Overview
Chennai Mathematical Institute, August 12 2020.
Revised e-board notes.
Online due to COVID-19, recorded on
YouTube.
(Caution: the talk is not to be taken unfailingly
literally, I often employ imprecise analogy and vague expressions
to convey ideas. I might also make mindless mistakes, which I’ll
be happy to hear about.)
Abstract.
Moduli spaces of simple (or somewhere injective in the closed
setting) pseudoholomorphic curves arise as smooth manifolds. In
this talk, we present some ideas for the multiply covered case.
That is the question of equivariant transversality – do generic
equivariant sections of an orbi-bundle intersect the zero section
transversally? In particular, we outline results like unbranched
multiple covers of closed curves are generically regular and
simple index 0 curves in dimension greater than four are
generically super-rigid. We also indicate some partial results for
regularity of branched covers, using a stratification argument for
spaces of multiple covers, framed in terms of a
representation-theoretic splitting of Cauchy Riemann operators
with symmetries.
This talk is based on
Wendl’s
paper “Transversality and super-rigidity for multiply covered
holomorphic curves” (arXiv:1609.09867).
Relevant background on pseudoholomorphic curves
We first present some background on pseudoholomorphic curves (details can be found in [Wen2] or [MS]) needed in the following discussion. For \((\Sigma, j)\) a closed Riemann surface and \((M,J)\) an almost complex manifold, a \(J\) -holomorphic or pseudoholomorphic (or often just holomorphic) curve \(u : (\Sigma, j) \to (M,J)\) is a smooth map satisfying \(\d u \circ j = J \circ \d u\). Such curves behave analogous to holomorphic curves in complex analysis, and have been widely studied ever since Gromov used them in 1985 to prove rigidity phenomena in symplectic topology (the most notable example being the non-squeezing theorem).
It is useful to take a global perspective and study the entire space of holomorphic curves in a manifold. Such a viewpoint is useful in topology since we are often only concerned with maps upto perturbations and the study of entire families often gives us information about the curves in that family. In this respect, let \(\mathcal{B} = \cinf(\Sigma, M)\) be the space of smooth maps from \(\Sigma\) to \(M\) and for a map \(u \in \mathcal{B}\), consider the space of \(u^*TM\) valued \((0,1)\) forms \(\mathcal{E}_u = \Omega^{0,1}(\Sigma, u^* TM)\). These vector spaces together form an “infinite-dimensional vector bundle” \(\mathcal{E} \to \mathcal{B}\). Using the definition of holomorphic curves, we arrive at a particular section \(\delbar_J(u) = 1/2 (\d u + J \circ \d u \circ j)\) of the bundle \(\mathcal{E} \to \mathcal{B}\). Under this formulation, holomorphic curves are precisely the zero set of this section, i.e., solutions to \(\delbar_J(u) = 0\) (this is the nonlinear Cauchy-Riemann equation).
It turns out that under suitable hypotheses, the space of such curves is indeed a smooth manifold. In finite dimensional differential geometry, we have that level sets of regular values for a smooth map between manifolds is a regular submanifold of the domain. Trying to imitate that in the global perspective described earlier, we find that \(0\) is in some sense a regular value for the \(\delbar_J\) operator (more formally, \(\delbar_J\) is a Fredholm section with respect to suitable Sobolev completions of \(\mathcal{E}\) and \(\mathcal{B}\)).
To outline the hypotheses, we introduce the notion of simple curves. Recall from complex analysis, that any holomorphic map between closed Riemann surfaces is either a biholomorphism or a cover of some kind. Similarly, a JH curve \(u : \Sigma \to M\) is multiply covered if there is a compact RS \((\Sigma', j')\), a JH curve \(v : \Sigma' \to M\) and a holomorphic branched covering \(\varphi : \Sigma \to \Sigma'\) such that \(u = v \circ \varphi\). The curve is simple if its not mulitply covered, the analogue of biholomorphisms from complex analysis. In the closed setting, these curves are the same as somewhere injective curves (curves \(u\) which are injective somewhere, that is, \(\exists z\) such that \(\d u(z) \ne 0\) and \(u^{-1}(u(z)) = \{z\}\)). It turns out that any holomorphic curve can be shown to factor through a simple curve.
Now, let \((M,\w)\) be a symplectic manifold. Then for a generic \(\w\) compatible almost complex structure \(J\) (that is to say, for \(J\) in a comeager subset of the space of all \(\w\) compatible acs), the space of simple JH curves representing a particular homology \(A \in H_2(M)\) is a smooth manifold of dimension given by an index formula (namely, the Riemann Roch fomula).
Now that we are done with the story of simple curves, a natural question which arises is what happens for multiple covers? Can we make similar statements about the space of multiple covers of a particular type?
The case of multiple covers
We begin with a discussion of unbranched covers. If \(u : (\Sigma, j) \to (M,J)\) is a closed JH curve and \(\varphi : (\Tilde{\Sigma}, \Tilde{j}) \to (\Sigma, j)\) is an unbranched \(d:1\) cover of closed surfaces, then the virtual dimensions of the moduli spaces containing \(u\) and \(u \circ \varphi : (\Tilde{\Sigma}, \Tilde{j}) \to (M,J)\), also known as the indices of these two curves (Achtung! The latter index would be the “expected” dimension of the space of all curves (simple or not) close to \(u \circ \varphi\), not only the \(d\) fold covers, as mentioned in the talk. Thanks to Ramadas for the question!), are related by \[ \ind(u \circ \varphi) = d \cdot \ind(u) \ge 0. \] Since \(\ind(u \circ \varphi) \ge 0\) if \(u\) were simple (and \(J\) regular, etc), there is no obvious reason why \(u \circ \varphi\) could not achieve transversality generically, but traditional methods in the theory for simple curves doesn’t generalize to this situation (the proof for simple curves uses the existence of an injective point to prove surjectivity of an operator). For branched covers, things are more subtle.
One of the ways to study the problem of multiple covers is to study the automorphism group of the cover itself. This leads us to the question of how symmetry interacts with transversality in general. We begin with an example in the finite dimensions.
Consider a smooth map \(f : \Z^2 \to \Z^2\) which is \(\Z_2\) equivariant in the sense \(f(x,-y) = -f(x,y)\). One can show that for every such map, there are no close \(\Z_2\) equivariant perturbations for which \(\vec{0}\) is a regular value (Achtung! Contrary to what was said in the talk, the method to prove that generic functions on a smooth manifold are Morse, fails to work in this case). This happens because the entire \(x\) axis \(\R \x \{0\} \In f^{-1}(0) \forall\) such \(f\), so \(\del_x f = \vec{0}\) along the \(x\) axis. As a result we donot have transversality in this case.
The best case scenario is that generic \(f\) intersect zero cleanly, i.e., all components of \(f^{-1}(0)\) are submanifolds with \[ T_x(f^{-1}(0)) = \ker{\d f(x)}. \]
The notion of clean intersection is the only other thing one could come up with for an intersection set to be a manifold other than transversality. Two submanifolds \(M, N \In X\) are said to intersect cleanly if \(T_x(M \cap N) = T_x M \cap T_x N\), in such a case, we again have that \(M \cap N\) is a smooth submanifold of both \(M\) and \(N\). The moral is that we cannot always have transversality and symmetry at the same time.
Because we took the global perspective of the space of holomorphic functions being the zero set of the \(\delbar_J\) section, a lot of the theory of JH curves is really about understanding bundles and sections of those bundles, when they intersect the zero section transversally or cleanly, etc. Let’s set up this language first in finite dimensions.
Equivariant transversality in finite dimensions
Fix an \(n\) dimensional orbifold \(M\) and an orbibundle \(E \to M\) of rank \(m\). Every \(x \in M\) has a finite group \(G_x\) and a neighborhood \(U_x \In M\) such that \[ E|_{U_x} \isom (\mathcal{O} \x \R^m)/G_x \]
for some linear action of \(G_x\) on \(\R^m\) and a neighborhood \(\mathcal{O} \In \R^n\) of \(0\).
The question to ask is if generic \(\sigma \in \Gamma(E)\) intersect the zero-section transversely (or at least cleanly). There are some theorems answering this question in some contexts:
- Sample theorem 1: If \(\dim{M} = \rk{E}\) and \(|G_x| \le 3\) for all \(x\), then generic sections of \(E\) intersect zero cleanly.
- Sample theorem 2: Generic smooth functions on an orbifold are Morse. (cf. Wasserman ’69, Hepworth ’09).
The perspective we shall take towards considering equivariant sections (i.e., sections of the orbibundle) is to break up our manifold into parts where the local isotropy groups act in the same way. This is the notion of stratification.
Ingredient A: Stratification via symmetry
For any finite group \(G\) and representations \(\rho : G \to
GL(n,\R)\), \(\tau : G \to GL(m,\R)\), define the
submanifold \[ M_{\rho, \tau} = \{x \in M \st G_x \isom
G, \text{ acting on} \\ T_x M \text{ as } \rho \text{ and on } E_x
\text{ as } \tau\} \]
and subbundle \[ E_{\rho, \tau} = \{v \in E|_{M_{\rho, \tau}} \st $G$ \text{ acts trivially on } $v$\}. \] Let \(\{\theta_i : G \to \Aut_{\R}(W_i)\}_{i=1}^N\) denote the real irreducible representations of \(G\), with \(\theta_1\) as the trivial representation, and let \(m_i(\rho) :=\) multiplicity of \(\theta_i\) in \(\rho\). Then \[ \dim(M_{\rho, \tau}) = m_1(\rho), \qquad \rk{E_{\rho, \tau}} = m_1(\tau). \] The orbifold \(M\) is thus a countable union of disjoint smooth submanifolds \(M_{\rho, \tau}\) with distinguished subbundles \(E_{\rho, \tau} \In E|_{M_{\rho, \tau}}\).
Notice: For all \(\sigma \in \Gamma(E), \sigma(M_{\rho,\tau}) \In E_{\rho, \tau}\), \[ \implies \sigma \not\pitchfork 0 \text{ at } x \in M_{\rho,\tau} \text{ unless } \tau \text{ is trivial}. \]
We now have:
Lemma (standard transversality arguments). For
generic \(\sigma \in \Gamma(E)\), \(\sigma|_{M_{\rho,
\tau}}\) is transverse to the zero section of
\(E_{\rho,\tau} \to M_{\rho,\tau}\) for every \(G, \rho, \tau\).
So we have partial transversality in the above sense. We still however, cannot conclude from this that \(\mathcal{M}(\sigma)\) is anything as nice as a smooth orbifold. Under certain circumstances, we can still push this to say that the space is “nice enough”.
Ingredient B: Splitting the linearization
At \(x \in \mathcal{M}_{\rho, \tau}(\sigma)\), there is a
linearization \[ D_x := D\sigma(x) : T_x M \to E_x. \] Recall
the irreps \(\{\theta_i : G_x \to \Aut_{\R}(W_i)\}_{i=1}^N\),
and denote \(d_i := \dim{W_i}\).
Since \(D_x\) is \(G_x\) equivariant, Schur’s lemma implies that it splits with respect to the isotypic decompositions \(T_x M = \oplus_{i=1}^N T_x M^i\) of \(\rho\) and \(E_x = \oplus_{i=1}^N E_x^i\) of \(\tau\), giving \[ D_x = D_x^1 \oplus \dots \oplus D_x^N, \quad \text{where} \quad D_x^i : T_x M^i \to E_x^i. \] These operators have Fredholm indices \[ \ind{D_x^i} = d_i[m_i(\rho) - m_i(\tau)], \] and we know \(D_x^1\) is surjective if \(\sigma\) is generic.
And finally we have the last ingredient:
Ingredient C:
Building walls (in the sense of “crossing”)
\(G_x\) acts on \(\ker{D_x^i}\) and \(\coker{D_x^i}\) as the
irrep \(\theta_i\) with some multiplicities, so their dimensions
are divisible by \(d_i\).
For non-negative integers \(\mathbf{k} = (k_1, \dots, k_N)\) and
\(\mathbf{c} = (c_1, \dots, c_N)\), let \[ \mathcal{M}_{\rho,
\tau}(\sigma; \mathbf{k}, \mathbf{c}) = \{x \in
\mathcal{M}_{\rho,\tau}(\sigma) \st \dim{\ker{D_x^i}} = d_i k_i,
\dim{\coker{D_x^i}} = d_i c_i \forall i\}. \] Then we have
Workhorse theorem. For generic \(\sigma \in \Gamma(E)\),
for all choices \(G, \rho, \tau, \mathbf{k}, \mathbf{c}\), \[
\mathcal{M}_{\rho, \tau}(\sigma; \mathbf{k}, \mathbf{c}) \In
\mathcal{M}_{\rho,\tau}(\sigma) \] is a smooth submanifold with
codimension \(\sum_{i=1}^N t_i k_i c_i\), where \(t_i :=
\dim_{\R}\End_{G}(W_i) \in \{1,2,4\}\).
A sketch of the proof of this workhorse theorem can be found in [Wen3].
Note that the above is an abstract theory. Considering multiply covered JH maps amounts to considering equivariance of \(\delbar_J\) section of \(\mathcal{E} \to \mathcal{B}\) under the action of the Deck groups of multiple covers. The problem of equivariant transversality arises as perturbing \(J\) perturbs \(\delbar_J\) equivariantly.
The main results and stratification theory
Theorem B (transversality, unbranched). “For generic \(J\),
for all simple JH curves \(u : (\Sigma,j) \to (M,J)\) and every
unbranched cover \(\varphi : (\Tilde{\Sigma}, \Tilde{j}) \to
(\Sigma, j)\) of closed RS, the curve \(u \circ \varphi\) is
Fredholm regular”.
Achtung! Note that Fredholm regularity of
\(u \circ \varphi\) means that \(u \circ \varphi\) lies in a
smooth manifold, which is the moduli space of all (simple
or not) curves, not only of covers. What was said in the
talk is correct, but unrelated (in a different ambient space).
In the general branched case, it is harder to achieve transversality as mentioned earlier. Let us consider \(u \circ \varphi\), where \(u\) is a closed simple curve and \(\varphi\) is a degree \(d\) branched cover. Let \(Z(\d \varphi)\) denote the algebraic count of branch points. Then, the Riemann Hurwitz formula \[ -\chi(\Tilde{\Sigma}) + d \chi(\Tilde{\Sigma}) = Z(\d \varphi) \] and the standard index formula for closed curves, give us \[ \ind(u \circ \varphi) = d \cdot \ind(u) -(n-3) Z(\d \varphi) \] where \(\dim_{\R}(M) = 2n\). If \(\ind(u) = 0\) and \(n > 3\), then \(\ind(u \circ \varphi) < 0\). So, there is no reason to expect transversality in this case. More seriously, if \(u\) is regular, then one can always find a smooth family of other multiple covers near \(u \circ \varphi\) obtained by varying both \(u\) and \(\varphi\) in their respective moduli spaces; since the latter lives in a space of real dimension \(2Z(\d \varphi)\), the condition \[ \ind(u \circ \varphi) \ge \ind(u) + 2Z(\d \varphi) \] is evidently necessary in order for \(u \circ \varphi\) to be Fredholm regular. One of the ways to achieve this is if \(\varphi\) has \(r \ge 0\) critical values, then the above condition is satisfied whenever \(\ind(u) \ge (n-1)r\) (refer [Wen1] pg 6 for a calculation). The following result says that this is also, in some sense, sufficient to achieve transversality.
Theorem C (transversality, branched). “For generic \(J\), branched covers satisfying the above condition can be \(\cinf\) approximated by Fredholm regular branched covers.” (Refer [Wen1] for the precise statement.)
Super-rigidity for index 0 immersed curves
Note that \(\ind(u) = \ind(D_u)\) that is, the index of the linearized CR operator. Observe that \(u\) must be immersed if \(J\) is generic, so it has a well-defined normal bundle \(N_u \to \Sigma\), and restricting the linearized CR operators for \(u\) and \(u \circ \varphi\) to the normal bundle and its pullback gives operators \(D^N_u := D_u|_{N_u}\) and \(D^N_{u \circ \varphi}\). Consider these normal operators, we have a new index relation similar to the one earlier, namely \[ \ind(D^N_{u \circ \varphi}) = d \cdot \ind(D^N_u) - (n-1)Z(\d \varphi). \] Note that for immersed curves, Fredholm regularity can also be characterized by surjectivity of the normal CR operator (see Prop. 2.2 of [Wen1]). If \(\ind(u) = 0\), then the latter is always nonpositive, so \(D^N_{u \circ \varphi}\) can be injective, and this condition has a geometric meaning (remember that sections of \(N_u\) can be considered as peturbations one can make to \(u\); alternatively, curves nearby \(u\) are perturbed by the sections of \(N_u\) via the exponential map): it implies that \(u \circ \varphi\) can never be the limit of a sequence of somewhere injective curves. In fact, the only other curves near \(u \circ \varphi\) are other branched covers of the form \(u \circ \varphi'\) for \(\varphi'\) near \(\varphi\). This is the notion of super-rigidity. We now state the theorem:
Theorem A (super-rigidity). If \(\dim{M} \ge 6\), for generic \(J\), every simple index 0 JH curve is super-rigid. (i.e., a closed, connected simple curve, which satisfies the properties:
- \(\ind(u) = 0\).
- \(u : \Sigma \to M\) immersion.
- \(\forall\) covers \(\Tilde{u} = u \circ \varphi\), \(D^N_{\Tilde{u}} \eta\) is injective.)
Considerable interest in super-rigidity has been motivated by the study of Gromov-Witten invariants in Calabi-Yau 3-folds, where all moduli spaces of holomorphic curves have virtual dimension 0. If \((M,J)\) denotes a CY 3-fold (i.e., \(c_1(TM) = 0\) and \(\dim{M} = 6\)) and \(\mathcal{M}_g(A,J)\) is the moduli space of unparametrized JH curves in \(M\) of genus \(g\) homologous to \(A \in H_2(M)\), then we have \(\virdim{\mathcal{M}_g(A,J)} = \ind(D(\delbar_J)) = 0\). We see trouble however once we consider mulitply covered curves. If \(v \in \mathcal{M}_h(A,J)\) and \(d \ge 2\), then \[ \mathcal{M}_g(dA, J) \supset \{u = v \circ \varphi \st \varphi : \Sigma_g \to \Sigma_h \text{ a holomorphic branched cover}\} \] has dimension \(2 Z(\d \varphi) > 0\) in general. But this contradicts transversality in the space of all (not only simple) curves as \(\ind(D(\delbar_J)) = 0\). This must mean \(\delbar_J \not\pitchfork 0\) for the full space of curves. This motivates the question if the intersection is atleast clean.
Conjecture (“super-rigidity”, Bryan-Pandharipande 2001). For generic compatible \(J\) in a symplectic CY 3-fold, \(\delbar_J\) intersects \(0\) cleanly.
Note that clean intersection of \(\delbar_J\) with \(0\) is equivalent to the definition of super-rigidity given above, in particular, the condition that \(D^N_{u \circ \varphi}\) is injective for all covers \(u \circ \varphi\) of \(u\).
The stratification theorem
The proofs of the above theorems use stratification (Theorem D). The idea is to study the local structure of spaces of the form \[ \mathcal{M}(k,c) := \{\Tilde{u} = u \circ \varphi \st \dim{\ker{D^N_{\Tilde{u}}}} = k \text{ and } \dim{\coker{D^N_{\Tilde{u}}}} = c\}, \] where \(k,c \ge 0\) are fixed integers, \(u\) varies in the moduli space of simple JH holomorphic curves and \(\varphi\) varies in the moduli space of holomorphic branched covers. Ideally, one would like to show that that these spaces are smooth manifolds for generic \(J\), and to compute their codimensions in the space of pairs \((u,\varphi)\). This turns Theorems A and B into “dimension counting” problems, as whenever one can show that the codimension of \(\mathcal{M}(k,c)\) is larger than the dimension of the ambient space for suitable values of \(k\) and \(c\), one may conclude that either \(\ker{D^N_{\Tilde{u}}}\) or \(\coker{D^N_{\Tilde{u}}}\) must be trivial. For a further discussion of wall-crossing arguments, refer [Wen1]. Note that this is a bootstrapping of the ideas of finite dimensional equivariant transversality mentioned earlier on top of equivariance of \(\delbar_J\) for multiple pseudoholomorphic covers. We now state the stratification result:
Theorem D (stratification). “For generic \(J\), and for all choices of finite group \(G\), branching data \(\mathbf{b}\), \(g,m \ge 0\), \(l_1, \dots, l_m \ge 1\), homology class \(A \in H_2(M)\) and tuples \(\mathbf{k} = (k_1, \dots, k_p)\) and \(\mathbf{c} = (c_1, \dots, c_p)\) of nonnegative integers corresponding to the \(p\) distinct irreducible real representations of \(G\), the subset \[ \mathcal{M}^d_{\mathbf{b},G}(\mathcal{M}_{g,m}(A,J; l_1, \dots, l_m); \mathbf{k}, \mathbf{c}) \In \mathcal{M}^d_{\mathbf{b}, G}(\mathcal{M}_{g,m}(A,J; l_1, \dots, l_m)) \] is a smooth submanifold of codimension \(\sum t_i k_i c_i\), where \(t_i := \dim_{\R}\mathbf{K}_i \in \{1,2,4\}\) depending on the type of the irreducible representations.”
The details on stratification can be found in [Wen1].
References
- [Wen1]: Transversality and super-rigidity for multiply covered holomorphic curves, Chris Wendl, arXiv:1609.09867.
- [Wen2]: Lectures on Holomorphic Curves in Symplectic and Contact Geometry, Chris Wendl, available on his website.
- [Wen3]: Slides of a 2018 talk “Symmetry and Transversality” by Chris Wendl at Amsterdam, available on his website.
- [MS]: J-holomorphic Curves and Symplectic Topology, Dusa McDuff & Dietmar Salamon (2012, AMS Colloquium Publications).