Let $(X,\omega)$ be a symplectic manifold, $L\subset X$ be a Lagrangian submanifold, $[L]$ denotes the Hamiltonian isotopy class. How to represent $L'\in[L]$ via $L$ (for example, a graph over $L$)? Is there an analogue to the $\partial\bar{\partial}$-lemma in Kähler geometry?
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3$\begingroup$ There is no general way of expressing Lagrangian submanifolds like this in general. In the special case of a cotangent bundle $T^*L$, any Hamiltonian deformation of the zero-section $L$ can be written in terms of a generating function quadratic at infinity: see work of Chaperon and the paper F. Laudenbach and J.-C. Sikorav, Persistence d'intersections avec la section nulle au cours d'une isotopies Hamiltonienne dans le fibr´e cotangent, Invent. Math., 82(1985), 349–357. $\endgroup$– Jonny EvansCommented Sep 9, 2014 at 16:06
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If the complex structure $J$ compatible with $\omega$ is integrable for $X$, then of course you have $\partial\bar{\partial}$ lemma. A natural generalization of the $\partial\bar{\partial}$ lemma in symplectic geometry is given by the "symplectic cohomology" introduced by Tseng and Yau: http://arxiv.org/pdf/0909.5418v2.pdf. Here you need to replace the operator $d^c$ by $d^\Lambda$, and the definition is analogous to that of the Bott-Chern cohomology. Just as the Bott-Chern cohomology measures how far the $\partial\bar{\partial}$ lemma is away from being true for a general complex manifold, the "symplectic cohomology" introduced by Tseng-Yau measures exactly the same thing for a general symplectic manifold. Of course the terminology here is a little ambiguous, because usually symplectic cohomology is defined by a direct limit of the Hamiltonian Floer cohomologies $HF^\ast(X,wH)$.
