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Formulae of some global CR invariants for Sasakian $η$-Einstein manifolds

Yuya Takeuchi

TL;DR

This work derives explicit formulas for Burns–Epstein and global CR invariants via renormalized characteristic forms on Sasakian $η$-Einstein manifolds, using Fefferman defining functions and a renormalized connection. It expresses the invariants $oldsymbol{ extmathscr{I}}_{oldsymbol{oldsymbol{ extvarSigma}}}$ as fiber integrals involving the Einstein constant $λ$ and a pulled-back Chern-type tensor, and shows that these invariants are algebraically independent while providing a triviality criterion when a polynomial vanishes modulo $ ext{ch}_1$. The Burns–Epstein invariant is computed in this setting, again reducing to base data through circle-bundle or tube constructions, with explicit formulas in terms of $λ$ and Chern classes. The paper also discusses a conjectural linear relation between $oldmu(M)$ and the $oldmathscr{I}$-invariants, verified in dimensions up to three (and partially in dimension four), highlighting deep connections between global CR invariants and classical curvature data in Sasakian geometry. Overall, the results advance computability and structural understanding of CR invariants for a broad and geometrically rich class of manifolds.

Abstract

In this paper, we give explicit formulae of the Burns-Epstein invariant and global CR invariants via renormalized characteristic forms introduced by Marugame for Sasakian $η$-Einstein manifolds. As an application, we show that the latter invariants are algebraically independent.

Formulae of some global CR invariants for Sasakian $η$-Einstein manifolds

TL;DR

This work derives explicit formulas for Burns–Epstein and global CR invariants via renormalized characteristic forms on Sasakian -Einstein manifolds, using Fefferman defining functions and a renormalized connection. It expresses the invariants as fiber integrals involving the Einstein constant and a pulled-back Chern-type tensor, and shows that these invariants are algebraically independent while providing a triviality criterion when a polynomial vanishes modulo . The Burns–Epstein invariant is computed in this setting, again reducing to base data through circle-bundle or tube constructions, with explicit formulas in terms of and Chern classes. The paper also discusses a conjectural linear relation between and the -invariants, verified in dimensions up to three (and partially in dimension four), highlighting deep connections between global CR invariants and classical curvature data in Sasakian geometry. Overall, the results advance computability and structural understanding of CR invariants for a broad and geometrically rich class of manifolds.

Abstract

In this paper, we give explicit formulae of the Burns-Epstein invariant and global CR invariants via renormalized characteristic forms introduced by Marugame for Sasakian -Einstein manifolds. As an application, we show that the latter invariants are algebraically independent.

Paper Structure

This paper contains 14 sections, 11 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. CR geometry
  3. CR structures
  4. Tanaka-Webster connection and pseudo-Einstein condition
  5. Sasakian manifolds
  6. Strictly pseudoconvex domains
  7. Graham-Lee connection
  8. Fefferman defining functions
  9. Global CR invariants via renormalized characteristic forms
  10. Burns-Epstein's renormalized connection
  11. Global CR invariants via renormalized characteristic forms
  12. Global CR invariants for Sasakian $\eta$-Einstein manifolds
  13. Burns-Epstein invariant for Sasakian $\eta$-Einstein manifolds
  14. Concluding remarks

Key Result

Theorem 1.1

Let $(S, T^{1, 0} S, \eta)$ be a closed $(2 n + 1)$-dimensional Sasakian $\eta$-Einstein manifold with Einstein constant $(n + 1) \lambda$. For a $GL(n + 1, \mathbb{C})$-invariant polynomial $\Phi$ of degree at most $n$, define a $GL(n, \mathbb{C})$-invariant polynomial $\Phi^{\prime}$ by Then where

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 4.1: Marugame2016*Proposition 3.5
  • Definition 4.2: Marugame2021*Theorem 1.1 and Proposition 4.9
  • Lemma 4.3
  • proof
  • ...and 14 more