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Tunneling, the Quillen metric and analytic torsion for high powers of a holomorphic line bundle

Robert J. Berman

TL;DR

The work analyzes tunneling phenomena in the asymptotic Dolbeault complex for high tensor powers $kL$ of an ample line bundle. By combining Bergman kernel asymptotics with variational identities, it derives a precise large-$k$ description of truncated analytic torsions and the Quillen determinant, showing that their limits are governed by the pluripotential energy $\mathcal{E}(P\phi_L,\phi_L)$ and that exponentially small eigenvalues necessarily appear when this energy is nonzero. The results provide a new proof of Quillen metric asymptotics in a non-Kähler setting and connect spectral tunneling to determinant-line geometry, with parallels drawn to Witten Laplacians and large deviation principles for fermions. The paper also discusses the distinction between small and exponentially small eigenvalues and outlines conjectures about limiting measures and localization phenomena in the Dolbeault context, offering a roadmap for future investigations in complex geometry and semi-classical analysis.

Abstract

Let L be a line bundle over a compact complex manifold X and endow L and TX with Hermitian metrics. Our main result provides a formula for the average distribution of the exponentially small eigenvalues of the corresponding Dolbeault Laplacians associated to high tensor powers of L; which in physics terminology is a measure of "tunneling" of the Dolbeault complex. Along the way a new proof of the asymptotics of the induced Quillen metric on the corresponding determinant line is obtained. A brief comparison with the tunneling effect for Witten Laplacians and large deviation principles for fermions is also made.

Tunneling, the Quillen metric and analytic torsion for high powers of a holomorphic line bundle

TL;DR

The work analyzes tunneling phenomena in the asymptotic Dolbeault complex for high tensor powers of an ample line bundle. By combining Bergman kernel asymptotics with variational identities, it derives a precise large- description of truncated analytic torsions and the Quillen determinant, showing that their limits are governed by the pluripotential energy and that exponentially small eigenvalues necessarily appear when this energy is nonzero. The results provide a new proof of Quillen metric asymptotics in a non-Kähler setting and connect spectral tunneling to determinant-line geometry, with parallels drawn to Witten Laplacians and large deviation principles for fermions. The paper also discusses the distinction between small and exponentially small eigenvalues and outlines conjectures about limiting measures and localization phenomena in the Dolbeault context, offering a roadmap for future investigations in complex geometry and semi-classical analysis.

Abstract

Let L be a line bundle over a compact complex manifold X and endow L and TX with Hermitian metrics. Our main result provides a formula for the average distribution of the exponentially small eigenvalues of the corresponding Dolbeault Laplacians associated to high tensor powers of L; which in physics terminology is a measure of "tunneling" of the Dolbeault complex. Along the way a new proof of the asymptotics of the induced Quillen metric on the corresponding determinant line is obtained. A brief comparison with the tunneling effect for Witten Laplacians and large deviation principles for fermions is also made.

Paper Structure

This paper contains 13 sections, 8 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Setup
  3. Statement of the main results
  4. Proofs of the main results
  5. Variational formulae
  6. Asymptotics
  7. Proof of Theorem \ref{['thm:quillen']}
  8. Proof of Theorem \ref{['thm:analyt torsion for ample']}
  9. Proof of Corollary \ref{['cor:exist of small eig']}
  10. Further remarks on tunneling and outlook
  11. Comparison with the tunneling effect for Witten Laplacians and large deviations for fermions
  12. "Small" vs. "exponentially small" eigenvalues for the Dolbeault complex
  13. Discussion and outlook

Key Result

Theorem 1.1

Let $L\rightarrow X$ be an ample line bundle equipped with a smooth Hermitian metric $h_{L}$ and let $h_{X}$ be a fixed smooth Hermitian metric on $X$ (which is not assumed to be Kähler). Let $\lambda_{k}$ be a sequence such that $\lambda_{k}=o(k)$ and such that $\lambda_{k}$ is not exponentially sm

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 5 more