Weighted Heights and GIT Heights

We investigate the relationship between Geometric Invariant Theory (GIT) heights and weighted heights, with a focus on their interaction in weighted projective spaces and their application to binary forms. Building on the weighted height framework developed in previous papers, we relate it to Zhang’s GIT height via the Veronese map. For a semistable cycle, we show that the GIT height decomposes into the logarithmic weighted height plus an Archimedean correction from the Chow metric.