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The full asymptotic expansion of analytic torsion on homogeneous spaces

Kai Köhler

TL;DR

This work provides explicit full asymptotic expansions for the equivariant complex analytic torsion $T_t(G/K,L^{\ell})$ on compact complex homogeneous spaces as $\ell\to\infty$, expressed via Lerch zeta data and representation-theoretic inputs. It treats both isolated fixed points and Hermitian-symmetric cases, and extends to torsion forms in fibrations, with rigorous error bounds. The expansions are checked against classical BV and Fi results and applied to lattice representations of Chevalley groups and to the Jantzen sum formula, yielding computable arithmetic data. The framework advances Arakelov geometry by providing precise coefficient formulas for torsion-related invariants.

Abstract

The full asymptotic expansion of the equivariant complex Ray-Singer torsion for high powers of line bundles on symmetric spaces is given in an explicit form. In the case of isolated fixed points this expansion is given for general complex homogeneous spaces. Furthermore the full asymptotic expansion is given for the complex analytic torsion form associated to fibrations by projective curves. The expansions are compared with results by Bismut-Vasserot, Finski and Puchol. The results are applied to lattice representations of Chevalley groups.

The full asymptotic expansion of analytic torsion on homogeneous spaces

TL;DR

This work provides explicit full asymptotic expansions for the equivariant complex analytic torsion on compact complex homogeneous spaces as , expressed via Lerch zeta data and representation-theoretic inputs. It treats both isolated fixed points and Hermitian-symmetric cases, and extends to torsion forms in fibrations, with rigorous error bounds. The expansions are checked against classical BV and Fi results and applied to lattice representations of Chevalley groups and to the Jantzen sum formula, yielding computable arithmetic data. The framework advances Arakelov geometry by providing precise coefficient formulas for torsion-related invariants.

Abstract

The full asymptotic expansion of the equivariant complex Ray-Singer torsion for high powers of line bundles on symmetric spaces is given in an explicit form. In the case of isolated fixed points this expansion is given for general complex homogeneous spaces. Furthermore the full asymptotic expansion is given for the complex analytic torsion form associated to fibrations by projective curves. The expansions are compared with results by Bismut-Vasserot, Finski and Puchol. The results are applied to lattice representations of Chevalley groups.
Paper Structure (10 sections, 24 theorems, 169 equations)

This paper contains 10 sections, 24 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Asymptotic expansion of a derivative of the Lerch zeta function
  3. Asymptotic expansion of sums over polynomials times logarithms
  4. Definition of analytic torsion and certain characteristic classes
  5. Asymptotic expansion of the torsion
  6. Comparison with other results
  7. Application to the Jantzen sum formula
  8. The case of isolated fixed points
  9. Lie-algebra-equivariant torsion on the projective plane
  10. Asymptotics of torsion forms

Key Result

Theorem 1.2

Let $M=G/K$ be a compact complex homogeneous space and let $L:=L_{\rho_K+\lambda}$ be a $G$-invariant holomorphic Hermitian line bundle on $M$. Assume that $L$ is positive in the sense that $\forall\alpha\in\Psi:\langle\alpha^\vee,\lambda\rangle>0$. Fix a Kähler metric $g_{X_0}$ on $M$. Let $t\in G$ with $|R_1|<\frac{(N-1)!}{(2\pi \ell c_2)^N}c_1$ for explicitly given $c_1,c_2\in{\mathbb R}^+$ whi

Theorems & Definitions (56)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 46 more