Example 13.2.96F4C920FEDCA

Example 13.2.9.6F4C920FEDCA Let $U=\{ (x,y)\in \mathbb C^2:|x|^2<1/2,\ |y|<\mathrm e^{-1}\} $, put $L=-\log |y|$, and define

On $U\setminus \{ y=0\} $ we have $L>1$, the function $u$ is smooth, and its complex second derivatives are

Since $u_{x\bar x}>0$ and $\det (u_{j\bar k})>0$, the complex Hessian is positive definite, so $u$ is plurisubharmonic off the axis $\{ y=0\} $ (proposition 5.9.2(1)). Moreover $u<-\sqrt L/2$ is bounded above and tends to $-\infty $ at the axis. By theorem 5.11.1, the plurisubharmonic extension across the axis takes the value $\varlimsup u=-\infty $ there, which is the displayed value; hence $u$ is plurisubharmonic on $U$.

Off the axis, $(\mathrm{dd}^{\mathrm{c}}u)^2$ is the smooth measure with density a fixed positive multiple of $\det (u_{j\bar k})$, by (7.2.4). The sets $O_ k=\{ u>-k\} $ are open, since $u$ is continuous as an extended-real function, and they increase to $U\setminus \{ y=0\} $; on $O_ k$ the truncation $\max \{ u,-k\} $ equals $u$, so $S_ k=\mathbf1_{O_ k}(\mathrm{dd}^{\mathrm{c}}u)^2$. Take $K=\{ |x|^2\leq 1/4,\ |y|\leq r\} $ with $0<r<\mathrm e^{-1}$. On $K$ we have $1-2|x|^2\geq 1/2$, and polar coordinates in $y$ give

Hence $\sup _ k\int _ KS_ k=\int _{K\setminus \{ y=0\} }(\mathrm{dd}^{\mathrm{c}}u)^2=\infty $, and (13.2.3) fails. Thus the non-pluripolar product of an arbitrary plurisubharmonic function need not be locally finite.