Exponents of a meromorphic connection on a compact Riemann surface (Q731180)

The notion of the exponents of a linear differential equation with regular singularities on the Riemann sphere was introduced by \textit{J. L. Fuchs} [J. Reine Angew. Math. 65, 74--111 (1866; ERAM 065.1691cj) and ibid. 66, 121--160 (1866; ERAM 066.1719cj)] (roughly speaking, these are complex numbers describing the behavior of solutions near singular points). He also obtained a relation on the sum of exponents (the celebrated Fuchs relation). The theory of exponents for linear differential systems with regular singularities was developed by \textit{A. H. M. Levelt} in [Hypergeometric functions. Amsterdam: Drukkerij Holland N. V. (1961; Zbl 0103.29502) and Nederl. Akad. Wet., Proc., Ser. A 64, 361--372; 373--385; 386--396; 397--403 (1961; Zbl 0124.03602)], he also established that the sum of exponents is a non-positive integer. In the article, the notion of exponents is generalized to the case of a meromorphic connection \(\nabla\) on a holomorphic vector bundle \({\mathcal E}\) over a compact Riemann surface \(X\) (a linear differential system can be thought of as a meromorphic connection on a holomorphically trivial vector bundle). For an unramified \(\nabla\) its exponents are defined as invariants of a vector bundle \({\mathcal E}_L\) canonically attached to \({\mathcal E}\) and called the Levelt bundle of \({\mathcal E}\). For a ramified \(\nabla\), the existence of a holomorphic cover \(\pi:\widetilde X\to X\) is shown such that the connection \(\pi^*\nabla\) on the bundle \(\pi^*{\mathcal E}\) over \(\widetilde X\) is unramified. Then, the exponents of \(\nabla\) are invariants of the Levelt bundle of J\(\pi^*{\mathcal E}\). Using this construction, the author defines the exponents of a linear differential equation with irregular singularities on a compact Riemann surface as the exponents of the meromorphic connection on some holomorphic vector bundle canonically attached to the equation (called the companion bundle of the equation). Finally, the author obtains upper and lower bounds for the degree of the Levelt bundle (which is the sum of the exponents of a connection) as well as the relation for the degree of the companion bundle (which is the sum of the exponents of an equation). The first result generalizes and improves previous results of the author obtained for a holomorphically trivial vector bundle over the Riemann sphere (Fuchs' inequalities) while the second one generalizes the relation for the sum of the exponents of a linear differential equation with irregular singularities on the Riemann sphere obtained by \textit{D. Bertrand} and \textit{G. Laumon} in 1985.