It is well-known that for holomorphic functions the Identity theorem holds: if two holomorphic functions agree on an open subset, they agree everywhere (assuming the manifold connected).
I would like to know if the same is true for holomorphic forms:
Let $X$ be a complex manifold of dimension $n$. Is it true that, if two holomorphic forms agree over an open subset then they agree everywhere?
It seems to me that this should follow by applying the Identity theorem to the the coordinate representation of the forms, and using the transition laws to ''propagate'' the identity. But I could not find this in books, so it is probably obvious or wrong.
One last question: is the same true for real-analytic manifolds and forms?
Thank you.
Yes. This is even true for a real analytic section of a real analytic bundle over a real analytic manifold.
To prove this, suppose $f=g$ on a set $U$ with nonempty interior (which we will henceforth also call $U$); if $U$ is not everything, then there is a point $x \in \partial U$. Work instead with $f-g$; we wish to show it vanishes in a neighborhood of $x$.
But in a small chart around $x$, there is a real analytic trivialization of the bundle. In this trivialization, our real analytic section becomes a real analytic function that vanishes on a nonempty open set. So it must vanish in the entire chart, hence showing that $x$ is in the interior, and hence our function vanishes everywhere.