Optimal Hermitian Metrics for Random Holomorphic Sections

Definitions and mathematical statement

Fix a holomorphic line bundle $L\to M$. The source asks whether some Hermitian metric minimizes the expected number of critical points of a random holomorphic section, and whether a minimizing metric is unique. It does not specify the base manifold, admissible metrics, probability ensemble or critical-point convention. These data are necessary despite the supplied verification calling the sentence internally complete.

For a chosen admissible metric class $\mathcal H$ and laws $\gamma_h$ on a section space, write schematically \[ \mathcal N(h)={\mathbb{E}}_{s\sim\gamma_h}\#\operatorname{Crit}_h(s). \] The existence target is \[ \exists h_*\in\mathcal H:\quad \mathcal N(h_*)=\inf_{h\in\mathcal H}\mathcal N(h), \] with uniqueness in an explicitly specified equivalence or normalization class. One common interpretation takes $M$ compact complex, a suitable positive line bundle, a Gaussian law from an $L^2$ inner product, and Chern-connection critical points $\nabla^h s=0$. Those are possible conventions, not extra hypotheses asserted to occur in the supplied report. The volume form, its dependence on $h$, section-space finiteness and conditions making critical counts finite and isolated must be fixed. Real critical points of a section norm are not silently identified with Chern critical points.

In the standard Chern/Gaussian model, constant rescaling of a metric leaves its Chern connection unchanged and rescales the Gaussian section by a scalar, which does not change the critical set. Literal metric uniqueness therefore needs a scale normalization or quotient, and possibly a specified treatment of bundle/base automorphisms. The record does not give those conventions; it is not claimed to prove unique metrics without them.

The supplied literature triage, checked 21 August 2026, reports asymptotic minimization in important settings, including by Calabi extremal metrics when they exist. It explicitly leaves existence and uniqueness of an exact minimizer for a fixed line bundle unresolved in full generality. An asymptotic statement, such as one for large tensor powers, is not an exact minimization theorem for the fixed bundle. No full expansion, theorem hypotheses or uniqueness result is supplied, and no general affirmative or negative answer is asserted by this draft.

Short English statement

The question needs a fixed random-section model and a normalization for metrics before existence or uniqueness is precise. The dated review reports asymptotic minimization results, while the exact fixed-bundle problem remains unresolved in its stated generality.

Sources

• Local UnsolvedMath registry, permanent ID 30000010, source code catalogue entry, supplied Oberwolfach Reports record; updated 16 September 2026.
• Supplied citation: Multiplier Ideal Sheaves in Algebraic and Complex Geometry (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/21, pp. 1108--1166: https://doi.org/10.4171/owr/2004/21. The proposed year is separately 2004.
• Supplied label "critical points and extremal metrics": https://arxiv.org/abs/math/0406089.
• Supplied label "metric dependence and asymptotic minimization": https://arxiv.org/abs/0802.1549.
• Dated literature triage checked 21 August 2026, derived from the supplied OWR curation; independently verify time-sensitive status and cited claims. No full author/theorem entries are supplied for the two references.