Complex Analytic Structure on Moduli Space of Stable Maps

Question

Suppose $(X,\omega,J)$ is a compact Kähler manifold, and $\beta\in H_2(X,\mathbb Z)$ is given. Then, we can form the space $\overline{\mathcal M}:=\overline{\mathcal M}_{0,0}(X,\beta)$ of stable maps $u:C\to X$ with $C$ a nodal curve of genus 0 and $u_*[C]=\beta$. Let $\mathcal M\subseteq\overline{\mathcal M}$ be the locus where the domain is nonsingular.

I know how to show that $\mathcal M$ is a space with a smooth ($C^\infty$) Kuranishi atlas (i.e., roughly, it locally has the structure of the zero set of a smooth vector valued function on a smooth finite dimensional manifold modulo a finite group). Near the points of $\overline{\mathcal M}\setminus\mathcal M$, I don't know how to put a smooth structure (instead, what I know amounts to only a topological implicit atlas, in the sense of https://arxiv.org/abs/1309.2370).

My question is the following: since I have assumed $X$ actually Kähler (and not simply symplectic with a compatible almost complex structure), can we show that $\mathcal M$ (or even better, $\overline{\mathcal M}$) has a complex analytic Kuranishi atlas? I would prefer approaches which stay within the realm of differential geometry (though for the case of projective $X$, I would be interested in hearing about algebro-geometric approaches as well). Any references would also be appreciated.

Answer 1

There's also a really beautiful approach by Salamon-Robbin-Ruan for integrable Js in their paper "The moduli space of regular stable maps"

https://people.math.ethz.ch/~salamon/PREPRINTS/smUW.pdf

building on the earlier paper by Salamon-Robbin "Construction of the Deligne-Mumford orbifold":

https://people.math.ethz.ch/~salamon/PREPRINTS/dmETH.pdf

They gives a smooth structure that extends across the boundary in an extremely natural way (I don't remember if they explicitly construct an analytic structure, but because they are working in the case of integrable J, I think it should be possible to extract it from their proof).

The idea is the following.

Consider the space of holomorphic maps $S^2\to X$. Since a holomorphic map on the sphere is determined by its values on the equator, you can think of this as a submanifold of the space of maps from the circle to $X$. In fact, it's precisely the intersection of the (infinite-dimensional) subspace of maps $S^1\to X$ which extend over the northern hemisphere with the (infinite-dimensional) subspace of maps which extend over the southern hemisphere (this is the kind of thing people mean when they say that Floer theory is the theory of intersections of semi-infinite cycles).

Now when your holomorphic map develops a bubble, this bubble will (generically) avoid the equator. So if you think of your moduli space as a space of maps $S^1\to X$ then you don't even notice bubbling.

I think their original hope was to use this for nonintegrable Js as well, and overcome all issues with smooth structures at the boundary of moduli spaces. The problem, I believe, is that they need to use Fourier theory and Hardy spaces to set things up in a way they can handle analytically. Why does Fourier theory show up? A simple heuristic: a map from the circle to $\mathbb{C}$ can be thought of as a power series in $z$ and $z^{-1}$ infinite in both directions; those which extend over the unit disc are those where the negative powers of $z$ are absent; those which extend over the north hemisphere of the Riemann sphere are those where the positive powers of $z$ are absent; the only holomorphic maps from $S^2$ to $\mathbb{C}$ are therefore the constants. The coefficients of $z^n$ can be thought of as Fourier coefficients. This kind of description of holomorphic maps as power series only makes sense in the integrable setting.

Answer 2

You can repeat the usual construction of a groupoid presentation of the Deligne-Mumford stack $\overline{\mathcal{M}}_{g,n}(X)$ from algebraic geometry, but instead using Douady spaces of compact complex analtyic spaces instead of Hilbert schemes of complex projective schemes.

Let $(X,J)$ be a compact, complex analytic space. Let $(C,p_1,\dots,p_n)$ be a projective, reduced, connected curve $C$ that is at worst nodal, and let $p_1,\dots,p_n$ be pairwise distinct, smooth points of $C$. For every embedding of $C$ into projective space such that the pullback of $\mathcal{O}(1)$ is sufficiently ample, an appropriate "slice" of the Hilbert scheme gives a versal deformation space of $(C,(p_1,\dots,p_n))$. This is a datum $$ \xi = (\pi:\mathcal{C}\to B, (\sigma_i:B\to \mathcal{C})_{i=1,\dots,n}), $$ of a projective, flat morphism $\pi$ of quasi-projective varieties -- both $\mathcal{C}$ and $B$ are smooth, as it turns out -- whose geometric fibers are reduced, connected, at-worst-nodal curves, together with an ordered $n$-tuple of sections $\sigma_i$ of $\pi$ whose images are disjoint and contained in the smooth / submersive locus of $\pi$. Moreover, $(C,p_1,\dots,p_n)$ is the fiber of the family over some $b\in B$, and the family is versal, i.e., for every $b'\in B$ with fiber $(C',p'_1,\dots,p'_n)$, the following Kodaira-Spencer map is surjective, $$ \kappa_{b'}:T_{b'} B \to \text{Ext}^1_{\mathcal{O}_{C'}}(\Omega_{C'}(\underline{p}'_1+\dots+\underline{p}'_n),\mathcal{O}_{C'}). $$ Since $B$ is quasi-projective, up to slicing $B$ by appropriate hyperplane sections, we can even assume that this family is miniversal at $b$, i.e., $\kappa_b$ is an isomorphism. Denote by $\omega_{\pi,\sigma}$ the invertible sheaf on $\mathcal{C}$, $$ \omega_{\pi,\sigma} := \omega_{\pi}\left(\sum_{i=1}^n \underline{\text{Image}(\sigma_i)}\right). $$

Form the product $\mathcal{C}\times X$ with the projection to $B$, $$ \mathcal{C}\times X \xrightarrow{\text{pr}_1} \mathcal{C} \xrightarrow{\pi} B. $$ This is a proper, flat morphism of complex analytic spaces. There is a relative Douady space of this morphism, i.e., a morphism of complex analytic spaces $$ \rho:D \to B, $$ and a closed analytic subspace of the fiber product, $$\iota:\mathcal{Z}\hookrightarrow D\times_B (\mathcal{C}\times X),$$ that is flat over $D$ and that is universal among pairs $(\rho,\iota)$ as above. The graph of $u$ gives a point in the fiber of $\rho$ over $b$. Near this point, the projection morphism, $$ \mathcal{Z}\xrightarrow{\iota} D\times_B \mathcal{C}\times X \xrightarrow{\text{pr}_{1,2}} D\times_B \mathcal{C}, $$ is an isomorphism. For complex analytic spaces over $D$ that are flat, for a $D$-morphism of such complex analytic spaces, the property of being a local isomorphism is an open condition on the domain of the morphism. The closed complement is proper over $D$, so the image in $D$ is a closed analytic subspace $W$ of $D$. By hypothesis, $[u]$ is in the open complement $V = D\setminus W$. Restricting over $V$, the projection morphism, $$ \mathcal{Z}_V\to V\times_B \mathcal{C}, $$ is proper and a local isomorphism, i.e., it is a finite covering map. The degree of a covering map is constant on connected components. Thus, there is an open and closed subset $V_1$ of $V$ on which the degree of the covering map equals $1$, i.e., the covering map is an isomorphism of complex analytic spaces. Thus, $V_1$ is precisely the relative Hom complex analytic space, i.e., the restriction of $\mathcal{Z}$ over $V_1$ is the graph of a universal morphism, $$ \phi:V_1\times_B \mathcal{C} \to X. $$
Denote the base change over $V_1$ of $\pi$, resp. of $\sigma_i$, by $$ \pi_{V_1} : \mathcal{C}_{V_1} \to V_1, \ \ \sigma_{V_1,i}:V_1\to \mathcal{C}_{V_1}, \ \ \mathcal{C}_{V_1} = V_1\times_B \mathcal{C}. $$ The datum $$ (\pi_{V_1}:\mathcal{C}_{V_1}\to V_1, (\sigma_{V_1,i}:V\to \mathcal{C}_{V_1})_{i=1,\dots,n}, \phi:\mathcal{C}_{V_1} \to X), $$ is a family of prestable maps. Finally, for the pullback $\omega_{V_1,\pi,\sigma}$ of $\omega_{\pi,\sigma}$ to $\mathcal{C}_{V_1}$, there is a maximal open subset $M$ of $V_1$ over which $\omega_{V_1,\pi,\sigma}$ is ample relative to the morphism $$ (\pi_{V_1},\phi):\mathcal{C}_{V_1} \to V_1\times X. $$
Concretely, the union of all irreducible curves in fibers of $(\pi_{V_1},\phi)$ on which $\omega_{V_1,\pi,\sigma}$ has nonpositive degree is a closed analytic subset of $\mathcal{C}_{V_1}$ whose image in $V_1$ is also a closed analytic subset (since $\pi_{V_1}$ is proper), and $M$ is the open complement of this closed analytic subset.

The restriction of the family over $M$, $$ \zeta_M = (\pi_M:\mathcal{C}_M\to M,(\sigma_{M,i}:M\to \mathcal{C}_M)_{i=1,\dots,n}, \phi_M:\mathcal{C}_M \to X), $$ is a family of stable maps to $X$. By openness of ampleness, the open subset $M$ of $V$ contains the point $[u]$.

Since the family $(\pi,(\sigma_i))$ is miniversal at $b$, and by universality of $D$, the family $\zeta_M$ is a local analytic chart of the complex analytic stack $\overline{\mathcal{M}}_{g,n}(X)$ at $[u]$. For any versal family, $$ \widetilde{\xi}= (\widetilde{\pi}:\widetilde{\mathcal{C}}\to \widetilde{B}, (\widetilde{\sigma}_i:\widetilde{B}\to \widetilde{\mathcal{C}})_{i=1\dots,n}), $$ that is miniversal at the corresponding point $\widetilde{b}$ of $\widetilde{B}$, for the family $\widetilde{Z}_{\widetilde{M}}$ over $\widetilde{M}$ constructed as above, for the two pullback families over the product $M\times \widetilde{M}$, i.e., $\text{pr}_M^*\zeta_M$ and $\text{pr}_{\widetilde{M}}^* \widetilde{\zeta}_{\widetilde{M}}$, the relative Isom of the two families, $$ I_{\zeta,\widetilde{\zeta}}=\text{Isom}_{M\times \widetilde{M}}(\text{pr}_M^*\zeta_M,\text{pr}_{\widetilde{M}}^* \widetilde{\zeta}_{\widetilde{M}}), $$ is a Hom complex analytic space constructed by the same method as above (beginning with Douady spaces). Since the two families are families of stable maps and $\xi$, $\widetilde{\xi}$ are miniversal, also the two projections, $$ q:I_{\zeta,\widetilde{\zeta}}\to M, \ \ \widetilde{q}:I\to \widetilde{M} $$ are both local isomorphisms. Altogether, the family of "charts" $(M,\zeta)$ and the family of Isoms $I_{\zeta,\widetilde{\zeta}}$ form a groupoid presentation for a complex analytic (Deligne-Mumford) stack $\overline{\mathcal{M}}_{g,n}(X)$ in the category of complex analytic spaces.

When $X$ happens to be projective, the connected components of $D$ are each projective over $B$, and $B$ is quasi-projective, so that also $M$ is quasi-projective. Moreover, every Isom is also quasi-projective. Thus, when $X$ is projective, this even gives a groupoid presentation in the category of complex algebraic spaces.