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 Refs. [1, 10], we relate it to Zhang’s GIT height via the Veronese map. For a semistable cycle \( \mathcal {Z}\subset \mathbb {P}_{\mathfrak {w}, \overline{\mathbb {Q}}}^N \) , we show that the GIT height decomposes into the logarithmic weighted height plus an Archimedean correction from the Chow metric. Specializing to degree- \( d \) binary forms \( f \in V_d \) , we define an invariant height \( \mathfrak {h}^{{{\,\mathrm{{\mathfrak {C}}}\,}}}(f) \) with respect to the Chow metric and prove that the moduli weighted height \( {{\,\mathrm{{{\,\textrm{h}\,}}_{\mathfrak {w}}}\,}}(\xi (f)) \) of \( f \) ’s invariants satisfies \( {{\,\mathrm{{{\,\textrm{h}\,}}_{\mathfrak {w}}}\,}}(\xi (f)) = \frac{1}{[K:\mathbb {Q}]} \, \mathfrak {h}^{{{\,\mathrm{{\mathfrak {C}}}\,}}}(f) + {{\,\textrm{h}\,}}_{{{\,\mathrm{{\mathfrak {C}}}\,}}}(f) \) thereby connecting GIT stability, moduli theory, and arithmetic complexity in a unified framework.
