D} : W^{k,p}(E) \to W^{k-1,p}(F) and \dot{\mathbf{D}} : W^{k,p,-\delta}(\dot{E}) \to W^{k-1,p,-\delta}(\dot{F}) have the same Fredholm index, and the map \eta \mapsto \eta|_{\dot{\Sigma}} defines an isomorphism \ker\mathbf{D} \to \ker \dot{\mathbf{D}}.

The proof of this is a straightforward application of elliptic regularity results, including the asymptotic formulas for Cauchy-Riemann type equations on half-cylinders due to Hofer-Wysocki-Zehnder et al (see the appendix of Siefring’s asymptotics paper). The main point is that if \eta \in W^{k,p,-\delta}(\dot{E}) satisfies \dot{\mathbf{D}}\eta = 0 and \delta > 0 is small enough so that a certain asymptotic operator has no eigenvalues between 0 and \delta, then asymptotic formulas imply that \eta does not actually grow exponentially, but is bounded well enough at infinity to deduce that it extends over the punctures as an element of W^{k,p}(E). In this context, it should not be very surprising that we impose an exponential growth condition instead of exponential decay: some exponential weight is necessary since \dot{\mathbf{D}} would otherwise fail to be Fredholm, but a decay condition would mean that we only see elements of \ker\mathbf{D} that happen to vanish at \Theta, which most of them will not.

The above discussion applies just as well to the pulled back operator \varphi^*\mathbf{D} as it does to \mathbf{D}: everything is fine as long as the exponential weight \delta > 0 is sufficiently small. We will assume from now on that both \dot{\mathbf{D}} and \varphi^*\dot{\mathbf{D}} are defined on weighted Sobolev spaces of this form. Notice that since branch points have been removed, the domain and target bundles of the latter can both be identified with pullback bundles:

\varphi^*\dot{\mathbf{D}} : W^{k,p,-\delta}(\varphi^*\dot{E}) \to W^{k-1,p,-\delta}(\varphi^*\dot{F}).

Regular presentations of branched covers

The next step is to encode the symmetries of \varphi : \dot{\Sigma}' \to \dot{\Sigma} in group-theoretic terms. The most obvious group to look at for this purpose is the finite automorphism group \text{Aut}(\varphi), but unless \varphi happens to be regular (i.e. normal), this group might be too small to contain all of the information we need—it might even be trivial. A more useful notion is the following: one can always identify \varphi : \dot{\Sigma}' \to \dot{\Sigma} with a map of the form

\left( \dot{\Sigma}'' \times \{1,\ldots,d\} \right) \Big/ G \to \dot{\Sigma} : [(z,i)] \mapsto \pi(z),

for some connected regular covering map \pi : \dot{\Sigma}'' \to \dot{\Sigma}, with a finite group G = \text{Aut}(\pi) acting on \dot{\Sigma}'' by deck transformations and acting transitively on \{1,\ldots,d\} via some injective homomorphism to the symmetric group,

\rho : G \to S_d.

Such a presentation can be derived from the following standard picture of covering spaces. Let \tilde{\pi} : {\mathscr U} \to \dot{\Sigma} denote the universal cover of \dot{\Sigma}, with \text{Aut}(\tilde{\pi}) = \pi_1(\dot{\Sigma}). After choosing an ordering for the lifts in \dot{\Sigma}' of a base point in \dot{\Sigma}, every based loop in \dot{\Sigma} lifts to paths in \dot{\Sigma}' that permute the lifted base points, thus defining a homomorphism \tilde{\rho} : \pi_1(\dot{\Sigma}) \to S_d such that \varphi : \dot{\Sigma}' \to \dot{\Sigma} can now be identified with the covering map

\left( {\mathscr U} \times \{1,\ldots,d\} \right) \Big/ \pi_1(\dot{\Sigma}) \to \dot{\Sigma} : [(z,i)] \mapsto \tilde{\pi}(z).

The group G := \pi_1(\dot{\Sigma}) / \ker\tilde{\rho} is now finite (with order at most d!) since \tilde{\rho} descends to an injection \rho : G \to S_d, and \tilde{\pi} likewise descends to \dot{\Sigma}'' := {\mathscr U} / \ker\tilde{\rho} as a regular cover \pi : \dot{\Sigma}'' \to \dot{\Sigma} with automorphism group G. If \Theta is chosen to contain only the critical values of \varphi : \Sigma' \to \Sigma, then we refer to G as the generalized automorphism group of \varphi, and one can identify it with the quotient of \pi_1(\dot{\Sigma}) by the normal core of the subgroup \varphi_*\pi_1(\dot{\Sigma}'). Notice that G is always a nontrivial group, with order at least d and at most d!. The lower bound is attained if and only if \varphi_*\pi_1(\dot{\Sigma}') equals its normal core, which just means that \varphi is regular and G is in this case \text{Aut}(\varphi). One can show that \varphi : \dot{\Sigma}' \to \dot{\Sigma} and \pi : \dot{\Sigma}'' \to \dot{\Sigma} are isomorphic covers if \varphi is regular; more generally, \pi always factors through another cover that is isomorphic to \varphi.

regular presentation of \varphi : \Sigma' \to \Sigma is defined in general to mean a choice of finite set \Theta \subset \Sigma containing the critical values, plus an identification of \dot{\Sigma}' with (\dot{\Sigma}'' \times \{1,\ldots,d\}) / G as described above for some regular cover \pi : \dot{\Sigma}'' \to \dot{\Sigma} with a finite automorphism group G that acts transitively on \{1,\ldots,d\} via a homomorphism \rho : G \to S_d. Here we need not generally assume that \rho is injective, but if this holds and \Theta is chosen to contain only the critical values of \varphi, then we say the regular presentation is minimal; in this case it is necessarily isomorphic to the one we constructed above, with G as the generalized automorphism group of \varphi. It is not hard to show that for any regular presentation, the regular cover \pi : \dot{\Sigma}'' \to \dot{\Sigma} extends to a regular holomorphic branched cover of closed Riemann surfaces

\pi : (\Sigma'',j'') \to (\Sigma,j)

such that \dot{\Sigma}'' = \Sigma'' \setminus \Theta'' where \Theta'' := \pi^{-1}(\Theta).

The stratification theorem I stated in the previous post requires fixing the minimal regular presentation of \varphi : \Sigma' \to \Sigma, but non-minimal presentations can also be useful in certain inductive arguments, e.g. in the proof of super-rigidity. For the construction below, we fix any regular presentation.

The twisted bundle construction

We now come to the heart of the matter. Identifying \dot{\Sigma}' with (\dot{\Sigma}'' \times \{1,\ldots,d\}) / G as described above, suppose

\boldsymbol{\theta} : G \to \text{Aut}_{\mathbb R}(W)

is a real finite-dimensional representation of G. We can use this to associate with \dot{E} \to \dot{\Sigma} a new complex vector bundle

E^{\boldsymbol{\theta}} := E \otimes W^{\boldsymbol{\theta}} \to \dot{\Sigma}

by defining W^{\boldsymbol{\theta}} \to \dot{\Sigma} to be the flat real vector bundle

W^{\boldsymbol{\theta}} := \left( \dot{\Sigma}'' \times W\right) \Big/ G,

with G acting on \dot{\Sigma}'' via deck transformations and on W via the representation \boldsymbol{\theta}. I am referring to W^{\boldsymbol{\theta}} as “flat” because the trivial connection on \dot{\Sigma}'' \times W \to \dot{\Sigma}'' descends naturally to a flat connection on W^{\boldsymbol{\theta}} \to \dot{\Sigma}. Defining F^{\boldsymbol{\theta}} := \dot{F} \otimes W^{\boldsymbol{\theta}} similarly, our Cauchy-Riemann operator \dot{\mathbf{D}} : \Gamma(\dot{E}) \to \Gamma(\dot{F}) now determines a “twisted” Cauchy-Riemann type operator

\mathbf{D}^{\boldsymbol{\theta}} : \Gamma(E^{\boldsymbol{\theta}}) \to \Gamma(F^{\boldsymbol{\theta}})

by setting \mathbf{D}^{\boldsymbol{\theta}}(\eta \otimes v) := (\dot{\mathbf{D}}\eta) \otimes v for any smooth local sections \eta of \dot{E} and v of W^{\boldsymbol{\theta}} such that v is flat.

The purpose of this construction becomes clearer when we consider the following choice of representation: the homomorphism \rho : G \to S_d determines a permutation representation

\boldsymbol{\rho} : G \to \text{GL}(d,{\mathbb R})

which acts on the standard basis vectors e_i \in {\mathbb R}^d for i=1,\ldots,d by \boldsymbol{\rho}(g) e_i := e_{\rho(g)(i)}. Using the obvious identification of E^{\boldsymbol{\rho}} with (\pi^*\dot{E} \otimes {\mathbb R}^d) / G, we can write global sections \eta \in \Gamma(E^{\boldsymbol{\rho}}) as G-equivariant sections \sum_i \eta^i \otimes e_i of \pi^*\dot{E} \otimes {\mathbb R}^d, where \eta^1,\ldots,\eta^d are now sections of \pi^*\dot{E} \to \dot{\Sigma}''. Denoting the action of G on \dot{\Sigma}'' by (g,z) \mapsto g \cdot z, the equivariance condition means

\eta^i(z) = \eta^{\rho(g)(i)}(g \cdot z)

for every z \in \dot{\Sigma}'', i \in \{1,\ldots,d\} and g \in G. It follows that we can now associate to \eta a section \hat{\eta} of \varphi^*\dot{E}, defined at [(z,i)] \in (\dot{\Sigma}'' \times \{1,\ldots,d\}) / G = \dot{\Sigma}' by

\hat{\eta}([(z,i)]) = \eta^i(z),

and the equivariance condition guarantees that this is well defined. Conversely, any section of \varphi^*\dot{E} written in this way gives rise to sections \eta^1,\ldots,\eta^d of \pi^*\dot{E} that automatically satisfy the equivariance condition, so that \eta = \sum_i \eta^i \otimes e_i forms a well-defined section of E^{\boldsymbol{\rho}}. Doing the same thing with the bundle F^{\boldsymbol{\rho}} and looking again at our Cauchy-Riemann operators, we conclude:

Proposition. The correspondence \sum_i \eta^i \otimes e_i \leftrightarrow \hat{\eta} described above defines natural bijections \Gamma(E^{\boldsymbol{\rho}}) \leftrightarrow \Gamma(\varphi^*\dot{E}) and \Gamma(F^{\boldsymbol{\rho}}) \leftrightarrow \Gamma(\varphi^*\dot{F}) which identify the Cauchy-Riemann type operators \mathbf{D}^{\boldsymbol{\rho}} and \varphi^*\dot{\mathbf{D}}.

Identifying \varphi^*\dot{\mathbf{D}} with \mathbf{D}^{\boldsymbol{\rho}} makes the operator easy to decompose: any splitting of {\mathbb R}^d into G-invariant subspaces produces a splitting of W^{\boldsymbol{\rho}} into subbundles such that \mathbf{D}^{\boldsymbol{\rho}} will respect the resulting splittings of E^{\boldsymbol{\rho}} and F^{\boldsymbol{\rho}}. The restriction of \mathbf{D}^{\boldsymbol{\rho}} to one of these subbundles will then be the same as \mathbf{D}^{\boldsymbol{\theta}} if \boldsymbol{\theta} is the representation defined by restricting \boldsymbol{\rho} to the corresponding invariant subspace.  To put it another way, if we fix a list \boldsymbol{\theta}_1,\ldots,\boldsymbol{\theta}_N of the distinct irreducible real representations of G, then

\boldsymbol{\rho} \cong \boldsymbol{\theta}_1^{\oplus k_1} \oplus \ldots \oplus \boldsymbol{\theta}_N^{\oplus k_N},

for uniquely determined integers k_1,\ldots,k_N \ge 0, and this produces splittings of the bundles E^{\boldsymbol{\rho}} and F^{\boldsymbol{\rho}} such that

\mathbf{D}^{\boldsymbol{\rho}} \cong (\mathbf{D}^{\boldsymbol{\theta}_1})^{\oplus k_1} \oplus \ldots \oplus (\mathbf{D}^{\boldsymbol{\theta}_N})^{\oplus k_N}.

It is often not easy to see directly which subspace of \Gamma(\varphi^*\dot{E}) a given subrepresentation of \boldsymbol{\rho} corresponds to, but the following example should be helpful to keep in mind. Since \boldsymbol{\rho} : G \to \text{GL}(d,{\mathbb R}) acts on {\mathbb R}^d by permuting coordinates, we can single out two subspaces that are always invariant: write

{\mathbb R}^d = W_+ \oplus W_-

where W_+ is the 1-dimensional subspace spanned by (1,\ldots,1), and W_- is its orthogonal complement, consisting of vectors whose coordinates add up to zero. This gives a decomposition \boldsymbol{\rho} = \boldsymbol{\theta}_+ \oplus \boldsymbol{\theta}_-, where \boldsymbol{\theta}_+ is the trivial representation. The subspace of \Gamma(\varphi^*\dot{E}) determined by \boldsymbol{\theta} is the one that we’ve previous called \Gamma_+(\varphi^*\dot{E}) in the degree 2 case: it consists of all sections of the form \eta \circ \varphi for \eta \in \Gamma(\dot{E}). The subspace \Gamma_-(\varphi^*\dot{E}) corresponding to \boldsymbol{\theta}_- consists of all sections \eta with the property that for every \zeta \in \dot{\Sigma},

\displaystyle \sum_{z \in \varphi^{-1}(\zeta)} \eta(z) = 0.

For example, if u = v \circ \varphi is a degree two cover, then \boldsymbol{\rho} : {\mathbb Z}_2 \to \text{GL}(2,{\mathbb R}) is the unique nontrivial permutation representation of {\mathbb Z}_2, whose decomposition into irreducible subrepresentations is simply \boldsymbol{\theta}_+ \oplus \boldsymbol{\theta}_-, producing the familiar decomposition

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

In the general case, this decomposition still exists, but one must expect W_- to admit smaller invariant subspaces, so that \boldsymbol{\theta}_- is reducible and gives rise to a further splitting of \mathbf{D}_u^-. Notice that since G acts transitively on \{1,\ldots,d\}, W_+ is the largest subspace of {\mathbb R}^d on which \boldsymbol{\rho} acts trivially, hence if we list the irreducible representations \mathbf{\theta}_1,\ldots,\mathbf{\theta}_N with the trivial representation first, then the first term in our general splitting of \mathbf{D}^{\boldsymbol{\rho}} always satisfies k_1 = 1 and matches \mathbf{D}^{\boldsymbol{\theta}_+}. Moreover, this operator is equivalent to \dot{\mathbf{D}} under the obvious bundle isomorphism between \dot{E} and E^{\boldsymbol{\theta}_+} = \dot{E} \otimes W^{\boldsymbol{\theta}_+} resulting from the fact that W^{\boldsymbol{\theta}_+} is canonically trivial.

Non-faithful representations

The equivalence of \mathbf{D}^{\boldsymbol{\theta}_+} with \dot{\mathbf{D}} is indicative of a more general phenomenon that is useful to be aware of: for a non-faithful representation \boldsymbol{\theta} : G \to \text{Aut}_{\mathbb R}(W), \mathbf{D}^{\boldsymbol{\theta}} can always be identified with a similar operator that comes from a branched cover of strictly smaller degree that \varphi factors through. The main idea is basically to take your regular presentation of \varphi and divide everything by the kernel of \boldsymbol{\theta}. I’ll refer you to the paper for details, rather than trying to explain them here, but the message is that for any given branched cover u = v \circ \varphi with its splitting of \mathbf{D}_u^N, the truly meaningful terms in the splitting are the ones that come from faithful representations. The rest are not actually telling you something about u, but rather about lower-degree covers of v that u factors through; or, in the case of the trivial representation, about v itself.

The regular case

I’ll conclude with one further observation about the splitting \mathbf{D}^{\boldsymbol{\rho}} \cong \bigoplus_{i=1}^N (\mathbf{D}^{\boldsymbol{\theta}_i})^{\oplus k_i}. Each integer k_i \ge 0 in this expression is the multiplicity with which the representation \boldsymbol{\theta}_i appears in \boldsymbol{\rho}, and while the trivial representation is always present with multiplicity k_1=1, some of the others can be zero in general. There is an important special case in which this cannot happen: if the cover \varphi is regular, then the permutation action of G on \{1,\ldots,d\} is not only transitive but also without fixed points, which means that \boldsymbol{\rho} : G \to \text{GL}(d,{\mathbb R}) is equivalent to the so-called regular representation. By a standard result in representation theory, every irreducible representation of G is a subrepresentation of the regular representation, so the integers k_i in this case are all positive. This has the consequence that everything one needs to know about the twisted operators \mathbf{D}^{\boldsymbol{\theta}_i} can be deduced by considering only regular covers. The proof of the stratification theorem—to be discussed in the next post—makes use of this.

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