# Compatible connection on the associated vector bundle

Assume $E\rightarrow X$ is a holomorphic vector bundle of rank $n$ with a linear connection $\nabla=\nabla^{1, 0}+\bar\partial_E$ which is compatible with the complex (somewhere the literature says "holomorphic") structure of $E$. Let $P\rightarrow X$ be the associated principal $GL_n(\mathbb{C})$-bundle (say the frame bundle of $E$).

There is a theorem which states that $\nabla$ induces a unique connection 1-form $\omega$ on $P$.

Let further $V$ be a representation of $GL_n(\mathbb{C})$ and let $P\times_{GL_n(\mathbb{C})}V$ be the associated vector bundle which should be in my opinion holomorphic as well. The principal connection $\omega$ induces always a linear connection $\tilde{\nabla}$ on the associated vector bundle $P\times_{GL_n(\mathbb{C})}V$. The question I want to pose is the following:

Is $\tilde{\nabla}$ compatible with the complex structure on $P\times_{GL_n(\mathbb{C})}V$?

P.S.:Thanks in advance for any clarifications, corrections and responses to that for me very interesting question.

• Yes. You can do everything in the holomorphic category: a holomorphic connection induces a holomorphic connection on $P$, and this induces a holomorphic connection on any associated vector bundle. – abx Apr 1 '14 at 17:03
• Thanks for your fast comment. $\nabla$ is NOT a holomorphic connection. It is just compatible with the complex structure (somewhere they say "holomorphic structure") of $E$. Compatible connection means $\nabla=\nabla^{1, 0}+\bar\partial_E$, where $\bar\partial_E^2=0$. – Flavius Aetius Apr 1 '14 at 17:11
• Yes. Write out the connection on $P \times_{GL_n(\mathbb{C})} V$, in a basis of local sections, in terms of the connection on $P$ and, you should see compatibility immediately. – Ben McKay May 2 '14 at 11:10
• Just to minimize confusion: in the common parlance, "complex" refers to topology, namely a vector bundle whose fiber is a complex vector space. The term "holomorphic bundle" refers to a holomorphic structure on a complex bundle, namely a way of picking out which sections are locally holomorphic vector valued functions on the base. The latter is most often described by a $\overline{\partial}$-operator as you've indicated above. Once you agree that it's these operators giving the holomorphic structure, the question you asked is largely reduced to a tautology. – Andy Sanders Jul 31 '14 at 1:58
• This question has been answered in a satisfactory way. – abx Nov 28 '14 at 6:32

According to your last comment, the holomorphic structure is irrelevant; you just have a $GL_n(\mathbb{C})$-bundle. I presume that your representation is complex, i.e., merely a homomorphism $H:=GL_n(\mathbb{C})\to G:=GL_m(\mathbb{C})$. (Otherwise, there is nothing to speak about.) This gives you a map $P\to P\times_HG$ of principal bundles, which is called extension of the structure group. (Unlike reduction of the structure group, it is well defined and involves no choices.) In this situation, any $H$-connection extends uniquely to a $G$-connection (e.g., via translations in the fibers, see the definition of connections as $1$-forms and their properties; BTW, a $1$-form is not induced by a connection: it is the connection). Since this is a $G$-connection, it is compatible with the complex structure just by definition.
• I don't follow here. Do you basically claim that $\tilde{\nabla}$ is compatible with the complex structure of $P\times_{GL_n(\mathbb{C})}V$? – Flavius Aetius Apr 1 '14 at 20:06
• Sure, $V$ is a complex vector space. So the representation is indeed a complex homomorphism as you pointed out. Ah, so basically it turns out that every connection on $P\times_{H}V$ coming from a connection 1-form on $P$ has to be compatible with the complex structure of $P\times_HV$? – Flavius Aetius Apr 1 '14 at 20:16
• Any induced connection is "compatible" with the new structure group, whatever this means. If the new group is in $GL_n(\mathbb{C})$, the connection is compatible with the complex structure (e.g., just because $J\in GL_n(\mathbb{C})$). – Alex Degtyarev Apr 1 '14 at 20:37