Explicit projective-space solution to item 9 of the 2016 complex Monge-Ampere list

Definitions and mathematical statement

This permanent record contains the entire thirteen-item workshop list, not a single Monge--Ampère question. Fix a normalization of $dd^c$; on a compact Kähler background $\omega$, distinguish $(\omega+dd^cu)^n$ from the local expression $(dd^cu)^n$. The source does not consistently state backgrounds, solution classes, or convergence topologies. Its recovered problems are:

1. Monge--Ampère mass on analytic sets (S. Dinew): seek a suitable definition for singular potentials, asking independence of bounded decreasing approximations with the same limit, and asking whether the weak limit for $u_j=\max(u,-j)$ defines $\operatorname{MA}(u)$. Seek $\operatorname{MA}(u)=\mu$ when $\mu$ is supported on an analytic set other than a point, possibly with prescribed Green-type singularities. Ask similar Dirichlet questions on $\Omega\Subset{\mathbb{C}}^n$, real/toric analogues, and the following degeneration questions. For a Fano $X$ with no KE metric, can normalized continuity-method potentials $\psi_t=\varphi_t-\sup\varphi_t$ have a subsequential limit with defined MA measure supported on the associated multiplier-ideal locus (example: ${\mathbb{P}}^2$ blown up at one point)? For Fano $X$ and smooth anticanonical $D$, study a subsequential limit of $\omega_\varepsilon$ satisfying $\operatorname{Ric}(\omega_\varepsilon)=\varepsilon\omega_\varepsilon+(1-\varepsilon)[D]$, and whether a suitably defined limiting Ricci current is supported on $D$ (H. Guenancia). The limiting parameter and singular-current conventions are unstated. The phrase "compact Kähler case" also gives ${\mathbb{P}}^n\setminus H$, which is not itself compact; the intended ambient/background setting needs clarification. Standard non-pluripolar MA products do not charge analytic pole sets, so they cannot be silently chosen as the requested charged-mass definition.
2. Find non-product smooth solutions of $(dd^cu)^n=0$ on a compact $M$, for example ${\mathbb{P}}^2$, without assuming $u$ psh (Y. Rubinstein). "Non-product" is undefined.
3. Find psh solutions of $(dd^cu)^n=0$ on ${\mathbb{C}}^n$ (W. He). Constants and lower-variable examples make the literal question too weak without further growth, nontriviality, or exhaustion conditions.
4. On a strongly pseudoconvex $\Omega$ with $C^{3,1}$ boundary, seek an analytic proof independent of Krylov of $u\in C^{1,1}(\Omega)$ for the intended Dirichlet solution $(dd^cu)^n=f\,dV$, $f\geq0$, $f^{1/n}\in C^{1,1}$, boundary value $\varphi\in C^{3,1}$.
5. Ask the same with $f^{1/(n-1)}\in C^{1,1}$, in dimension $n\geq2$. The source says this is not covered by Krylov and records Pliś's affirmative special case for the ball with zero boundary value. These are historical qualifications, not a current complete status assessment.
6. Can Ross--Witt-Nystrom's HCMA examples without a foliation yield a counterexample to the maximal-rank question (M. Paun)? The exact maximal-rank statement is absent.
7. Solve $(\pi^*\omega+i\partial\overline\partial u)^{n+1}=0$ on $X\times A$, $\pi$ the projection and $A$ an annulus, with boundary data $\varphi_0,\varphi_1$ on the circles of radii $1,e$, these data $\omega$-psh with unspecified regularity, $\omega$ possibly degenerate, and $\int_X\omega^n>0$ (E. Di Nezza).
8. Find a PDE proof of Kołodziej's $L^\infty$ estimate and the optimal constant for a ball (Z. Blocki); the data and normalization are unstated.
9. Find a KE Fano $X$ and $u\in\Lambda_1$ with $\int_Xu^3\omega_{\mathrm{KE}}^n\ne0$ (H. Macbeth). The eigenspace/Laplacian convention must be specified.
10. Does $(dd^cu)^n=1$ together with $\Delta u\in L^{n(n-1)}$ imply $u\in C^\infty$ (T. Collins)? The source omits the solution class and interior/boundary scope.
11. Seek a complex analogue of Pogorelov's estimate; no precise inequality is given.
12. For $p:X\to D$, $X$ Kähler and $K_X$ nef, study solutions on $X_t$ of $\operatorname{Ric}(\omega_{t,\varepsilon})=-\omega_{t,\varepsilon}-\varepsilon\beta_t$ (M. Paun). The base, $\beta_t$, degeneration, and desired estimates are unspecified.
13. Find an analytic proof of the ACC conjecture/theorem (T. Collins); the invariant and precise ACC statement are not identified.

The report's explicit candidate concerns only item 9. On ${\mathbb{P}}^m$, $m\geq2$, normalize the Fubini--Study KE form by $\operatorname{Ric}(\omega)=\omega$, and use the first positive complex-Laplacian convention $\Delta u=u$. It asserts that \[ u([Z])=\frac{|Z_0|^2}{\sum_{j=0}^m|Z_j|^2}-\frac1{m+1}\in\Lambda_1, \qquad \frac{\int_{{\mathbb{P}}^m}u^3\omega^m}{\int_{{\mathbb{P}}^m}\omega^m} =\frac{2m(m-1)}{(m+1)^3(m+2)(m+3)}>0. \] For $m=2$ this equals $1/135$. The report also claims a divisorial truncation formula relevant to item 1, a proper-exhaustion obstruction for item 3, and a conservative status map for all thirteen items; none of those formulas or hypotheses is supplied, so no complete additional result is reconstructed.

Short English statement

The record merges thirteen Monge--Ampère workshop questions and must retain them all. The report's explicit projective-space moment formula addresses item 9 only; the many omitted definitions and hypotheses in the remaining items are recorded for review.

Sources

• Complex Monge--Ampère Equation Workshop: Open problems, edited by M. Dellatorre, September 8, 2016: https://aimath.org/pastworkshops/mongeampereproblems.pdf. The supplied verification says it is a two-page thirteen-item list; no PDF or report was accessed here.
• Local canonical registry aim-analysis-notes.json, notes[58], tag problem; permanent UnsolvedMath ID $20000527$, source code catalogue entry. The extraction merges all items, makes item 5 "15" by joining a footer, loses the fractional exponents and $\varepsilon$, and appends page 2. The clean statement is absent.
• Source attributions: S. Dinew, H. Guenancia, Y. Rubinstein, W. He, M. Paun, E. Di Nezza, Z. Blocki, H. Macbeth, and T. Collins; source comparisons name Krylov, Pliś, Ross--Witt-Nystrom, Kołodziej, and Pogorelov. No full bibliographic entries or precise ACC reference are supplied. Original remarks and literature field are empty.
• Supplied research attempt 1 proposes the projective-space family and mentions additional truncation/exhaustion results. It is machine generated, not peer reviewed, and requires independent verification before citation.