ScalingStacks

Remark 5.4 [0369]

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Remark 5.4

We obtain the same sheaf of differential forms on Xan{X^{\rm an}} as in [CD12], §3. In the latter reference, all analytic moment maps were used to define differential forms on Xan{X^{\rm an}} and so it is clear that our differential forms here are also differential forms in the sense of [CD12]. To see the converse, we argue as follows: By Proposition 4.16, tropical charts (V,φU)(V,\varphi_{U}) form a basis in Xan{X^{\rm an}}. It follows from Proposition 7.2 that an analytic moment map φ:V→(𝔾mr)an\varphi:V\rightarrow({\mathbb{G}}_{m}^{r})^{\rm an} may be locally in x∈Vx\in V approximated by an algebraic moment map φ′:U′→𝔾mr\varphi^{\prime}:U^{\prime}\rightarrow{\mathbb{G}}_{m}^{r} such that (φ′)trop=trop∘φ(\varphi^{\prime})_{\rm trop}={\rm trop}\circ\varphi in an open neighbourhood of xx in VV. Here, U′U^{\prime} is a suitable very affine open subset of UU with x∈(U′)anx\in(U^{\prime})^{\rm an}. It follows from [CD12], Lemma 3.1.10, that we may use algebraic moment maps to define differential forms in the sense of [CD12]. Using that φU′\varphi_{U^{\prime}} factorizes through φ′\varphi^{\prime} (see 4.12), we get the claim.

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