Technical Summary: Asymptotics of Small Eigenvalues on Degenerations of Kähler Manifolds

Problem Statement

The paper investigates the asymptotic behavior of the small eigenvalues of the Laplacian operator on the smooth fibers $(X_s, \omega_s)$ of a one-parameter degeneration of compact Kähler manifolds $\pi: X \to S$ as the parameter $s$ approaches the singular fiber $X_0$.

Specifically, let $X_0$ be the unique singular fiber. After a suitable base change and normalization, the central fiber decomposes into a reduced divisor $Z = \bigcup Z_j$ with $N$ irreducible components. While the smooth fiber $X_s$ is connected (yielding exactly one zero eigenvalue), the limiting central fiber $Z$ has $N$ connected components, resulting in $N$ zero eigenvalues in the limit. Consequently, exactly $N-1$ eigenvalues of the Laplacian on $X_s$, denoted $\lambda_1(s) \leq \dots \leq \lambda_{N-1}(s)$, must converge to zero as $s \to 0$. These are termed "small eigenvalues."

The central problem is to determine the precise rate at which these eigenvalues vanish. While Yoshikawa established the continuity of the spectrum, the exact asymptotic rate for higher-dimensional degenerations, particularly those with non-reduced or reducible singular fibers, remained an open question. Previous work by Dai and Yoshikawa addressed this for Riemann surfaces (dimension $n=1$) assuming reduced singular fibers, utilizing analytic torsion. This paper aims to generalize these results to arbitrary dimensions $n$ and more general singular fibers.

Methodology

The author employs a combination of complex-analytic techniques to overcome the lack of uniform lower bounds on the Ricci curvature, which typically blows up as $s \to 0$. The proof strategy is divided into establishing a lower bound and an upper bound for the small eigenvalues.

1. Lower Bound: Rescaled Uniform Skoda Estimates and Auxiliary Monge-Ampère Equations

To establish the lower bound, the paper avoids classical Riemannian estimates (which fail due to curvature blow-up) and instead utilizes a purely complex-analytic route:

• Rescaled Uniform Skoda Inequality: The author adapts Li's recent uniform Skoda inequality for plurisubharmonic functions to the setting of degenerating Kähler manifolds. A key step involves rescaling the relative Kähler form by a factor of $|\log|s||^{-1}$ to control the volume growth near the singular fiber.
• Method of Auxiliary Monge-Ampère Equations: Following the framework of Guo–Phong–Song–Sturm–Tong and Guedj–Tô, the author constructs auxiliary complex Monge-Ampère equations. These equations provide uniform $L^\infty$ estimates for solutions to Poisson equations.
• Synthesis: By combining the rescaled Skoda estimate with the Laplacian estimates derived from the auxiliary Monge-Ampère method, the author derives a uniform bound on the $L^\infty$ norm of the eigenfunctions relative to the eigenvalue, leading to the lower bound estimate.

2. Upper Bound: Quantitative Retraction and Test Functions

To establish the upper bound, the paper generalizes the test-function construction used by Dai and Yoshikawa for Riemann surfaces:

• Quantitative Retraction: The author constructs a family of diffeomorphisms $F_s$ that retract the smooth part of the singular fiber $Z_{reg}$ onto the nearby smooth fibers $X_s$. This construction is made quantitative, providing precise estimates on the distance distortion and the convergence of the pulled-back metric and volume forms.
• Test Functions on the Central Fiber: Using the local structure of the simple normal crossing (snc) singular fiber, the author constructs a family of smooth test functions supported on each irreducible component of the central fiber. These functions are designed to be constant on the bulk of each component and transition smoothly to zero near the singular locus.
• Min-Max Principle: These test functions are pushed forward to the smooth fibers $X_s$ via the retraction $F_s$. By applying the min-max principle to the resulting orthonormal system, the author derives the upper bound for the eigenvalues.

Key Results

Main Theorem

The paper establishes that the small eigenvalues decay at a logarithmic rate. Specifically, if $N \geq 2$ (i.e., the singular fiber has at least two irreducible components contributing to the volume), there exist constants $C_1, C_2 > 0$ such that for all $0 < |s| \leq 1/2$:
$$C_1 |\log|s||^{-1} \leq \lambda_1(s) \leq \dots \leq \lambda_{N-1}(s) \leq C_2 |\log|s||^{-1}$$
This result holds for general degenerations, including those with non-reduced and reducible singular fibers, generalizing the Dai–Yoshikawa result for Riemann surfaces.

Technical Contributions

1. Generalization to Higher Dimensions: The paper resolves the question of small eigenvalue asymptotics for Kähler degenerations in arbitrary complex dimensions $n$, removing the dimension restriction of previous works.
2. Relaxation of Singularity Assumptions: Unlike the Dai–Yoshikawa result which assumed reduced singular fibers, this work handles general singular fibers (including non-reduced cases) by utilizing semistable reduction and analyzing the number of irreducible components with positive volume.
3. New Analytic Tools: The paper successfully adapts the method of auxiliary Monge-Ampère equations and Li's uniform Skoda inequality to the degeneration setting, providing a robust alternative to analytic torsion methods which are specific to dimension one.

Applications and Significance

Geometric Analysis Estimates

The precise control over the small eigenvalues allows for refined estimates in geometric analysis:

• Green Function Bounds: The author derives a lower bound for the fiberwise Green function $G_s(x, y)$ of the form $G_s(x, y) \geq -C |\log|s||$. This bound is crucial for analyzing families of plurisubharmonic functions and solutions to Poisson equations on degenerating families.
• Poisson Equation Estimates: The results yield uniform estimates for the solutions of fiberwise Poisson equations $\Delta_{\omega_s} \phi_s = g$, showing that $\|\phi_s\|_{L^\infty} \leq C |\log|s|| \|g\|_{L^\infty}$.

Examples in Canonical Metrics

The paper discusses the phenomenon in specific geometric contexts:

• Calabi-Yau Degenerations: The behavior of eigenvalues depends on the dimension of the essential skeleton. If the skeleton has dimension $\geq 1$, the diameter grows as $|\log|s||^{1/2}$, leading to small eigenvalues of order $|\log|s||^{-1}$.
• Hyperbolic Curves: For degenerations of hyperbolic curves, the results recover the known $N-1$ small eigenvalues of order $|\log|s||^{-1}$.
• Fano Manifolds: The paper notes that for Fano manifolds, the Lichnerowicz theorem ensures a uniform positive lower bound for the first non-zero eigenvalue, meaning no small eigenvalues occur.

Non-Archimedean Perspective

The author proposes a conjectural link between the complex-analytic spectral behavior and the non-Archimedean limit (hybrid convergence). The paper suggests that the $N-1$ small eigenvalues correspond to the positive eigenvalues of a weighted graph Laplacian defined on the dual graph of the central fiber, where the weights are determined by the limiting non-Archimedean Monge-Ampère measures. This provides a spectral convergence principle where the rescaled eigenvalues $|\log|s|| \lambda_k(s)$ are expected to converge to the eigenvalues of the graph Laplacian.

Conclusion

The paper provides a complete characterization of the asymptotic rates of small eigenvalues for degenerations of compact Kähler manifolds. By combining rescaled Skoda inequalities with auxiliary Monge-Ampère equations, it establishes the optimal logarithmic decay rate $|\log|s||^{-1}$ for all dimensions and singular fiber types. This work not only generalizes previous results on Riemann surfaces but also opens the door to further applications in geometric analysis and non-Archimedean geometry.