Pardon Gromov topology on stable J-holomorphic map moduli

Fix a compact smooth symplectic manifold $(X,\omega )$, a smooth $\omega$-tame almost-complex structure $J$, integers $g,n\ge 0$ with $2g-2+n>0$, and $\beta \in H_2(X;\mathbb{Z})$. A stable $J$-holomorphic map of type $(g,n,\beta )$ is an isomorphism class of data $(C,p_1,\dots ,p_n,u)$ where $C$ is a compact connected nodal Riemann surface of arithmetic genus $g$, the ordered marked points are distinct smooth points of $C$, and the maps on the normalization components are smooth and satisfy $du\circ j=J\circ du$. The branch values at every node agree, so the component maps define a continuous map $u:C\to X$. Require $u_{\ast }[C]=\beta$ and require the automorphism group of $(C,p_1,\dots ,p_n,u)$ to be finite. Isomorphisms are biholomorphisms of the nodal domains preserving the ordered markings and intertwining the maps.

Write ${\bar{\mathcal{M}}}_{g,n}$ for the proper moduli space of stable pointed curves and ${\mathcal{C}}_{g,n+\mathrm{\ell }}\to {\bar{\mathcal{M}}}_{g,n+\mathrm{\ell }}$ for its universal curve. To define the Gromov topology near $(C,\mathbf{p},u)$, choose finitely many additional distinct smooth points $\mathbf{q}=(q_1,\dots ,q_{\mathrm{\ell }})$ so that $(C,\mathbf{p},\mathbf{q})$ is a stable pointed curve, use a local uniformizing chart of ${\bar{\mathcal{M}}}_{g,n+\mathrm{\ell }}$ around that pointed curve, and form the graph ${\mathrm{\Gamma }}_{u,\mathbf{q}}=\{(u(p),p):p\in C\}$ in $X\times {\mathcal{C}}_{g,n+\mathrm{\ell }}$. A neighborhood basis is obtained by requiring graphs of nearby stable maps, represented after forgetting the extra markings, to be close to ${\mathrm{\Gamma }}_{u,\mathbf{q}}$ in the Hausdorff metric induced by any compatible metric on this compact product. Different auxiliary points and local uniformizing charts give the same topology. This is the topology used here; it is not the fixed-domain $C^{\mathrm{\infty }}$ topology on maps.

Symplectic and almost-complex curve data

Stable n-pointed algebraic curves and the moduli functor

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