A bound on the multiplicity of horizontal sections for a connection on a Riemann surface (Q2576898)

Consider a compact Riemann surface \(T\) endowed with a holomorphic vector bundle \(V\) of rank \(n\), together with a meromorphic connection \(\nabla\). The latter possesses a finite set of poles, denoted by \(\Sigma\subseteq T\). This setting provides a collection of linear differential systems on \(T\): for any trivializing open set \(U\) of the vector bundle \(V\), the connection \(\nabla\) is represented by a matrix flux of meromorphic 1-forms on \(U\). Consider a horizontal section \(s\) for \(\nabla\), away from the singular locus \(\Sigma\). Take a meromorphic section \(q_0\) of the dual bundle \(V^\ast\), with polar set \(\Sigma'\). The purpose is to provide information on the multiplicity at a point \(t_0\) in the complement of \(\Sigma\cup \Sigma'\) of the natural pairing \((q_0,s)\). Note that globally, \((q_0,s)\) is a meromorphic function on \(T\setminus\Sigma\). If \(t_0\in T\) is neither a singularity of \(\nabla\) nor a pole of \(q_0\) then \((q_0,s)\) is holomorphic at \(t_0\). Define the degree of the meromorphic section \(q_0\) by: \(\text{deg}\,q_0 = -\sum_{t\,\text{pole of }q_0}\text{ord}_t\,q_0\). The author produces an explicit upper bound on the order of its Taylor expansion at \(t_0\). Geometrically, this order measures the contact between the integral curve \(t\mapsto s(t)\) of a non-autonomous linear vector field and a linear hyperplane of \(\mathbb C^n\), whose coefficients vary meromorphically with \(t\). Suppose the monodromy representation associated to \((V,\nabla)\) is irreducible. Let \(t_0\in T\setminus(\Sigma\cup\Sigma')\). Then \(\text{ord}_{t_0}(q_0,s)\leqslant n - 1 + n \cdot \text{deg}\,q_0 +\frac{n(n-1)}{2} \cdot [\text{deg}\,\nabla -\chi(T) + p]-c_1(\Lambda^nV).\) The question turns out to be of a global nature: indeed, this estimate involves total order of the poles, rank of the vector bundle, Chern class of the determinant bundle, Euler characteristic of the Riemann surface. It does not depend on the particular choice of horizontal section, nor on the coefficients of the connection matrices. Note that this statement does not assume any restriction on the singularities of \(\nabla\). On the other hand, if we allow arbitrary monodromy representation, then we should prescribe the type of singularities for \(\nabla\). Under the regularity assumption, the author arrives at the same upper bound.