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Let $X$ be a smooth projective variety over a subfield $k$ of $\mathbb{C}$. Then one has a short exact sequence

$$ 0 \to H^0(X, \Omega^1_X) \to H^1_{dR}(X/k) \to H^1(X, \mathcal{O}_X) \to 0 $$

One knows that, after tensoring by $\mathbb{C}$, this sequence becomes split, as $H^1_{dR}(X/k) \otimes_k \mathbb{C}$ is canonically isomorphic to the singular cohomology $H^1(X(\mathbb{C}), \mathbb{C})$ which carries a Hodge structure.

Question: Are there criteria for this sequence to be already split over $k$?

Already the case of curves seems pretty interesting.

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    $\begingroup$ Any short exact sequence of vector spaces splits. $\endgroup$
    – Fernando Muro
    Commented Jun 19, 2014 at 9:05
  • $\begingroup$ Do you want to consider a flat family of varieties? $\endgroup$
    – S. Carnahan ♦
    Commented Jun 19, 2014 at 9:14
  • $\begingroup$ not an expert, but isn't that just the degeneration of the Hodge-dR spectral sequence ? $\endgroup$
    – meh
    Commented Jun 19, 2014 at 12:11
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    $\begingroup$ I think the OP means: "When is the canonical Hodge splitting already defined over $k$ ?". This happens rarely and the failure to split over $k$ is basically encoded in the Mumford-Tate group, about which there are many deep conjectures. For instance if $X$ is a CM abelian variety and $k$ contains the Galois closure of the CM field, then the sequence splits (and the Mumford-Tate group is a torus). $\endgroup$
    – Damian Rössler
    Commented Jun 20, 2014 at 7:53

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