Higher Schwarzians

Abstract

The classical Schwarzian can be computed directly from the coefficients of a second-order linear equation. We extend this coefficient viewpoint to monic operators of arbitrary order over unital associative differential algebras over the rationals. The normalized coefficients freely generate the universal gauge covariants under covariant differentiation. Projective normalization in the scalar case, followed by symmetrization, gives higher Schwarzians with compatible finite laws under changes of frame and parameter. On a complex curve, these invariants, together with the coefficient connection and quadratic datum, reconstruct the operator. On a compact curve of genus $g\ge2$, canonical Szegő kernels turn the construction into a tool for recovering bundle geometry. A bracket of the cubic invariant recovers marked extensions of acyclic line bundles with distinct cubics. Higher invariants through degree $4g-1$ separate acyclic line bundles and detect their self-extensions. For acyclic bundles of arbitrary rank, the full invariant jets at a point generate the associative span of the canonical monodromy. The same universal coefficient calculus gives automorphic tensors on Siegel modular varieties. For genus-two filtered symplectic connections with invertible Kodaira--Spencer map, matrix Schwarzian invariants determine where scalar equations selected by the Hodge filtration acquire apparent singularities. Exact Picard--Fuchs calculations exhibit these divisors and express their equations in Siegel modular forms. The exact calculations are supplied in a reproducible ancillary package.