The good analytic properties of semi-positive metrics allow to extend the definition of the local height pairing to the case of admissible line bundles. Indeed, when one approximates uniformly a semi-positive line bundle by a sequence of smooth semi-positive line bundles, one can prove that the corresponding sequence of local height pairings converges, the limit being independent on the chosen approximation.

The proof is inspired by Zhang’s proof of the global case in [59] and goes by induction. Let us consider, for each $j$, two smooth semi-positive metrics on the line bundle $L_{j}$ and assume that they differ by a factor $e^{-h_{j}}$. Then, the corresponding local height pairings differ from an expression of the form

$$\sum_{j=0}^{k}\int_{\mathrm{Z}}h_{j}c_{1}(\overline{L}_{0})\dots\widehat{c_{1}(\overline{L}_{j})}\dots c_{1}(\overline{L}_{k}),$$

where the written curvature forms are associated to the first metric for indices $<j$, and to the second for indices $>j$. This differential forms are positive by assumption, so that the integral is bounded in absolute value by

$$\sum_{j=0}^{k}\left\|{h_{j}}\right\|_{\infty}\int_{\mathrm{Z}}c_{1}(\overline{L}_{0})\dots\widehat{c_{1}(\overline{L}_{j})}\dots c_{1}(\overline{L}_{k})=\sum_{j=0}^{K}\left\|{h_{j}}\right\|_{\infty}(c_{1}(L_{0})\dots\widehat{c_{1}(L_{j}))}\dots c_{1}(L_{k})|{\mathrm{Z}}),$$

where the last expression is essentially a degree. (In these formulae, the factor with a hat is removed.) This inequality means that on the restriction to the space of smooth semi-positive metrics, with the topology of uniform convergence, the local height pairing is uniformly continuous. Therefore, it first extends by continuity. on the space of continuous semi-positive metrics, and then by multilinearity to the space of admissible metrics.