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 1. Suppose is a real-linear Cauchy-Riemann type operator over a Riemann surface , such that the bundle map defined by the complex-antilinear part of is invertible on some open subset . Then satisfies Petri’s condition on .
Recall from the previous post: the words “satisfies Petri’s condition on ” mean that the natural map
is injective, where sends each to the section of restricted to . Put another way, this means that if we fix bases and , then for every nontrivial set of real numbers , the section
is guaranteed to be nonzero somewhere in . 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 and come with complex structures, the operators and are in general not complex linear, so all tensor products in this discussion must be understood to be real tensor products, even if and happen in some cases to be complex vector spaces. That makes the following an example in which Petri’s condition fails: take to be the trivial line bundle over , and with formal adjoint . Then
is a nontrivial element in the kernel of . 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 are power series in while are power series in .
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 or are also complex-linearly independent, so that one might realistically hope for properties of to carry over to . 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 on a Riemann surface and a fixed open subset with compact closure, there is a Baire subset in the space of all smooth linear bundle maps supported in , consisting of perturbations such that satisfies Petri’s condition on .
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 translates into nontrivial pointwise linear dependence relations among some linearly independent local solutions and 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 that vanishes on some open set , then one could pair it with any nontrivial and call 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 and are Fredholm. That sounds good at first, because it seems much more likely for the “Petri map” 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: and do not depend continuously on in a straightforward way, as their dimensions can jump suddenly downward. One cannot therefore just set up some kind of “universal moduli space”
and try to apply the Sard-Smale theorem to the obvious projection because 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 which are equivariant under the action of some finite symmetry group , so that the perturbations are also required to be -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 , nor should we assume that is Fredholm… whatever can be proven should be provable by considering small zeroth-order perturbations of the standard Cauchy-Riemann operator . 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 does vary smoothly with the operator 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 and complex vector bundle , giving rise to the bundle and the affine space of real-linear Cauchy-Riemann type operators , which map to . Fix a point and let denote the vector space of -jets of sections of at . Each then descends to a linear map
for every , and usefully, this map is always surjective. The latter can be deduced from standard local existence results for solutions to the equation , 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 . 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 is equivalent (via choices of local coordinates and trivializations near ) to an arbitrarily small perturbation of the standard operator .
Let denote the space of linear maps that are induced by operators in . Since the -jet of a section depends on the zeroth-order perturbation only up to its -jet, is an affine space over the finite-dimensional vector space . We can now consider the k-jet Petri map
defined by letting the natural map descend to quotient spaces. We will be interested particularly in the restriction of to the subspace for each . There is a trivial reason why this map will never actually be injective: if vanishes to order and vanishes to order with , then their product vanishes to order at least and is thus trivial in . The fancy way to say this is that jet spaces carry natural filtrations,
where we can identify -jets with Taylor polynomials in coordinates to define as the space of Taylor polynomials that are . Under the natural tensor product filtration that inherits from the filtrations of and , the Petri map preserves filtrations and thus vanishes on . This observation motivates considering for each and each with the space
Notice how we’ve just quietly reinserted unique continuation into this discussion. If we can find sequences such that for a given operator , then we’ve proven that the only possible counterexamples to Petri’s condition for are nontrivial elements that vanish to all orders at the point . 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 via a dimension count. My initial naive hope had been that this expected dimension would turn out to be negative, perhaps after choosing sufficiently large, and one could then argue via Sard’s theorem that is empty for almost every . But the expected dimension isn’t negative. In fact, for all choices , and , turns out to be a nonempty open subset in a nontrivial vector space that depends smoothly on . 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 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 means that one can associate to every Cauchy-Riemann operator and integer a sequence of tensor products of sections
such that for each , does not vanish to order at , but and vanish to order and the Petri map takes to a section of that also vanishes to order at . It is very far from obvious whether can be made to converge to something as , 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 may be unbounded. One can rephrase this as follows: given two vector spaces and , say that an element has rank if one can write for two linearly-independent sets and . 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 with a Hilbert space tensor product of local sections. But that is not what we are doing; the sequence described above only has any chance of converging to a counterexample if the rank of stays bounded.
With this in mind, let’s modify our definition of : define for each the space
One should view this as a subset of
which is a smooth submanifold of the vector space , 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 and produces a general formula for the dimension of that grows linearly with . On the other hand, the extra condition cuts out a subset whose expected codimension is the dimension of ; that is the number of distinct Taylor polynomials up to degree in and with values in a fiber of , and it grows quadratically with . As a result, the expected dimension of becomes negative as soon as 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 , there exists such that for all and almost every .
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 , namely
which one might hope should be a smooth submanifold of the manifold
e.g. because the smooth map 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 with respect to changes in has its rank bounded below by some quadratic function of . As a measure of plausibility for this claim, notice that since the space of perturbations is an affine space over , its dimension is also a quadratic function of . The bound on the rank does not prove that is a submanifold, but it does prove that it’s something I like to call a -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 when is large enough, that for almost every , itself is locally contained in submanifolds that have negative dimension, meaning 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 with respect to variations in at an arbitrary point , 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 ; it suffices instead to establish this bound only for the special case , for which it is a bit tedious but not very hard in principle to write down and the linearized map explicitly. Once you’ve done that, the rank bound carries over to an open neighborhood of such pairs in , and since every Cauchy-Riemann operator is (up to choices of coordinates and trivializations) an arbitrarily small perturbation of , 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 and to be the trivial line bundle over and consider the operators
where and 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 and . 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 for functions on a half-cylinder , with an -family of real-linear transformations on , and the equation then has a special solution of the form
whenever is an eigenfunction of the operator with eigenvalue . Now, 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 to be periodic in , which makes arbitrary real numbers possible for the eigenvalue . 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 and for by
If we identify the fibers of and with so that the fibers of become the space of real 2-by-2 matrices, then feeding into the Petri map gives constant sections,
These all take values in the 3-dimensional vector space of matrices of the form , 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 .
Acknowledgement: A substantial proportion of what I understand about the subject of this post emerged from conversations with Aleksander Doan and Thomas Walpuski.