The exponential decay of gluing maps for \(J\)-holomorphic map moduli space (Q2423254)

Exponential decay of the derivative of the gluing maps with respect to the gluing parameter is proved (Theorem 1.1, Corollary 1.2 and Theorem 5.2, Theorem 5.3). The authors say that based on these estimates, it shown that the Gromov-Witten invariants can be defined integral over top strata of virtual neighborhood, and prove such invariants satisfy all the Gromov-Witten axioms of Kontsevich and Manin [the authors, ``Virtual neighborhood technique for holomorphic curve moduli spaces'', Preprint, \url{arXiv:1710.10581}]. To state the main results, estimates of the linearized operators \(D_u\) and \(D\mathcal{S}\), we review the definitions of these operators described in \S1. This paper treats a marked nodal Riemann surface \(\Sigma\) of genus \(g\), with marked points \(\mathbf{y}=(y_1,\ldots,y_n)\) and nodal point \(q\) such that \(\Sigma=\Sigma_1\wedge\Sigma_2\), \(\mathbf{y}=(\mathbf{y}_1,\mathbf{y}_2), q=(q_1,q_2)\), where \(\Sigma_i\), \(i=1,2\), are Riemann surfaces of genus \(g_i\) with marked points \(\mathbf{y}_i\) and nodal point \(q_i\). Set \[\begin{aligned} &\Sigma_1^\circ=\Sigma_1\setminus\{q_1\}\cong\Sigma_{10}\cup\{[0,\infty)\times S^1\}, \\ &\Sigma_2^\circ=\Sigma_2\setminus\{q_2\}\cong\Sigma_{20}\cup\{(-\infty,0]\times S^1\}. \end{aligned} \] and set \(\Sigma^\circ=\Sigma\setminus{(q_1,q_2)}=\Sigma_1^\circ\cup\Sigma_2^\circ\). Let \(\mathbf{A}_i\) be an open set of the Teichmüller space of genus \(g_i\) and \(n_i\), \(i=1,2\). A complex structure, \((a_1,a_2)\in\mathbf{A}_1\times\mathbf{A}_2\) is chosen. The gluing parameter is given as \(((a_1,a_2),r,\tau)\). A metric \(\mathbf{g}\) on \(\Sigma^\circ\) is defined for metrics \(\mathbf{g}_i\) in the given conformal class as the standard cylindrical metric. Then a closed smooth symplectic manifold \((M,\omega,J)\), \(J\) an \(\omega\)-tame almost complex structure, is prepared and \((j_i,J)\)-holomorphic maps \(u_i:\Sigma_i\to M\) are taken such that \(u_1(q_1)=u_2(q_2)\). The local coordinates \((x^1,\dots,x^{2m})\) in a neighborhood of \(u(q)\) are taken to be \[ (x^1,\dots,x^{2m})(u(q))=0, J(0)\frac{\partial}{\partial x^i}|_0=\frac{\partial}{\partial x^{m+i}}|_0, J(0)\frac{\partial}{\partial x^{m+i}}|_0=-\frac{\partial}{\partial x^i}|_0. \] Then the linearized operator \(D_u=D_u^{j}\) of \(\bar{\partial}_{j,J}\) is explicitly computed (\S6. Appendix Theorem 6.1, cf. (1.8)). \(D_u\) is not a Fredholm operator, when acting on the ordinary Sobolev space. To overcome this problem, a weighted space \(W^{k,2,\alpha}(u^\ast TM)\) is used, where \(e^{\alpha|s|}\), \(\alpha\) not in the spectrum of \(J_0\frac{d}{dt}\), is used as the weight function. Then \[ D_u:W^{k,2,\alpha}(\Sigma^\circ;u^\ast TM)\to W^{k-1,2,\alpha}(\Sigma^\circ;u^\ast TM\otimes\wedge^{0.1}_jT^\ast \Sigma^\circ) \] is a Fredholm operator. Let \(\mathcal{B}_i=\{v_i=\exp_{u_i}(h_i)|h_i\in \mathcal{W}^{k,2,\alpha}_{u_i}\}\), \(\mathcal{W}_u^{k,2,\alpha}=W^{k,2,\alpha}(\Sigma^\circ, u^\ast TM)\), and \(\mathcal{E}_i\) be the Banach bundle over \(\mathbf{A}_i\times \mathcal{B}_i\) whose fiber at \((a_i, v_i)\) is \(W^{k-1,2,\alpha}(\Sigma^\circ, v_i^\ast TM\otimes \wedge^{0.1}_{j_{a_i}}T^\ast \Sigma^\circ)\). \((\mathbf{A}_i\times\mathcal{B}_i,\mathcal{E}_i,\bar{\partial})\) is a Fredholm system. Then by using parallel transport along the geodesics \(s\to \exp_{u(r)}(s\xi)\) with respect to the connection \(\tilde{\nabla}\) (cf. [\textit{D. McDuff} and \textit{D. Salamon}, \(J\)-holomorphic curves and symplectic topology. 2nd ed. Providence, RI: American Mathematical Society (AMS) (2012; Zbl 1272.53002)], a finite dimensional subspace \(K_{b_0}=(K_{b_{01}}, K_{b_{02}})\subset \mathcal{E}|_{b_0}=(\mathcal{E}_{b_{01}},\mathcal{E}_{b_{02}})\) is chosen. It is considered a subspace of \(L^{k-1.,2,\alpha}_{r,u(r)}=W^{k-1,2,\alpha}(\Sigma^\circ, u(r)^\ast TM\otimes \wedge^{0,1}_{j_0}T^\ast\Sigma^\circ)\). Using the map \(\bar{\Phi}\) induced from parallel transport, the operator \(\mathcal{S}\); \[ \mathcal{S}(\kappa,b)=\bar{\partial}_{j_0,J}v+\bar{\Phi}_{b_0,b}\kappa, \quad \kappa\in K_{b_0}, \] is defined. \(D\mathcal{S}\) is its linearized operator. Let \(Q_{0,b_0}\) be a right inverse of \(D\mathcal{S}_{0,b_0}|_{K_{b_0}\times W^{k,2,\alpha}}\) and let \(r=(r,\tau)\) be the gluing parameter. Then \(I_r:\mathrm{ker}D\mathcal{S}_{(\kappa_0,b_0)}\to \mathrm{ker}D\mathcal{S}_{(\kappa_0,b_{(\kappa_0,b_{(r)})}}\) is defined by \[ I_r(\kappa,h+\hat{h}_0)=(\kappa,h_{(r)})-Q_{(\kappa,b_{(r)})}\circ D\mathcal{S}_{(\kappa_0,b_{(r))}(\kappa,h_{(r)})}. \] The main result of this paper is {Corollary 1.2}. Let \(\ell\) be a fixed positive integer. There exist positive constants \(\mathrm{C}_{o,\ell}, \mathrm{d}, R_0\) such that for any \((x,\zeta)\in \mathrm{ker}D\mathcal{S}_{(\kappa_0,b_0)}\) with \(\|(\kappa,\zeta)\|_{\mathcal{W}^{k,2,\alpha}}\le\mathrm{d}\), restricting to the compact set \(\Sigma(R_0)\), the following estimate holds \[ \|\frac{\partial}{\partial r}(I_r(\kappa,\zeta)+Q_{\kappa_0,b_{(r)})}\circ f_{(r)}\circ I_r(\kappa,\zeta)\|_{C^\ell(\Sigma(R_0))}\le \mathrm{C}_{o,\ell}e^{-(c-5\alpha)\frac{r}{4}(\mathrm{d}+1)}. \] This corollary follows from the same estimate but with the norm replaced by the weighted Sobolev norm (Theorem 1.1), which is proved in \S4. In \S5, Corollary 1.2 is extended to an estimate including the derivatives with respect to \(\tau_i\) (Theorem 5.1) and estimates of higher derivatives (Theorem 5.3).