# Over which schemes can there exist non-trivial G_a bundles?

The group scheme G_a here is the one-dimensional additive group.

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## 1 Answer

Principal Ga-bundles on a scheme X, in any of the Zariski, etale, or flat topologies, are classified by the coherent cohomology group H^1(X,OX). For a smooth complex projective variety, this is the antiholomorphic component of the de Rham group H^1(X,C), which is a topological invariant. So (in this smooth Kahler setting) the existence of nontrivial Ga-bundles depends only on the topological type of X.

I omitted some underscores for typesetting reasons.

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You can escape the underscores with a backslash so that they don't cause a problem (i.e. type \\_ to get _). Better yet, instead of typing H^1(X,O\\_X), you can type H<sup>1</sup>(X,O<sub>X</sub>) to get really pretty-looking output. –  Anton Geraschenko Oct 28 '09 at 5:16