Lemma 3.13 . [0501] Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.
Source coverage notes Source fidelity gap: the original arXiv HTML omits an author TeX footnote attached to equation e:def-omega, including its reference to Remark r:error-function. The retained HTML is preserved as published; the original author TeX remains available. TeX correspondence is not complete. Complete original source context · Original author HTML
Lemma 3.13 .
Denote by ∗ : Ω k ( Q ) → Ω m − k ( Q ) *:\Omega^{k}(Q)\to\Omega^{m-k}(Q) ∗ *
(1)
(3.95) ∗ dvol T = ( − 1 ) m + 1 dvol N ⋅ ( 1 − H α y α + O ~ ( r 2 ) ) , *\dvol_{T}=(-1)^{m+1}\dvol_{N}\cdot(1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2})),
(2)
For any α ∈ { 1 , 2 , 3 } \alpha\in\{1,2,3\} ,
(3.96) ∗ ( η α ∧ dvol T ) = λ 1 η α ^ + λ 2 η α + 1 ^ + λ 3 η α + 2 ^ , *(\eta_{\alpha}\wedge\dvol_{T})=\lambda_{1}\eta_{\widehat{\alpha}}+\lambda_{2}\eta_{\widehat{\alpha+1}}+\lambda_{3}\eta_{\widehat{\alpha+2}},
where λ 1 = 1 − H α y α + O ~ ( r 2 ) \lambda_{1}=1-H^{\alpha}y_{\alpha}+\widetilde{O}(r^{2}) , λ 2 = O ~ ( r 2 ) \lambda_{2}=\widetilde{O}(r^{2}) , λ 3 = O ~ ( r 2 ) \lambda_{3}=\widetilde{O}(r^{2}) .