PomeDefinition 10.6.8.9CC891317C85 A Fubini–Study metric on $L$ is a metric $h_0\mathrm e^{-\varphi }$ with
\[ \varphi =\frac1m\max _{1\leq i\leq N} \bigl (\log |s_ i|_{h_0^ m}^2+\lambda _ i\bigr ), \qquad m,N\in \mathbb Z_{\geq 1},\quad \lambda _ i\in \mathbb Q, \]
where $s_1,\ldots ,s_ N$ are holomorphic sections of $L^ m$, not all zero. We denote the set of these metrics by $\mathrm{FS}(L)$. The set of generalized Fubini–Study metrics is the smallest set of plurisubharmonic metrics that contains $\mathrm{FS}(L)$ and is closed under minima of metrics and under convex combinations of weights with rational coefficients. A convex combination of weights corresponds to the weighted geometric mean of the quadratic metrics, with value zero on the zero vector and the value $+\infty $ on a nonzero vector over a pole of a factor with positive coefficient.
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