Quillen metric for singular families of Riemann surfaces with cusps and compact perturbation theorem

1. Problem Statement

The paper investigates the asymptotic behavior of the Quillen metric on determinant line bundles associated with families of Riemann surfaces that degenerate. Specifically, it addresses the scenario where:

• The fibers of a holomorphic family $\pi: X \to S$ are Riemann surfaces with cusps (punctures).
• The family degenerates, creating singular fibers with ordinary double points (nodes).
• The goal is to understand how the renormalized Quillen metric behaves as one approaches the singular locus in the base $S$, and to relate the metric on the singular fiber to the metric on the normalization of that fiber.

Previous work (by the author and others) established that a renormalized Quillen metric extends continuously over the singular locus. However, the precise geometric interpretation of this extension and the exact universal constants involved in the relationship between the singular and normalized metrics were not fully determined.

2. Methodology

The author employs a combination of complex geometry, spectral theory, and analytic torsion techniques:

• Determinant Line Bundles & Quillen Norms: The study focuses on the determinant line bundle $\lambda(j^*(\xi \otimes \omega_{X/S}(D)^n))$, where $\omega_{X/S}$ is the relative canonical bundle and $D$ represents the divisor of cusps. The Quillen norm is defined as the product of the analytic torsion and the $L^2$-norm.
• Analytic Torsion for Cusped Surfaces: The paper utilizes a specific definition of analytic torsion for surfaces with cusps (from previous work [18]), which involves subtracting a universal contribution from the heat trace to handle the non-compactness of the fibers.
• Degeneration Analysis (Plumbing & Grafting): The author uses the plumbing construction (gluing surfaces via annuli) to model the degeneration. A key technique is the model grafting of metrics (following Wolpert), where the hyperbolic metric near the node is approximated by a "grafted" metric that interpolates between the hyperbolic metric on the smooth parts and a hyperbolic cylinder metric near the node.
• Anomaly Formulas: The proof relies heavily on the Bismut-Gillet-Soulé anomaly formula, which relates the change in the Quillen metric to the change in the underlying metrics and vector bundles via Bott-Chern forms.
• Reduction to Universal Cases: The proof strategy involves reducing the general case to a "universal" local model near the singular points. By constructing specific auxiliary metrics (trivial near the singularity and grafting to hyperbolic metrics), the author isolates the universal constants governing the degeneration.

3. Key Contributions and Results

A. Restriction Theorem (Theorem 1.2)

The main result establishes an explicit formula relating the continuous extension of the Quillen metric on the singular fiber to the Quillen metric on its normalization.

• Setup: Let $\pi: X \to S$ be a family degenerating to a singular fiber $X_0$ with $k$ nodes. Let $\rho: Y_0 \to X_0$ be the normalization.
• Result: Under the canonical isomorphism between the determinant line bundle of the singular fiber and that of the normalization (twisted by specific factors), the norms satisfy:
  $$\|\cdot\|_{L_n}|_{S'} = \exp(k \cdot \text{rk}(\xi) \cdot C_{-n}) \cdot \|\cdot\|_{L'_n}$$
  where $C_{-n}$ is an explicit universal constant depending on $n$ (given by $C_0 = -6\log(\pi)$ and a specific formula for $k \ge 1$).
• Significance: This proves that the Quillen metric is compatible with the adjunction of cusps. It shows that the "loss" of information or the "jump" in the metric during degeneration is entirely captured by a universal constant, independent of the specific geometry of the family.

B. Compatibility with Takhtajan-Zograf (Theorem 1.4)

The paper proves that the author's definition of analytic torsion (based on heat kernel regularization) coincides exactly with the definition proposed by Takhtajan and Zograf (based on the Selberg zeta function) for surfaces with cusps.

• Result: $T(g_{hyp}, \dots) = T_{TZ}(g_{hyp}, \dots)$.
• Significance: This unifies two major approaches to analytic torsion on non-compact hyperbolic surfaces, confirming the consistency of the renormalization procedure used in the paper.

C. Compatibility with Clutching Morphisms (Theorem 1.6)

Applied to the moduli space of pointed stable curves $\overline{\mathcal{M}}_{g,m}$, the result describes the behavior of the Quillen metric under clutching morphisms (gluing maps that create nodes).

• Result: The isomorphisms induced by clutching maps are isometries up to the explicit universal factor $\exp(m \cdot C_{-n})$.
• Significance: This provides a precise description of the Quillen metric's behavior on the boundary of the moduli space, which is crucial for understanding the geometry of the Hodge line bundle over $\overline{\mathcal{M}}_{g,m}$.

D. Refined Compact Perturbation Theorem (Theorem 1.8)

The paper provides an explicit relation between the Quillen metric associated with a cusped metric and one associated with a metric on the compactified surface.

• Result: It gives a formula involving Bott-Chern forms and a regularized integral that relates the two metrics, pinning down the previously unknown universal constant.
• Significance: This refines the relative compact perturbation theorem, offering a concrete tool for calculating metric changes when transitioning between compact and cusped geometries.

4. Technical Details of the Proof

The proof of the main theorem (Theorem 1.2) proceeds in three steps:

1. Trivialization of Coordinates: Using the anomaly formula, the author shows that the behavior of the metric is independent of the specific choice of Poincaré-compatible coordinates near the cusps.
2. Reduction to $m=0$: The problem is reduced to the case where the fibers have no cusps ($m=0$) by modifying the metric near the punctures. This allows the use of results from Bismut regarding smooth families.
3. Universality and Constant Calculation: By constructing a specific "model" family (using plumbing and grafting) and comparing the asymptotics of the Quillen norm for the grafted metric versus the hyperbolic metric, the author calculates the universal constant $C_{-n}$. This involves detailed analysis of the heat kernel expansion and integrals over the degenerating cylinder.

5. Significance and Impact

• Moduli Space Geometry: The results provide a rigorous foundation for the behavior of the Quillen metric (and thus the Hodge metric) on the Deligne-Mumford compactification of the moduli space of curves. This is essential for arithmetic geometry and string theory (where these metrics appear in partition functions).
• Unification of Definitions: By proving the equivalence with the Takhtajan-Zograf definition, the paper bridges the gap between spectral theory (heat kernels) and number-theoretic approaches (zeta functions) in the context of non-compact surfaces.
• Explicit Constants: Unlike many asymptotic results in complex geometry which often leave constants as "universal but unknown," this paper provides explicit formulas for these constants, making the results computationally applicable.
• Generalization: The work extends Bismut's results on smooth families to families with cusps and singular fibers, handling the additional analytic difficulties posed by non-compactness and nodal singularities simultaneously.

In summary, Finski's paper resolves the precise asymptotic behavior of the Quillen metric in degenerating families of cusped Riemann surfaces, providing explicit universal constants and confirming the compatibility of different definitions of analytic torsion.