Theorem 3.5 . [02E1]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

Theorem 3.5.

Let $X$ be projective algebraic complex manifold, $\omega_{0}$ a smooth semi Kähler form that is Kähler outside a complex subvariety $S\subset X$ , and fix $\Omega$ be a Kähler form on $X$ . Assume that $\omega_{o}^{n}=D\Omega^{n}$ , where $D^{-\varepsilon}$ is in $L^{1}(\Omega^{n})$ , and that $[\omega_{0}],[\Omega]\in NS_{\mathbb{R}}(X)$ .

Let $s_{1},...,s_{p}$ (resp. $t_{1},...,t_{q}$ ) be holomorphic sections of some line bundle $L$ (resp $L^{\prime}$ ) on $X$ . Fix $k\in\mathbb{R}_{\geq 0}$ , $l\in\mathbb{R}_{\geq 0}$ and $F\in{\mathcal{C}}^{\infty}(X,\mathbb{R})$ . Assume that

\int_{X}\frac{1}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}\Omega^{n}<\infty\text{ and }\int_{X}\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}=\int_{X}\Omega^{n}.

Then the unique continuous function $\varphi\in PSH(X,\omega_{0})$ such that

(\omega_{0}+dd^{c}\varphi)^{n}=\frac{|s_{1}|^{2k}+\ldots+|s_{p}|^{2k}}{|t_{1}|^{2l}+\ldots+|t_{q}|^{2l}}e^{F}\Omega^{n}\,\text{ and }\sup_{X}\varphi=-1

is smooth outside $B=S\cup\cap_{i}\{s_{i}=0\}\cup\cap_{i}\{t_{i}=0\}$ .