PomeDefinition 18.28.46.C3C647253911 For any $\varphi \in \mathrm{PSH}(X,\theta )_{>0}$, we introduce the partial Bergman kernels of $\varphi $ (with respect to $(K,v)$) as follows: For any $k\geq 1$, we introduce
14.50
\begin{equation} \begin{aligned} B^ k_{v,\varphi , \mu }(x) \coloneqq \sup \left\{ h^ k \otimes h_ T(s,s)\mathrm{e}^{-kv}(x):s \in \mathrm{H}^0\left(X,L^ k \otimes T \otimes \mathcal{I}(k\varphi )\right),\right.\\ \left. N^ k_{v,\mu }(s) \leq 1 \right\} ,\quad x\in K. \end{aligned} \end{equation}
We extend $B^ k_{v,\varphi , \mu }$ to the whole $X$ by setting it to be $0$ outside $K$.
The partial Bergman measures of $\varphi $ (with respect to $(K,v)$) are defined as
14.53
\begin{equation} \beta ^ k_{v,\varphi ,\mu } \coloneqq \frac{n!}{k^ n} B^ k_{v,\varphi ,\mu } \, \mathrm{d}\mu \end{equation}
for each $k\geq 1$.
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.