Pome

Theorem 13.29.3151BAA2FA1AB

Theorem 13.29.3.151BAA2FA1AB Let $\theta _0,\theta _1$ be smooth real $(1,1)$-forms on $Y$ with positive areas $A_ i=A_{\theta _ i}$, and put $\lambda =A_1/A_0$. The map

\[ F_{10}:\mathcal M(\theta _0)\longrightarrow \mathcal M(\theta _1), \qquad F_{10}(p_{\theta _0}(\mu ))=p_{\theta _1}(\lambda \mu ), \]

is a bijection and satisfies

7ADEC26E190A 13.29.4
\begin{equation} \begin{split} d_{\theta _1}(F_{10}(p),F_{10}(q))& =\lambda d_{\theta _0}(p,q),\\ U_{\theta _1}(F_{10}(p))& =\lambda U_{\theta _0}(p),\\ P_{\theta _1}[F_{10}(p)]_{\mathcal I} & =F_{10}(P_{\theta _0}[p]_{\mathcal I}). \end{split} \end{equation}

For a third smooth background $\theta _2$ of positive area, the same construction gives $F_{21}F_{10}=F_{20}$.

Proof. Multiplication by $\lambda $ is a bijection from the polar measures of mass at most $A_0$ onto those of mass at most $A_1$. By proposition 13.29.1, it gives the asserted bijection, and the distances satisfy

\[ \| \lambda \mu -\lambda \nu \| _{\mathrm{TV}} =\lambda \| \mu -\nu \| _{\mathrm{TV}}. \]

The volume identity follows from $A_1-\lambda \mu (Y)=\lambda (A_0-\mu (Y))$. Taking the atomic part commutes with multiplication by $\lambda $, which gives the ideal-envelope identity. The area ratios multiply, proving the composition rule. $\square$

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