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not the first time in history that an attempted proof of the super-rigidity conjecture for holomorphic curves has been withdrawn—in fact, it is not even the first time in history that such a proof has been withdrawn (or at least drastically downgraded) by *me*.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 and are real Euclidean vector bundles over a manifold with a fixed volume form, is a real-linear partial differential operator and denotes its formal adjoint. We say that satisfies

**Petri’s condition**on some region if the natural map

defined by 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 -holomorphic curves whose normal Cauchy-Riemann operators all satisfy Petri’s condition on some open subset where 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 on a trivial line bundle over the disk, one can find complex-linearly independent sections and such that the section

is identically zero. Evidently, Petri’s condition can only hold for *real* Cauchy-Riemann type operators, and only under some condition preventing and 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 defined as the complex-antilinear part of to be invertible at some point , a condition that holds for all normal Cauchy-Riemann operators of -holomorphic curves if is generic. When that condition holds, it forces and to be totally real subspaces of and 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 are power series in while sections in are power series in . 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 and 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 is a complex-linear Cauchy-Riemann type operator on a vector bundle over a Riemann surface , and is a point. Then there exists a generic condition on the -jet of a complex-antilinear bundle map at such that for all satisfying this condition, the operator satisfies Petri’s condition on neighborhoods of .*

## 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:

- 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. - 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.
- 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.
- 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.
- There is also a weaker condition that does hold for all Cauchy-Riemann type operators: say that satisfies Petri’s condition “up to rank ” on a region if for all subspaces and of dimensions at most , the natural map 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.