PomeDefinition 18.28.58.A0F71B40AAE1 Take $k\geq 1$ and $\varphi \in \mathrm{PSH}(X,\theta )$, let $\operatorname{Norm}\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$ be the space of norms on the vector space $\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$.
Let $\mathcal{L}_{k,\varphi } \colon \operatorname{Norm}\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )) \to \mathbb R$ be the partial Donaldson functional:
14.62
\begin{equation} \mathcal{L}_{k,\varphi }(H) = \frac{n!}{k^{n+1}} \log \frac{\operatorname{vol}\{ s \in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) : H(s) \leq 1\} }{\operatorname{vol}\left\{ s\in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right) : N^ k_{0,\omega ^ n}(s) \leq 1\right\} }, \end{equation}
where $\operatorname{vol}$ is a Lebesgue measure on $\mathrm{H}^0(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi ))$. The normalization does not matter.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.