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Let $f:X \rightarrow Y$ be an (unramified) holomorpic covering map between two (maybe non compact) complex manifolds.

Q: Does every infinitesimal deformation of Y lift faithfully to an infinitesimal deformation of X, (i.e. is there a canonical injective map $l:H^1(Y, \Theta_Y) \rightarrow H^1(X,\Theta_X)$?

If not, do you know a counterexample?

Thanks!

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I think that if you look at the universal covering of a compact Riemann surface of genus at least two, i.e. covering by the disk, there are no infinitesimal deformations of the disk, so the lifting is not faithful. –  Ben McKay Aug 18 '11 at 16:24
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1 Answer

Since $f$ is unramified, $f^*\Theta_Y = \Theta_X$ so the map you want is just the map induced by pullback. The map will be injective if $f$ is a finite covering since then the natural inclusion $\Theta_X \to f_*f^*\Theta_X$ splits by using the trace.

In general it will not be injective. For an example you can consider an elliptic curve $Y = \mathbb{C}/\Lambda$ with $\Lambda$ a lattice (and $X = \mathbb{C}$). $H^1(Y,\Theta_Y)$ is $1$ dimensional but the corresponding space for $X$ is $0$.

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