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2 votes
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When does the rational Hodge structure determine the integral Hodge structure?

If the image of the usual period map is an open subset of the period domain, this map is certainly not generically injective: For a typical point in the image, its translate by any element of $GL_n(\...
Will Sawin's user avatar
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6 votes
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Prefactor $2\pi i$ for Tate-Hodge structure

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(...
Donu Arapura's user avatar
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13 votes
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Understanding the Hodge filtration

The naive Hodge filtration of a smooth affine variety is, indeed, the whole thing. We always have the short exact sequence of complexes: $$0 \to \Omega^{\bullet, \geq p} \to \Omega^{\bullet} \to \...
David E Speyer's user avatar
7 votes

How does complex conjugation act on the Hodge filtration?

I thought it would be useful to give an explicit example to supplement Olivier Benoist's answer; I will use the same notation as in his answer. Let $f(x)\in \mathbb{R}[x]$ be cubic with distinct ...
Donu Arapura's user avatar
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12 votes
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How does complex conjugation act on the Hodge filtration?

Let us first consider the singular cohomology group $H^i(X(\mathbb{C}),\mathbb{C})$. There are three ways to let the group $G:=\mathrm{Gal}(\mathbb{C}/\mathbb{R})\simeq\mathbb{Z}/2$ act on it : by ...
Olivier Benoist's user avatar

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