Remark 5.4 [0369]
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Remark 5.4
We obtain the same sheaf of differential forms on as in [CD12], §3. In the latter reference, all analytic moment maps were used to define differential forms on 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 form a basis in . It follows from Proposition 7.2 that an analytic moment map may be locally in approximated by an algebraic moment map such that in an open neighbourhood of in . Here, is a suitable very affine open subset of with . 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 factorizes through (see 4.12), we get the claim.