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for multiply covered holomorphic curves has just been uploaded to my homepage, and will be replacing the previous (withdrawn) version on the arXiv within the next couple of days.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approach differs from earlier work by other authors… in particular the three or four previous attempted proofs (later withdrawn) that super-rigidity holds for generic J in Calabi-Yau 3-folds.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.