Bergman-Kernel Asymptotics at Curvature-Vanishing Points

Definitions and mathematical statement

Let $L\to X$ be a holomorphic line bundle with a smooth Hermitian metric $h^L$, and let $R^L$ denote its Chern curvature. Semipositivity means that the real $(1,1)$-form $\sqrt{-1}R^L$ is nonnegative. Let $(F,h^F)\to X$ be a Hermitian holomorphic vector bundle. Once a volume form on $X$ has been specified, the metrics define an $L^2$ inner product on sections of $L^{\otimes p}\otimes F$, for positive integers $p$. In the compact setting, let $P_p$ be the orthogonal projection onto $H^0(X,L^{\otimes p}\otimes F)$, and write $P_p(x,x)\in\operatorname{End}(F_x)$ for its diagonal Bergman kernel, using the natural identification of endomorphisms of the one-dimensional factor $L_x^{\otimes p}$.

Requested higher-dimensional extension. Establish a full pointwise asymptotic expansion of $P_p(x,x)$ as $p\to\infty$ at curvature-vanishing points for semipositive line bundles when $\dim_{{\mathbb{C}}}X>1$. Here a full pointwise expansion means an asymptotic description to arbitrarily high order at each fixed point under consideration. The supplied record does not specify the higher-dimensional hypotheses, the permitted asymptotic powers or other terms, or the remainder estimates. Thus this is a research target requiring further formulation, rather than a completely specified universally quantified conjecture. Compactness is inherited from the motivating surface theorem below; it is not explicitly imposed by the supplied higher-dimensional statement. The volume form and any additional geometric assumptions must also be specified before fixing the kernel normalization.

Supplied motivating surface result. The excerpt states that if $X$ is a compact Riemann surface, $(L,h^L)\to X$ is semipositive with curvature vanishing to finite order at every point, and $(F,h^F)\to X$ is a Hermitian holomorphic vector bundle, then, at each fixed $x\in X$, \[ P_p(x,x)=\sum_{j=0}^{N}c_j(x)p^{(2-2j)/r_x} +O\!\left(p^{-2N/r_x}\right) \qquad(p\to\infty) \] for every truncation index $N$. The excerpt describes the $c_j$ as sections of $\operatorname{End}(F)$, states that $c_0(x)>0$, and says that this leading coefficient is determined by the Bergman kernel of a model Kodaira Laplacian on $T_xX$. Positivity here is positivity of the Hermitian endomorphism. The parameter $r_x$, including its precise relation to curvature vanishing order, is not defined in the supplied excerpt. This formula is recorded as the supplied surface result, not asserted as the desired higher-dimensional formula.

The excerpt also supplies a surface spectral-gap assertion under the same hypotheses, with positive constants $c_1,c_2$ and lower threshold $c_1p^{2/r}-c_2$, but does not define its global parameter $r$ or clearly identify the operator in the extracted text. It reports that a corresponding spectral gap fails in higher dimensions, citing Donnelly's counterexample to Siu's eigenvalue conjecture. Consequently, the surface spectral-gap assertion cannot be silently adopted as a hypothesis-free higher-dimensional tool.

The higher-dimensional target leaves unresolved whether curvature vanishing means $R^L(x)=0$ as a tensor or includes points where a nonzero semipositive curvature form has deficient rank. It also leaves unresolved whether finite-order vanishing is required at every point, only at the point of expansion, or in specified directions, and whether twisting by arbitrary $F$ is intended.

Short English statement

The problem asks for arbitrarily detailed pointwise Bergman-kernel asymptotics at curvature-vanishing points in complex dimension greater than one. A supplied theorem provides the motivating expansion on compact Riemann surfaces with finite-order curvature vanishing. The higher-dimensional geometric assumptions and the form of the expansion remain unspecified, so this draft records the target and its formulation gaps without asserting a new theorem.

Sources

• Local registry: UnsolvedMath permanent ID $30005538$, source code catalogue entry, title Bergman-Kernel Asymptotics at Curvature-Vanishing Points; supplied complete source record. No individual proposer is supplied; the proposed year is 2023.
• Supplied report citation: Hypoelliptic Operators in Geometry (2023), Oberwolfach Reports, pp. 1323--1396, DOI 10.4171/owr/2023/24, https://doi.org/10.4171/owr/2023/24. The record attributes the problem to workshop-report material.
• Supplied literature source, labeled "OWR 2023 24": https://ems.press/content/serial-article-files/47017.
• Supplied literature source, labeled "Marinescu Savale": https://doi.org/10.1007/s00208-023-02750-3. No article title or fuller bibliographic record is supplied.
• The supplied excerpt labels its surface results Corollary 3 and Theorem 4, both attributed there to reference [13], and points to [9, Problem 4.8] for the broader expansion problem. It also mentions [9, Theorem 1.5.5] and methods from [5, 9, 10]. The numbered bibliography is not included, so these labels cannot be fully resolved from the record.
• The excerpt attributes an earlier asymptotic estimate for semipositive line bundles to Berndtsson; an expansion on the positive part and away from the augmented base locus, assuming ampleness, to Berman; and an expansion on the positive part with canonical-bundle twisting $F=K_X$ to Hsiao--Marinescu. It attributes the higher-dimensional spectral-gap counterexample to Donnelly, reference [6]. These are supplied contextual claims without complete individual citations.
• Literature assessment dated 2026-08-22: the record labels the problem open and says that the Riemann surface case is proved while the corresponding higher-dimensional semipositive expansion remains unresolved. This is supplied dated curation, not an independently verified literature assessment. The record expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official report. These are supplied provenance claims; no independent source inspection or mathematical verification is asserted here.