Holomorphic connections on filtered bundles over curves (Q2439219)

Let \(X\) be a compact Riemann surface. By a well-known result, due to \textit{A. Weil} [J. Math. Pures Appl. (9) 17, 47--87 (1938; Zbl 0018.06302)] and \textit{M. F. Atiyah} [Trans. Am. Math. Soc. 85, 181--207 (1957; Zbl 0078.16002)], a holomorphic vector bundle \(E\to X\) admits a holomorphic connection \(D: E\to E\otimes \Omega^1_X\) precisely when all of its irreducible summands have degree zero. Suppose \(D\) is a holomorphic connection on \(E\). If \(E\) carries a filtration \[ 0= E_0\subset E_1\subset \dots\subset E_{l-1}\subset E_l=E \] which is \(D\)-horizontal, i.e., \(D (E_k)\subset E_k\otimes \Omega^1_X\), then each summand of the associated graded bundle \(\mathrm{Gr}E=\oplus E_k/E_{k-1}\) inherits a connection, i.e., \(D(E_k/E_{k-1})\subset (E_k/E_{k-1})\otimes \Omega^1\). Hence for all \(1\leq k\leq l\), \(E_k/E_{k-1}\) has irreducible components, which are topologically trivial. It is then natural to look for (other) conditions, under which there exists a connection, preserving the given flag. More generally, one can formulate an analogue of this question in the context of holomorphic principal bundles. Let \(G\) be an affine complex reductive group and \(P\subset G\) a parabolic subgroup, with Levi factor \(L(P)\). Let \(E_P\to X\) be a holomorphic principal \(P\) bundle, and \(E_{L(P)}\) the \(L(P)\)-bundle, obtained by extension of structure groups via \(P\to L(P)\). Let also \(E_G\) be the principal \(G\)-bundle, obtained after extending the structure group from \(P\) to \(G\). The reduction \(E_P\subset E_G\) is called \textit{rigid} if \(H^0(X, \mathrm{ad}E_G/\mathrm{ad}E_P)=0\), and, consequently, \(H^1(X,\mathrm{ad}E_P )\hookrightarrow H^1(X,\mathrm{ad}E_G )\). By analysing the Atiyah sequences of all bundles involved, the authors prove the following result: { Theorem 2.1}. Assume that \(E_{L(P)}\) admits a holomorphic connection, and that the reduction \(E_P\subset E_G\) is rigid. Then \(E_P\) admits a holomorphic connection. The authors then proceed to find sufficient conditions for the existence of a connection on \(E_{L(P)}\). In particular, they prove: { Proposition 3.1}. Suppose that \(E\) and \(F\) are holomorphic vector bundles, admitting holomorphic connections. Suppose, moreover, that every global section of \(E^\vee\otimes F\) is horizontal (with respect to the natural induced connection). Then for any extension \[ 0\longrightarrow E\longrightarrow W\longrightarrow F\longrightarrow 0 \] the holomorphic vector bundle \(W\) admits a holomorphic connection, preserving \(E\subset W\).