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Ops, sorry, I was indeed careless in reading, answering and commenting, now I see why the answer could (should?) be yes. I'll stop adding nonsense and leave that to people that know what they're talking about!
I'm sorry, I didn't note that in the first version of the question, thanks for the added note. Would you mind pointing to a definition of the arc length measure? Because the only reference I found by googling is page 199 in "The Theory of Subnormal Operators" by John B. Conway (on google books) where it says that the arc-length measure on a (geometric) curve is defined precisely by taking any function $u$ and any parametrization $\gamma$ and imposing $\int_{\partial D}u d\mu = \int_0^1u(\gamma(y))|\gamma'(t)|dt$. This would give the usual notion, thus 0 as in my answer...
Obviously you're right, and just as obviously the fact was much easier than anything I was thinking of... And suddenly I realize that most of the questions written above are meaningless! Well, let's hope that they will be useful to other confused grad students as me!
Okay, so it seems that this is turning to a discussion on my thesis... I will stop before the discussion on MO overtake my work on the subject, otherwise I risk to become out of work ;-) I take from the absence of an answer that there is no hope of having a satisfying one? Indeed, I doubt of it. But it still puzzles me if it is really true that for any projective manifold $X$ (if needed, with $H_1(X)$ torsion-free) it is true that the homomorphisms in $H^1(X, \mathbb{C})$ with values in $\lbrace k \pi i \rbrace_{k \in \mathbb{Z}}$ correspond to form of type $(0,1)$, that seems an odd result!
