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I am trying to read the proof of torelli theorem by Henrik H.Martens "A new proof of torelli's theorem" Annals of mathematics vol78 no. 1 .The proof seems to me like using mysterious combination of 3 lemmas and techniques .Is there any way to find motivation behind the proof.Although I can understand it line by line, I cannot get how the proof is working.This is the proof given in Raghavan Narasimhan's book on compact riemann surface.

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    $\begingroup$ I, personally, don't think your question has a good answer. I think that this genre of proofs of Torelli are somewhat mysterious and technical. It was and is a really important theorem, so people have reproved it in various ways. $\endgroup$
    – meh
    Commented Oct 29, 2014 at 14:20
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    $\begingroup$ If you need a proof of Torelli that you can understand, I'll recommend Andreotti's. Now, at the moment, I'm not able to access JSTOR at the moment, so I can't look into Martens's proof in more detail, but if it hasn't been answered by the weekend, send me an @ reply so that I spot it and I'll take a more careful look. $\endgroup$
    – Charles Siegel
    Commented Oct 29, 2014 at 14:39
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    $\begingroup$ I believe that Geometry of Algebraic Curves contains a version of the Andreotti proof. I personally find it more approachable. In addition there is a paper of Griffiths called something like "infitesimal Variations of Hodge Structure III" , which is in vol 3 of his collected works. I like that proof a lot. But this is probably 'one mans poison' sort of a thing. $\endgroup$
    – meh
    Commented Oct 31, 2014 at 0:17
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    $\begingroup$ I do not think that the proof is really mysterious: The idea is to identify the image $W^1$ in the intersection of $W^{g-1}$ with some of its shifts. I think it would be a good idea to do the proof for the genus 3 case by hand. $\endgroup$
    – Sebastian
    Commented Oct 31, 2014 at 9:08
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    $\begingroup$ Since, this is based on Weil's proof, it might help to understand that. In Mumford's Curves and their Jacobians, he says such proofs are related to the lemma that the intersection of Theta with its translate by a, is contained in the union of two other translates, iff a is the difference of two points on the Abel curve W1. He also points out a much easier proof using this lemma, since it implies that Theta determines the surface of differences of points of the Abel curve, and the tangent cone to the origin of that difference surface is exactly the curve. $\endgroup$
    – roy smith
    Commented Nov 1, 2014 at 0:03

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