Timeline for Deformation long exact sequence of GW theory in the analytical setting
Current License: CC BY-SA 3.0
17 events
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| May 14, 2016 at 2:40 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Apr 14, 2016 at 2:31 | answer | added | Mohammad Farajzadeh-Tehrani | timeline score: 2 | |
| Mar 24, 2016 at 17:23 | comment | added | Jason Starr | There is a holomorphic and fiberwise linear map from the complex line bundle $T_\Sigma(-\sum_i p_i)$ to the complex line bundle $T_\Sigma$. That induces an injective map of sheaves $A(T_\Sigma(-\sum_i p_i)) \to A(T_\Sigma)$. Now compose that map with $du$. By abuse of notation, I am also calling that composition $du$. | |
| Mar 24, 2016 at 15:42 | comment | added | Mohammad Farajzadeh-Tehrani | Thanks Jason, I think this sheaf theoretic description of yours was the key to make sense of Q2. Now I see where I was struggling: to make sense of the map $du\colon \mathbb{C}^\infty(T\Sigma(-p)) \to \mathbb{C}^\infty (u^*TX)$. Originally, $du$ is a map from $T\Sigma$ to $u^*TX$, how do the marked points affect it then? | |
| Mar 24, 2016 at 11:53 | comment | added | Jason Starr | One clarification: for a holomorphic vector bundle $E$, $E(-\sum_i p_i)$ is a holomorphic vector bundle. When I write $E(-\sum_i p_i)$, I really mean the fine, $C^\infty$ sheaf $A(E(-\sum_i p_i))$. As a subsheaf of $A(E)$, this is different from the subsheaf of $C^\infty$ sections that vanish at each $p_i$. For instance, on the complex plane, the real coordinate $x$ is a section of $A(\mathbb{C})$ that vanishes at $0$, but it does not equal a $C^\infty$, $\mathbb{C}$-valued function times the complex coordinate $z=x+iy$, which is a generator of $A(\mathbb{C}(-\underline{0}))$. | |
| Mar 24, 2016 at 11:43 | comment | added | Jason Starr | Yes, all of the sheaves in those complexes are the fine, $C^\infty$ sheaves, not the holomorphic sheaves. Of course for a holomorphic vector bundle $E$ on $\Sigma$ with associated fine, $C^\infty$ sheaf $A(E)$ of $C^\infty$ sections, the complex in degrees $[0,1]$, $\overline{\partial}:A(E)\to \Omega^{1,0}\otimes A(E)$ is an acyclic resolution of the sheaf $\mathcal{O}(E)$ of holomorphic sections,. Hence the complex has hypercohomologies equal to $H^0(\Sigma,\mathcal{O}(E))$, resp. $H^1(\Sigma,\mathcal{O}(E))$. | |
| Mar 24, 2016 at 11:25 | comment | added | Mohammad Farajzadeh-Tehrani | 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? | |
| Mar 24, 2016 at 10:23 | comment | added | Jason Starr | 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 $\text{Lie}(\text{Auto}(\Sigma,p_1,\dots,p_k))$. | |
| S Mar 24, 2016 at 1:56 | history | suggested | Silvia Ghinassi | CC BY-SA 3.0 |
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| Mar 24, 2016 at 1:50 | comment | added | Jason Starr | 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. | |
| Mar 24, 2016 at 1:43 | review | Suggested edits | |||
| S Mar 24, 2016 at 1:56 | |||||
| Mar 24, 2016 at 1:38 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 3.0 |
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| Mar 24, 2016 at 1:02 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 3.0 |
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| Mar 24, 2016 at 0:38 | comment | added | Mohammad Farajzadeh-Tehrani | 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. | |
| Mar 24, 2016 at 0:32 | comment | added | Jason Starr | 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? | |
| Mar 24, 2016 at 0:27 | history | edited | Mohammad Farajzadeh-Tehrani | CC BY-SA 3.0 |
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| Mar 24, 2016 at 0:22 | history | asked | Mohammad Farajzadeh-Tehrani | CC BY-SA 3.0 |