a Hermitian metric $G$ on $\mathrm{H}^0(X, L^k)$ and an orthonormal basis $\{s_i\}$ with respect to $G$, one defines the Fubini-Study metric $h= \mathrm{FS}(G)$ on the line bundle $L^k$ by the requirement that $\sum_i |s_i|_{h}^2 = 1$. The result is independent of the orthonormal basis chosen. Now given this metric $h$ on $L^k$ we define a new Hermitian metric on $\mathrm{H}^0(X, L^k)$, denoted $\mathrm{Hilb}(h)$, by $$\| s \|_{\mathrm{Hilb}}^2 = R\int_{X} |s|_h^2 \omega_h^n/n!$$ where $\omega_h$ is the Kähler form $-\sqrt{-1}\partial\overline{\partial}\log h$ and where $R$ is the constant $$R = \frac{\mathrm{dim}\,  \mathrm{H}^0(X, L^k)}{\mathrm{Vol}(X, \omega_h^{n}/n!)}.$$ This defines the $T$ map: $T(G)=\mathrm{Hilb}(\mathrm{FS}(G))$.

The $T_{\nu}$ map is defined analogously, but instead of the volume form $\omega_h^n/n!$ we fix a volume form $\nu$ of our choosing. As above we set $$\| s \|_{\mathrm{Hilb}_{\nu}} = R_{\nu} \int_X |s|_h^2 \nu,$$ where $$R_{\nu} = \frac{\mathrm{dim}\,  \mathrm{H}^0(X, L^k)}{\mathrm{Vol}(X, \nu)}.$$ Then we define $T_{\nu}(G) = \mathrm{Hilb}_{\nu}(\mathrm{FS}(G)).$

The $T_K$ function is defined in case $L^k = K^{-p}$, where $K$ is the canonical bundle. Again we only modify the volume form, this time choosing $$\omega_{G,K} = \left( \sum s_i \otimes \overline{s}_i \right)^{-1/p}.$$ The resulting metric on $\mathrm{H}^0(X, L^k) = \mathrm{H}^0(X, K^{-p})$ is given as above: $$\| s \|_{\mathrm{Hilb}_K} = R_K \int_X |s|_h^2 \omega_{G,K}$$ where 