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The Archimedean height pairing for differential forms on degeneration of Riemann surfaces

Junyu Cao

Abstract

We define the Archimedean height pairing for fiberwise cohomologically trivial differential forms on a one-parameter degeneration of Riemann surfaces, and we study its asymptotic behavior. The proof relies on recent work by Dai--Yoshikawa on the asymptotics of small eigenvalues. As an application, we relate this pairing to the current-valued pairing of Filip--Tosatti, extending their construction to broader geometric settings.

The Archimedean height pairing for differential forms on degeneration of Riemann surfaces

Abstract

We define the Archimedean height pairing for fiberwise cohomologically trivial differential forms on a one-parameter degeneration of Riemann surfaces, and we study its asymptotic behavior. The proof relies on recent work by Dai--Yoshikawa on the asymptotics of small eigenvalues. As an application, we relate this pairing to the current-valued pairing of Filip--Tosatti, extending their construction to broader geometric settings.
Paper Structure (18 sections, 44 theorems, 180 equations)

This paper contains 18 sections, 44 theorems, 180 equations.

Table of Contents

  1. Technical Preparation
  2. Laplacian on singular varieties
  3. Continuity of fiber integral à la Barlet
  4. Convergence on the singular fiber
  5. Smoothness of spectral projection
  6. Spectral geometry of degeneration of algebraic curves
  7. Properties of small eigenvalues
  8. Thick-thin decomposition of nearby fibers
  9. Properties of small eigenfunctions
  10. Estimates of preferred potentials
  11. Estimate of the high frequency part
  12. Estimate of the low frequency (I)
  13. Estimate of the low frequency (II)
  14. Asymptotics of Archimedean height pairing
  15. Application to K3 surfaces with parabolic automorphisms
  16. ...and 3 more sections

Key Result

Theorem 2

There is a constant $c_{\alpha, \beta} \in {\mathbb R}$ such that $\langle \alpha, \beta \rangle - c_{\alpha, \beta} \log\left\lvert s\right\rvert^2$ extends continuously to $S$. In particular, $\langle \alpha, \beta \rangle \in L^\infty(S^\circ)$ if and only if $\langle \alpha, \beta \rangle \in C^

Theorems & Definitions (88)

  • Definition 1: Archimedean height pairing, \ref{['dfn:pairing_of_forms']}
  • Theorem 2: \ref{['thm:continuity_of_pairing_full']}, \ref{['cor:continuity_of_pairing_general_fiber']}
  • Remark 3
  • Remark 4
  • Proposition 5: \ref{['prop:continuity_of_current_val_pairing']}
  • Corollary 6: \ref{['cor:dynamical_consequences']}
  • Remark 7
  • Remark 8
  • Theorem 1.1: Li_Tian_Heat_Kernel_1995*Section 4
  • Remark 1.2
  • ...and 78 more