Timeline for DGLA controlling deformation of holomorphic curves
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2018 at 12:08 | comment | added | Grisha Papayanov | arxiv.org/abs/math/0507287 and arxiv.org/abs/math/0601312 might be exactly what you're interested in, as well as the subsequent works of the authors; these two papers give an explicit $L_{\infty}$-structure on the cone of Kodaira-Spencer dg-Lie algebras for two manifolds and describe the corresponding Maurer-Cartan functor | |
| S Sep 12, 2018 at 12:17 | history | bounty ended | Mohan Swaminathan | ||
| S Sep 12, 2018 at 12:17 | history | notice removed | Mohan Swaminathan | ||
| Sep 12, 2018 at 12:17 | vote | accept | Mohan Swaminathan | ||
| Sep 7, 2018 at 21:21 | comment | added | Phil Tosteson | Kapranov shows in "Rozansky–Witten invariants via Atiyah classes" that the Atiyah class makes $T_X[-1]$ into a lie algebra in $D(X)$. The relationship of this Lie algebra to deformation theory is discussed here: mathoverflow.net/questions/143269. Unfortunately, I don't have enough expertise to give a detailed answer. | |
| Sep 7, 2018 at 19:49 | comment | added | Mohan Swaminathan | I'm not familiar with the Kapranov bracket. Could you point me to a reference where it is defined? Also, it would be nice if you can expand this comment into a more detailed answer. | |
| Sep 7, 2018 at 16:24 | comment | added | Phil Tosteson | You can see that part of this is right by considering the fiber sequence of formal moduli problems which forgets the map to $f$. This corresponds to a short exact sequence of $dg$ lie algebras, where the quotient governs deformations of $C$ and the sub governs deformations of maps to $X$ where the curve is fixed. | |
| Sep 7, 2018 at 16:22 | comment | added | Phil Tosteson | A natural guess is that $\Omega^{0}(f^* T_X) \otimes \Omega^{0} (f^* T_X) \to \Omega^{0,1}(f^*T_X)$ is given by pulling back a representative for the Kapranov bracket, while $\Omega^{0,1}(T_C) \otimes \Omega^0(T_X) \to \Omega^{0,1}(T_X)$ is induced by the Lie bracket of vector fields and the rest of the degree 1 map is zero. | |
| Sep 7, 2018 at 8:21 | answer | added | Jon Pridham | timeline score: 10 | |
| S Sep 6, 2018 at 23:45 | history | bounty started | Mohan Swaminathan | ||
| S Sep 6, 2018 at 23:45 | history | notice added | Mohan Swaminathan | Canonical answer required | |
| Sep 4, 2018 at 23:11 | history | asked | Mohan Swaminathan | CC BY-SA 4.0 |