Theorem 11.41.17281EDB812F1

Proof. The Poincaré–Lelong formula gives $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =[\Gamma ]+\beta $, so every function on the right is $\theta $-plurisubharmonic and has the type of $\varphi $. Conversely, let $u\in \mathrm{PSH}(X,\theta )$ with $u\sim \varphi $. Then $u$ has the same generic Lelong numbers as $\varphi $ along the components of $\Gamma $, namely the coefficients of $\Gamma $, and lemma 5.38.1 gives

The locally integrable difference $w=u-\varphi $ therefore has a $\beta $-plurisubharmonic representative. It is bounded off the divisor because $u\sim \varphi $, and the local submean property preserves both bounds across the divisor. Since $u$ and $\varphi +w$ agree almost everywhere, they are equal as quasi-plurisubharmonic functions.

Off the divisor, the currents $\theta +\mathrm{dd}^{\mathrm{c}}(\varphi +w)$ and $\beta +\mathrm{dd}^{\mathrm{c}}w$ agree, and locality identifies their Monge–Ampère products there. The extended measure on the left gives the divisor zero mass, and so does the bounded product on the right, which is finite and charges no analytic set by section 5.28. This proves (11.41.2). In the background $\beta $, the equation with bounded density has a continuous solution and a unique constant by theorem 14.19.2, proposition 14.19.34. The density $\mathrm{e}^ F$ is bounded above and below by positive constants, so corollary 14.24.17 gives uniqueness modulo constants. The correspondence transfers these assertions to the whole ordinary type and identifies the functionals. $\square$