I have already posted this question in math.stackexchange here, but didn't get any response, so I'm posting my question here as well.
Let $X$ be a smooth projective variety over $\mathbb{C}$, and $\pi:E\rightarrow X$ be a holomorphic vector bundle of rank $n$. Given an orthogonal form on $E$, i.e. a symmetric non-degenerate bilinear form $\varphi : E\times E\rightarrow \mathcal{O}_X$, one can show that the frame bundle $Fr(E)$ has a reduction of structure group to $O(n)$, ,i.e. there exists an $O(n)$-invariant subspace $P\hookrightarrow Fr(E)$ which is a principal $O(n)$-bundle over $X$. Conversely, given such an $O(n)$-reduction $P\subset Fr(E)$, we have that the associated vector bundle $P\times_{O(n)}\mathbb{C}^n\simeq E$, which shows that the transition functions of $E$ take values in $O(n)$, which enables us to define an orthogonal form on $E$ by defining it locally on each trivializing cover as the 'canonical' bilinear form $U\times (\mathbb{C}^n\times \mathbb{C}^n)\rightarrow U\times \mathbb{C}$ , $(u,(x_1,\cdots,x_n),(y_i,\cdots,y_n))\mapsto (u,\sum x_i y_i)\cdots\cdots(*)$,
which are compatible on overlaps and give rise to a global form $\varphi: E\times E\rightarrow \mathcal{O}_X.$
My question now is the following: suppose I consider orhogonal forms with values in a line bundle, i.e. I fix a line bundle $L$, and look at symmetric non-degenerate bilibear maps $E\times E\rightarrow L$.
Question: Can one say that the orthogonal forms with values in $L$ correspond to $O(n)$-reductions of $Fr(E)$?
My calculations seem to suggest that it should not be true; specifically, given an $O(n)$-reduction of $Fr(E)$, when I consider a common trivializing cover of $E$ and $L$, the transition functions of $L$ seem to mess up the compatibility of the 'canonical' orthogonal forms as in $(*)$.
Can somone help me with this? Thanks in advance!
