Higgs-Demailly System and Positivity of Higgs Bundles

Abstract: We prove that the Higgs-Demailly system on a compact Riemann surface admits a smooth admissible solution at its terminal parameter if and only if the Higgs bundle is H-ample. This gives an independent analytic characterization of H-ampleness by 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.