Holomorphic quadratic differential ($q\in H^0(X,K_X^2)$)

A quadratic differential with holomorphic local coefficients. If it is nonzero, its zeros have total order $4g-4$ on a compact genus-$g$ Riemann surface. Away from its zeros, a nonzero differential has flat coordinates whose changes of coordinate have the form $w\mapsto \pm w+c$.