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Given a regular scheme $X$, projective and flat over $Spec(\mathbb{Z})$, i.e. an arithmetic variety and a vector bundle $E$ on $X$. We get an associated vector bundle $F$ on the associated complex manifold $M:=X(\mathbb{C})$. On $M$ one has complex conjugation $c$.

Following Kobayashi's "Differential geometry of complex vector bundles", the set $H(F)$ of hermitian metrics on $F$ can be identified with the symmetric space $GL(F)/U(F)$.

How to find those metrics $h \in H(F)$, which are invariant under $c$, i.e. $c^{*}h=h$? Is there a nice description of this set?

I am interested in this metrics, because they are often mentioned in articles about Arakelov geometry, especially Soule's book. This condition comes somehow out of nowhere. I would be very happy to understand why this metrics are the right ones to consider. Another question in this context: is "invariant under $c$" a strong restriction to the metrics in $H(F)$, i.e. are there many of them or rather a few who fulfill this condition.

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I've thought about similar things recently. Perhaps this should really be a linear algebra question first! Take $X$ to be a point, so you are simply working with a complex vector space $F$ endowed with a real structure, i.e., a descent datum isomorphism from $F$ to its conjugate space. You want to classify Hermitian forms on $F$ which are compatible, in a sense, with the descent datum. If this is what I think it is, the answer should be something like $GL(F_R) / O(F_R, B)$, where $F_R$ is the conjugation-invariant subspace, and $O(F_R, B)$ is a certain orthogonal group. –  Marty Mar 1 '10 at 21:58
    
In general, whenever one has a compact group acting on a manifold, one can turn a non-invariant Riemannian metric h into a invariant one by averaging over all g*h for g in G. This gives again a Riemannian metric. But I don't think this is, what you were looking for. –  HenrikRüping Mar 2 '10 at 10:38
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