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A rather basic question. What was the original reason to consider the underlying $\mathbb{Z}$-module of the - as canonical object regarded - Tate-Hodge structure $\mathbb{Z}(1)$ to be given as $2 \pi i \mathbb{Z} (\subset \mathbb{C}$) endowed with this notorious factor $2 \pi i$.

Looks like a (historically arising) normalization factor, but with respect to what/ which form (...which presumably should be expressed in terms of certain integrals over loops).

Clearly, as Hodge structure is an additional datum not extractible only from the underlying $\mathbb{Z}$- module structure, it is choiceless to define those Hodge structures $H$ with one dimensional $H \otimes_{\mathbb{Z}} \mathbb{C}$ to have a $r \cdot \mathbb{Z}, r \in \mathbb{C}-\mathbb{R}$ as underlying $\mathbb{Z}$-module structure, but which original condition precisely motivated to take the factor $2 \pi i$ for THS $\mathbb{Z}(1)$?

Does anybody know reason or a source adressing this rather natural question ( for which suprisingly searching on the web I found no source discussing this issue even though this question may permanently pop up after becoming acquainted with this arguably first example for a pure Hodge structure which one encouters as a student). As remarked it seems that this should presumable come from certain normalization condition for Hodge structure of deRham cohomology groups of compact complex manifolds (one may feel free to focus to $X=\mathbb{PC}^n$ only). But which one? Does somebody where this could be found elaborated?

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You are right that it is in some sense a matter of convention, but I claim it's a natural one. Perhaps the easiest example to explain is $H=H_1(X)$, where $X=\mathbb{C}^*$. By Deligne, this carries a(n a priori mixed) Hodge structure, which is one dimensional and pure of weight $-2$, and therefore isomorphic to $\mathbb{Z}(1)$. In fact, let's take this as the definition. Then the natural generator of $H_1(X,\mathbb{Z})$ is the unit circle $\gamma$ oriented counterclockwise. The natural generator of $H^{-1.-1}$ is dual to $dz/z$. The change of basis matrix is just $\int_\gamma dz/z=2\pi i$.

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  • $\begingroup$ probably I see. So if I understand it correctly then the historical motivation for this factor $2 \pi i$ comes from the desire to describe $H_1(X, \mathbb{Z})$ explicitly as dual space of "integral" deRham classes via explicit isomorphism $V \cong V^*$ (ie explicit identification of a space with it's dual) via pairing/bilinear form induced by usual integration, right? $\endgroup$
    – user267839
    Commented May 8 at 23:34
  • $\begingroup$ But if that's the whole reason behind multiplying with this factor, one could ask provocatively why we should modify $H_1(X,\mathbb{Z})$ by multiplying with this factor $2 \pi i$ and not conversely instead modifying all integer deRham forms by multiplying them by factor $1/2 \pi i$? Ie why modifying $H_1(X,\mathbb{Z})$ instead of modifying integer deRham forms is "more natural"? $\endgroup$
    – user267839
    Commented May 8 at 23:41
  • $\begingroup$ (1) I'm not a historian, I imagine that $\mathbb{P}^1$ was the first example understood along these lines. But it is closely related to the example I just gave. (2) Sure, the $2\pi i$ has to go somewhere... $\endgroup$
    – Donu Arapura
    Commented May 9 at 0:10

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