Let $(X, \omega)$ be a closed symplectic manifold of dimension $2n$ and $\pi: X \rightarrow Q$ a Lagrangian torus fibration. Let $F_q$ denote the fiber at $q \in Q$. It is claimed in a paper of Abouzaid "The Family Floer Functor is Faithful" that the Arnol'd-Liouville theorem furnishes an isomorphism $T_q Q \cong H^1(F_q, \mathbb{R})$. Has anyone seen an argument along these lines or is familiar with this paper? I do not see why Arnol'd-Liouville should apply here.
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4$\begingroup$ The Arnold-Liouville theorem gives you a canonical isomorphism of the trivial bundle $T_q(Q)\times F_q$ over $F_q$ with the cotangent bundle $T^*(F_q)$. Since $F_q$ is a torus, this cotangent bundle is naturally isomorphic to the trivial bundle $\Omega \times F_q$, where $\Omega $ is the space of invariant differential 1-forms on $F_q$. But the de Rham theorem gives a canonical isomorphism $\Omega \overset{\sim}{\longrightarrow}H^1(F_q,\mathbb{R})$. $\endgroup$– abxCommented Jul 9, 2019 at 4:12
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$\begingroup$ Ah, thank you very much! That makes perfect sense. $\endgroup$– John RachedCommented Jul 9, 2019 at 4:18
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1 Answer
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By the time I finished writing this answer someone has explained the whole idea in a comment, but I thought I'd post it anyway as there is some more detail here. I assume the version of Arnold-Liouville you're familiar with says something like this: of you have a complete system of commuting Hamiltonians $(H_1,\ldots,H_n)$ with compact, connected simultaneous level sets then, near some regular value, you can find some functions $(G_1(H_1,\ldots,H_n),\ldots,G_n(H_1,\ldots,H_n))$ which generate periodic flows with standard period lattice (so that in particular the regular level sets are tori).
Given a Lagrangian fibration $\pi:X\to Q$ and a regular value, pick local coordinates $(q_1,\ldots,q_n)$ on $Q$ near this regular value. Considered as functions on the total space of the fibration, the $q_i$ Poisson commute because their simultaneous level sets are Lagrangians. Therefore they give you a complete commuting system of Hamiltonians to which you can apply the Arnold-Liouville theorem.
Since we know the fibres are Lagrangian tori, fibres near to $F_q=\pi^{-1}(q)$ can be written as graphs of closed 1-forms in a Weinstein neighborhood of $F_q$. By taking cohomology class of this class 1-form you get a map from a neighbourhood of $q\in Q$ to $H^1(F_q;R)$. Since nearby fibres don't intersect, the 1-forms cannot be cohomologous (cohomologous 1-forms differ by $df$ for some function $f$ which necessarily has a critical point), so actually $Q$ is locally modelled on $H^1(F_q;R)$. This is all canonical up to the action of $Diff(F_q)$ on cohomology, so $Q$ inherits an integral affine structure as a result. Arnold-Liouville also gives a way of thinking of this integral affine structure: the Hamiltonians generating the periodic flows give local coordinates on $Q$ which are canonical up to changing Hamiltonians by an integral affine transformation (as such a transformation does not change the period lattice).
So actually Arnold-Liouville gives much more than this identification of tangent space with cohomology: it gives an identification of an open set in the base with an open set in the cohomology. That's what gives the integral affine structure.
There are many good places to read about this, but in addition to those, I have some notes about this:
answered Jul 9, 2019 at 4:39