Kähler–Ricci Flow from Currents with Positive Lelong Numbers

Definitions and mathematical statement

The source asks a qualitative question and does not supply enough hypotheses to determine a unique initial-value problem. The following definitions describe the conventional setting, with the missing qualifications made explicit.

Let $X$ be a complex manifold. In the usual interpretation of Kähler--Ricci flow from currents, the initial datum $T_0$ is a closed positive current of bidegree $(1,1)$. Positivity means that $T_0$ evaluates nonnegatively on compactly supported smooth strongly positive test forms of complementary bidegree. Closedness and bidegree are conventional here; the supplied statement says only "positive current."

Locally write $T_0=(i/\pi)\partial\bar\partial u$, where $u$ is a plurisubharmonic potential. Using this normalization, its Lelong number at $x\in X$ is \[ \nu(T_0,x)=\liminf_{z\to x}\frac{u(z)}{\log\lVert z-x\rVert}, \] computed in local holomorphic coordinates centered at $x$. This number is independent of the coordinates and the choice of local potential. Write \[ E_+(T_0)=\{x\in X:\nu(T_0,x)>0\}. \] The phrase "has positive Lelong numbers" is interpreted here as $E_+(T_0)\ne\varnothing$; the source gives no more specific condition on this locus.

For a smooth family of Kähler forms $\omega(t)$, the unnormalized Kähler--Ricci flow equation is \[ \frac{\partial\omega(t)}{\partial t}=-\operatorname{Ric}(\omega(t)), \qquad \operatorname{Ric}(\omega)=-i\partial\bar\partial\log\det(g_{j\bar k}), \] where $(g_{j\bar k})$ is the associated Hermitian metric matrix. A current-valued evolution starting from $T_0$ would ordinarily have the initial trace \[ T(t)\rightharpoonup T_0\quad\text{as }t\downarrow0, \] meaning convergence against every compactly supported smooth test form of complementary degree. Neither this initial-trace convention nor a definition of weak solution is supplied in the record; the smooth equation alone does not define the Ricci form of an arbitrary singular current.

Source question. What is the behavior of Kähler--Ricci flow with initial positive current $T_0$ having positive Lelong numbers? In the conventional interpretation above, the question concerns weak evolutions with $E_+(T_0)\ne\varnothing$. The source does not specify the ambient manifold, compactness, initial cohomology class, normalized versus unnormalized flow, time interval, weak-solution concept, or a particular behavior to prove. Thus it supplies a research question rather than a uniquely quantified conjecture; the displayed conventions do not resolve those omissions.

Short English statement

The problem asks how Kähler--Ricci flow behaves when its initial current has positive Lelong numbers, which measure local singularity concentration. The source leaves the geometric hypotheses, weak-solution framework, and intended conclusion unspecified, so a unique precise conjecture cannot be extracted without further source clarification.

Sources

• Local UnsolvedMath registry: permanent ID $30002916$, exact source problem code catalogue entry, title "Kähler--Ricci Flow from Currents with Positive Lelong Numbers," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the registry gives proposed year $2015$.
• Supplied report citation: Differentialgeometrie im Großen (2016), Oberwolfach Reports, DOI 10.4171/owr/2015/31, pp. 1759--1807. https://doi.org/10.4171/owr/2015/31. The record describes these reports as documenting mathematical workshops and their open-problem discussions. The title year $2016$, DOI year component $2015$, and proposed year $2015$ are retained as supplied; their relationship is unverified.
• Supplied literature reference, labeled "Di Nezza and Lu primary paper": https://doi.org/10.1016/j.aim.2016.10.011. No paper title or specific result is supplied.
• Supplied literature reference, labeled "Di Nezza Guedj and Lu primary paper": https://arxiv.org/abs/2606.15816. No paper title or specific result is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and reports that weak Kähler--Ricci flow from positive currents is understood in several regularity and uniqueness settings, including divisorial singularities, while the full positive-Lelong-number behavior remains unsettled. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that mathematical content was preserved while notation, typography, or omitted definitions were repaired for self-contained publication. These are supplied editorial assertions, not independent mathematical verification of this draft.