dhy
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Projective automorphisms of a plane cubic curves
You can rephrase this as looking for the automorphisms of $E$ which preserve the degree $3$ line bundle coming from the embedding of $D$ into $\mathbb{P}^2.$ If $E$ is generic then its automorphism group is generated by translations and multiplication by $-1$ and you can explicitly check that the subgroup preserving $L$ is generated by $3$-torsion points and multiplication by $-1$.
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Is the pushforward of a closed immersion ever fully-faithful at the level of Derived Categories?
It will almost never be fully faithful (more precisely, I suspect that if $Z$ and $X$ are of finite type over a field, it will be fully faithful iff $Z$ is a connected component of $X$.) Here's one way to see that it's not in your example of projective space: the endomorphism ring of the skyscraper sheaf at a smooth point $p\in X$ is isomorphic to $\operatorname{Sym}(\Omega_X|_p[1])$ (proof: use the Koszul complex). This remembers the dimension of the ambient variety $X$.
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What is the proof for any non trivial zero?
I don't think this question is appropriate for MO, but here is a related Math.SE post which contains the answer to your question: math.stackexchange.com/questions/1903742/….
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Roadmap to geometric Langlands for a mathematical physics student
AFO on the other hand does require more mathematical background. This is not so close to what I do so take everything I say with a grain of salt, but my impression is that this is closely related to the general story of quantum enumerative geometry of symplectic resolutions, of which the prototypical case is the paper "Quantum Cohomology of the Springer Resolution" by Braverman-Maulik-Okounkov. After that I would suspect the paper to read is "Quantum Groups and Quantum Cohomology" by Maulik-Okounkov?
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Roadmap to geometric Langlands for a mathematical physics student
@DanielWaters My point is that neither Hecke eigensheaves (which is just a definition) nor D-modules should be considered background for Kapustin-Witten, because they only really use the definitions (which they provide and explain.) There is of course independent value in learning about D-modules and why they appear in GRT (the book of Hotta-Takeuchi-Tanisaki is one place to learn this if you are interested) but I don't think knowing this theory will help for reading Kapustin-Witten.
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Roadmap to geometric Langlands for a mathematical physics student
@DanielWaters In particular if Kapustin-Witten is your goal, $\infty$-categories, DAG, "classical" geometric representation theory, etc., are certainly not necessary prerequisite knowledge.
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Roadmap to geometric Langlands for a mathematical physics student
@DanielWaters Is there a specific mathematical point you get stuck at when you try to read Kapustin-Witten? I would say those papers are written to require quite a bit of physics background and not very much math background (e.g. the introduction contains the sentence "No prior familiarity with the Langlands program is assumed; instead, we assume a familiarity with subjects such as supersymmetric gauge theories, electric-magnetic duality, sigma-models, mirror symmetry, branes, and topological field theory.")
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Roadmap to geometric Langlands for a mathematical physics student
By the way, there are many different approaches to GL. If you have a concrete goal in mind (e.g., "I want to understand this paper _______ by ______") I can write a longer form answer giving a roadmap.
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Roadmap to geometric Langlands for a mathematical physics student
I would actually actively recommend against reading any source on $\infty$-categories (unless you are intrinsically interested in them). The power of $\infty$-categories is that, once you understand the philosophy, they are very convenient to use, without having to delve into their foundational details. For the purposes of geometric Langlands, you will learn much more from seeing how they are used than actually reading about them...
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Singular cohomology to cohomology of quasi-coherent sheaf
To be more explicit: For a complex variety, two sensible cohomology theories you can consider are a) the sheaf cohomology (in the Zariski topology) of quasi-coherent sheaves and b) singular cohomology, which can be identified with the sheaf cohomology (in the complex topology) of the constant sheaf. You seem to be trying to mix the two, considering the (Zariski) sheaf cohomology of the constant sheaf - as Zhen Lin mentioned, this only gives you trivial cohomology groups. However, there is a more indirect link given by the Hodge-de Rham spectral decomposition.