Proposition 6.27 . [02WL]

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Proposition 6.27.

With the previous hypothesis and notations, the equality

(6.28)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{1}}(X_{\Sigma_{1}})=(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}(A^{\ast}\psi_{{\overline{L}},s})^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\psi^{\vee}_{{\overline{L}},s}-u)\,\text{\rm d}\operatorname{vol}_{M_{1}}

holds. Moreover

(6.29)

\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}_{1}}(X_{\Sigma_{1}})-\operatorname{h}^{\operatorname{tor}}_{{\overline{L}}}(Y)=(d+1)!\lambda_{K}\int_{H^{\vee}(\Delta_{\Psi})}(A^{\ast}\Psi)^{\vee}\,\text{\rm d}\operatorname{vol}_{M_{1}}\\ =(d+1)!\int_{H^{\vee}(\Delta_{\Psi})}H^{\vee}_{\ast}(\iota_{\Delta_{\Psi}}-\lambda_{K}u)\,\text{\rm d}\operatorname{vol}_{M_{1}},

where $\iota_{\Delta_{\Psi}}$ is the indicator function of $\Delta_{\Psi}$ (Example 3.16).