Open Problems in Pluripotential Theory — Question 11

Definitions and mathematical statement

Let $\pi\colon X\to\mathbb{CP}^{2}$ be the blow-up of the complex projective plane at one point. Thus $X$ is a del Pezzo surface with positive first Chern class. Fix a Kähler form $\omega$ with $[\omega]=c_1(X)$, and a smooth anticanonical divisor $D\subset X$, meaning a smooth effective divisor with $\mathcal O_X(D)\simeq K_X^{-1}$. Write $[D]$ for the current of integration over $D$. Set \[ V:=\int_X\omega^2=\int_D\omega, \qquad \mu_D:=\frac{\omega\wedge[D]}{V}. \] The equality of the two integrals follows from $[D]=c_1(X)=[\omega]$ in cohomology. In particular, $\mu_D$ is a probability measure characterized by \[ \int_X f\,{\,\mathrm{d}}\mu_D=\frac1V\int_D f|_D\,\omega|_D \qquad(f\in C(X)). \] To make the notation explicit, adopt the convention $dd^c=\frac{i}{\pi}\partial\bar\partial$. An $\omega$-plurisubharmonic function is an upper semicontinuous, locally integrable function $u\colon X\to[-\infty,\infty)$, not identically $-\infty$, which is locally the sum of a plurisubharmonic function and a smooth function and satisfies $\omega+dd^c u\geq0$ as a current. Denote this class by $\operatorname{PSH}(X,\omega)$. For bounded $u\in\operatorname{PSH}(X,\omega)$, use the Bedford--Taylor product to define \[ \operatorname{MA}_{\omega}(u):=\frac{(\omega+dd^c u)^2}{V}. \] Here $\operatorname{DMA}(X,\omega)$ is interpreted as the global domain of definition under decreasing bounded approximation: it consists of those $u\in\operatorname{PSH}(X,\omega)$ for which there exists a probability measure $\nu_u$ such that, for every sequence $u_j\in\operatorname{PSH}(X,\omega)\cap L^\infty(X)$ decreasing pointwise to $u$, \[ \operatorname{MA}_{\omega}(u_j)\rightharpoonup\nu_u. \] Set $\operatorname{MA}_{\omega}(u)=\nu_u$ for such $u$.

Question (attributed in the source to Phong). For every choice of $\omega$ and $D$ as above, does there exist $\varphi_D\in\operatorname{DMA}(X,\omega)$ such that \[ \operatorname{MA}_{\omega}(\varphi_D)=\mu_D? \] The supplied excerpt does not define $V$, $\operatorname{DMA}$, the normalization of $\operatorname{MA}$, or the convention for $dd^c$. The definitions above give an explicit standard interpretation requiring confirmation against the source's surrounding definitions. In particular, the operator here is defined by decreasing bounded approximation; it is not being interpreted as the non-pluripolar Monge--Ampère product.

Short English statement

On the projective plane blown up at one point, fix a Kähler form representing the first Chern class and a smooth anticanonical divisor. The question asks whether normalized area measure on that divisor is the Monge--Ampère measure of a potential in the global domain of definition. The operator and normalization conventions made explicit above require source review.

Sources

• Local UnsolvedMath registry, permanent ID $800880$, exact source code catalogue entry, title: Open Problems in Pluripotential Theory --- Question 11. The supplied registry labels this problem open; that status is not independently verified here.
• Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 11. Supplied source URL: https://arxiv.org/abs/1511.00705. The question itself carries the attribution [Phong]; the supplied record does not specify a full name or a separate reference for that attribution.
• Supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance claims, not evidence of independent source consultation for this draft. The supplied upstream revision is d371ee917ac9647fbb06cd6fc529deec6f3dda82.