Higgs-Demailly System and Positivity of Higgs Bundles

Abstract

We prove that the Higgs--Demailly system on a compact Riemann surface admits, for suitable fixed parameters, a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. Here H-ampleness is understood in the modified sense of Biswas--Misra--Ray. In particular, the argument gives an analytic proof of the equivalence between this H-ampleness and Griffiths-positive Hitchin--Simpson curvature. The proof extends the Demailly--Pingali--Murakami approach using a priori estimates and Leray--Schauder degree theory. The missing scalar lower bound follows from a Higgs-compatible quotient construction: a blow-up sequence produces a nonzero Higgs quotient of nonpositive degree, contradicting H-ampleness.