Pome

25.23.2 Open problems8B52D455AC39

We give a list of important open problems in this theory.

We do not repeat the conjectures mentioned in the main text.

Conjecture 25.23.3.F96A00F2C17A Let $X$ be a connected compact Kähler manifold and $Y$ be a submanifold. Fix a Kähler class $\alpha $ on $X$. For each Kähler current $S\in \alpha |_ Y$, we can find a Kähler current $T\in \alpha $ such that

\[ \operatorname{Tr}_ Y(T)\sim _{\mathcal{I}}S. \]
If we formally view $\operatorname{Tr}_ Y$ as an analogue of the trace operator in the theory of Sobolev spaces, then this conjecture corresponds exactly to the Dirichlet problem.

Problem 25.23.4.6DBA7A1F707B Is it possible to extend the definition of the trace operator $\operatorname{Tr}_ Y$ to the case where the ambient variety is only unibranch?

The difficulty lies in the lack of Demailly type regularization theorems.

Problem 25.23.5.6BD8E77C1AAF Is there a natural definition of the transcendental Okounkov body of a closed positive $(1,1)$-current $T$ with $0$-mass so that its dimension is equal to the numerical dimension of $T$?

See [Cao14] for the definition of the numerical dimension of a current. The following two problems are proposed by Witt Nyström.

Problem 25.23.6.0C91C5C2794A Consider a compact Kähler manifold $X$ and a connected submanifold $Y$. We have defined the trace operator $\operatorname{Tr}_ Y$ from a subset of $\mathrm{QPSH}(X)/\sim _\mathcal {I}$ to $\mathrm{QPSH}(Y)/\sim _\mathcal {I}$. Is it possible to refine this operator to one from a subset of $\mathrm{QPSH}(X)/\sim _ P$ to $\mathrm{QPSH}(Y)/\sim _ P$?

Problem 25.23.7.7FAD0FAF17B7 Consider a connected compact Kähler manifold $X$ of dimension $n$ and a smooth flag $Y_{\bullet }$ on $X$. Consider a closed smooth real $(1,1)$-form $\theta $ on $X$ representing a big cohomology class and $\varphi \in \mathrm{PSH}(X,\theta )$ with $\int _ X \theta _{\varphi }^ n>0$.

Can one define a refined notion of partial Okounkov bodies $\Delta '_{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ contained in $\Delta _{Y_{\bullet }}(\theta +\mathrm{dd}^{\mathrm{c}}\varphi )$ with volume given by $\frac{1}{n!}\int _ X \theta _{\varphi }^ n$?

Note that a satisfactory solution to the latter problem is not very likely, as can be easily seen from examples on $\mathbb {P}^1$. We also look for generalizations of our theory to more general settings.

Problem 25.23.8.856DCCA91F34 To what extent can the results in the current book be generalized to the non-Kähler setting?

The non-pluripolar products in the non-Kähler setting were recently studied by Boucksom–Guedj–Lu in [BGL24]. See also the references therein.

Problem 25.23.9.18561C2CB4F4 To what extent can the results in the current book about closed positive $(1,1)$-currents be generalized to closed positive currents of higher bidegree?

A fundamental issue is the lack of a strong enough Demailly type approximation for general currents. The regularization theorem of Dinh–Sibony [DS04] seems too weak for our purposes.
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