On the finite principal bundles (Q1024794)

Let \(X\) be a compact connected Kähler manifold. Given a holomorphic vector bundle \(E\to X\) and a polynomial \(\varphi(z)=\sum_{i=0}^na_i\,z^i\) with nonnegative integral coefficients, define \(\varphi(E)\) to be the holomorphic vector bundle over \(X\) \[ \varphi(E):=\bigoplus_{i=0}^n(E^{\otimes i})^{\oplus a_i}. \] A holomorphic vector bundle \(E\to X\) is said to be finite if there are two distinct such polynomials \(\varphi_1\) and \(\varphi_2\) such that \(\varphi_1(E)\) is holomorphically isomorphic to \(\varphi_2(E)\). Next, let \(G\) be a connected linear algebraic group defined over the complex numbers and fix a finite-dimensional faithful \(G\)-module \(V_0\). If \(E_G\) is a holomorphic principal \(G\)-bundle over \(X\), then it is called finite if for each subquotient (i.e., the quotient \(G\)-module \(V_2/V_1\) for any two submodules \(V_1,V_2\) of the \(G\)-module \(V_0\) with \(V_1\subset V_2\)) \(W\) of the \(G\)-module \(V_0\), the holomorphic vector bundle \(E_G(W)\) over \(X\) associated to \(E_G\) for \(W\) is finite. In this paper, the author proves the following four equivalences. {\parindent=4mm \begin{itemize}\item[{\(\bullet\)}] The principal \(G\)-bundle \(E_G\) admits a flat holomorphic connection whose monodromy group is finite. \item[{\(\bullet\)}] There is a finite étale Galois covering \(f: Y\to X\) such that the pullback \(f^*E_G\) is a holomorphically trivializable principal \(G\)-bundle over \(Y\). \item[{\(\bullet\)}] For any finite-dimensional complex \(G\)-module \(W\), the holomorphic vector bundle \[ E_G(W):=E\times^{G}W \] over \(X\) associated to the principal \(G\)-bundle \(E_G\) for the \(G\)-module \(W\) is finite. \item[{\(\bullet\)}] The principal \(G\)-bundle \(E_G\) is finite. \end{itemize}} Observe that when \(X\) is a projective variety, the same result was proven by \textit{M. V. Nori} [Proc. Indian Acad. Sci. Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)].