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I recently came across the following generalization of the Darboux-Weinstein lemma:

Let $N$ be a manifold endowed with two symplectic forms $\omega_1, \omega_2$, and let $P$ be a compact submanifold of $N$ along which $\omega_1 = \omega_2 + O_p(k)$, for some $k \in \mathbb{N}^* \cup \{+\infty\}$. Then there exists open neighbourhoods $U$ and $V$ of $P$ in $N$ and a diffeomorphism $f : U \rightarrow V$ such that $f = \text{id}_N + O_p(k+1)$ and $f^* \omega_2 = \omega_1$.

Here I do not know what $O_p(k)$ means, and I cannot find any references to it online. Any help would be much appreciated.

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    $\begingroup$ It would help to give a reference to where you came across the notation, especially since you seem to have switched between capital $P$ and lowercase $p$, and it's not clear whether they are meant to be the same; but it seems plausible that $O_P(k)$ means the functions vanishing on $P$ to order at least $k$. $\endgroup$
    – LSpice
    Commented Nov 18, 2021 at 3:49
  • $\begingroup$ arxiv.org/pdf/1504.07112.pdf This is the reference on page 42. Thank you. $\endgroup$
    – KXJ
    Commented Nov 18, 2021 at 4:06
  • $\begingroup$ Hi, could you also explain how to define functions vanishing on $P$ to order at least $k$, or point me to a resource? Thank you. $\endgroup$
    – KXJ
    Commented Nov 18, 2021 at 5:08

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