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Let $f\!=\!(u\colon (\Sigma,p_1,\ldots,p_k) \to X)$ be an element of the moduli space of genus $g$ $k$-marked degree $A$ $J$-holomorphic maps $\mathcal{M}_{g,k}(X,A,J)$. For simplicity assume $C=(\Sigma,p_1,\ldots,p_k)$ is stable.

In the algebraic (or holomorphic) setting, deformation long exact sequence of $\mathcal{M}_{g,k}(X,A,J)$ around $f$ has the form

\begin{align} 0\to \operatorname{Def}(u) &\to \operatorname{Def}(f) \to \operatorname{Def}(C) \\ \to \operatorname{Obs}(u) &\to \operatorname{Obs}(f) \to 0\;; \end{align}

see Section 24 of "Mirror Symmetry" book by Hori, Katz, Klemm, etc.

In the analytical (symplectic) setting, the first column corresponds to kernel and cokernel of linearization of Cauchy Riemann operator $$ D_u\bar\partial\;\colon \Gamma(u^*TX)\to \Gamma(\Omega^{0,1}_\Sigma\otimes u^*TX); $$ i.e. $\operatorname{Def}(u)=\ker(D_u\bar\partial)$ and $\operatorname{Obs}(u)=\operatorname{coker}(D_u\bar\partial)$.

$\operatorname{Def}(C)$ is as in the algebraic case: $$ \operatorname{Def}(C)=H^1(T\Sigma(-p_1\cdots -p_k)) $$

Question 0: Does such long exact sequence even always exist in the analytical setting?

Question 1: What is the analytical description of the map $\operatorname{Def}(C)\to \operatorname{Obs}(u)$? Whether the answer to Q0 is Yes or No, this map should be naturally definable.

Question 2: What are the analytical descriptions of $\operatorname{Obs}(f)$ and $\operatorname{Def}(f)$?

Question 3: Do you know of any reference where this sequence is explained for the analytical setup of GW theory?

Comments: In the case of no-marked points, $\operatorname{Def}(C)=H^1_{\bar\partial}(T\Sigma)$ and we can get the map via $du:T\Sigma \to TX$. In the case there are marked points, the map should be similar, I just have hard time visualizing it. In question 2, if the map is immersion and no-marked points the spaces are similar to that of $u$ with $N_{u(\Sigma)}X$ instead of $TX$.

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  • $\begingroup$ Mohammad, is there any reason to expect that the analytic description of this connecting map is different from the algebraic description? In particular, just to check, are you deforming the Cauchy-Riemann equation? $\endgroup$ – Jason Starr Mar 24 '16 at 0:32
  • $\begingroup$ I am just trying to write down its analytical description explicitly. Since I don't well understand the short exact sequence from which such long exact sequence is constructed (in the analytical setting) I can't simply follow the algebraic definition of connecting map. $D_u\bar\partial$ is of the form $\bar\partial+A$ where $\bar\partial$ defines a holomorphic structure on $u^*TX$ and $A$ is some degree 0 map (which depends on Nijenhueis tensor). So $Def(u)$ and $Obs(u)$ are deformations of $H^0_{\bar\partial}(TX)$ and $H^1_{\bar\partial}(TX)$. That makes me a bit confused. $\endgroup$ – Mohammad Farajzadeh-Tehrani Mar 24 '16 at 0:38
  • $\begingroup$ Your first column is the hypercohomology of the complex of sheaves on $\Sigma$ in degrees $[0,1]$, $f^*T_X\to \Omega^{1,0}_\Sigma\otimes f^*T_X$. Assume $\Sigma$ is smooth. Form the new complex concentrated in degrees $[-1,1]$, $T_\Sigma(-\sum_i p_i)\to f^*T_X \oplus (\Omega^{1,0}_{\Sigma}\otimes T_\Sigma(-\sum_i p_i)) \to \Omega^{1,0}\otimes f^*T_X$. The first map is $(du,\overline{\partial})$, and the second map is $(Du\overline{\partial},-\text{Id}\otimes du)$. The hypercohomology is $\text{Def}(f)$ and $\text{Obs}(f)$. There is a map of these complexes giving your long exact sequence. $\endgroup$ – Jason Starr Mar 24 '16 at 1:50
  • $\begingroup$ Typo correction: Every $f^*T_X$ should be $u^*T_X$. Also, the visible chain homomorphism of complexes from the first complex to the second complex is injective, and the cokernel is the complex concentrated in degrees $[-1,0]$, $T_\Sigma(-\sum_i p_i) \to \Omega^{1,0}_\Sigma\otimes T_\Sigma(-\sum_i p_i)$. This short exact sequence of complexes gives the long exact sequence in your question. Also, this shows that in the case that $(\Sigma,p_1,\dots,p_k)$ is not stable, then your long exact sequence should begin with $\text{Lie}(\text{Auto}(\Sigma,p_1,\dots,p_k))$. $\endgroup$ – Jason Starr Mar 24 '16 at 10:23
  • $\begingroup$ Thanks Jason. By $T{\Sigma}(-\sum p_i)$ you mean sheaf of smooth (and not the usual holomorphic) sections of $T\Sigma$ that vanish at those points, correct? $\endgroup$ – Mohammad Farajzadeh-Tehrani Mar 24 '16 at 11:25
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In addition to the nice description of Jason in the comments, there is a fairly detailed description of the deformation long exact sequence in Section 3.2 of the article of Siebert-Tian in "Symplectic 4-manifolds and algebraic surfaces".

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