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Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for their choice of complex structure and exact isotopy $\phi$. \par

To do this they take a $\xi \in T_{u}\mathcal{P}$, where $u\in \mathcal{M}_{J,\phi}$, and claim that $E_u(\xi)=\nabla_{s}|_{s=0} \bar \partial _J(u_s)$. What does this actually mean , since $\bar \partial _J(u_s)$ is a family of vector fields $\xi_s$ along $u_s$.

Also at some point they claim $\nabla_{s}\frac{\partial u_s}{\partial \tau}|_{s=0}= \nabla_s\frac{\partial u}{\partial \tau }|_{s=0} = \nabla_{\tau}\frac{\partial u}{\partial s}|_{s=0}$, where I don't know why we have the first inequality.

The notation is also the notation from https://stacks.stanford.edu/file/druid:bz202yk0512/immersed-augmented.pdf of page $37$.

Any help or just poiting a reference where I could check this would be appreciated. Thanks in advance.

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  • $\begingroup$ You might get a better response to these questions you're posting if you explain some of the notation you're using (for one thing, the paper you allude to is behind a paywall). $\endgroup$
    – Jonny Evans
    Commented Jul 2, 2021 at 22:49
  • $\begingroup$ Yeah , but that is kinda of my question I am unsure what the author means by $\nabla_{s}|_{s=0}\bar \partial_J(u_s)$. I am trying to understand if this is a normal derivative or a covariant one. $\endgroup$
    – user174565
    Commented Jul 3, 2021 at 7:20
  • $\begingroup$ But there is other notation here: what are P and E_u? What is u_s? What is tau? And your moduli space: how is the exact isotopy involved in its definition? Presumably this is Lagrangian HF you're talking about, but not specifying your setting makes it hard for us to help! $\endgroup$
    – Jonny Evans
    Commented Jul 3, 2021 at 12:02

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