TITLE: Almost complex structures in Floer theory
QUESTION [6 upvotes]: When defining the Floer cohomology  $HF(L_0,L_1)$ of 2 Lagrangians in a symplectic manifold $(M,\omega)$, one first has to choose some extra data such a 1-parameter family of almost complex structures $(J_t)$.  Usually one requires that $J_t$ be compatible with $\omega$, ie that $g(u,v)=\omega(u,J_tv)$ defines a Riemannian metric.
However there is also the related notion of a tame $J$, one such that $\omega(u,Ju)>0$ for all nonzero $u$.  My question is:
What goes wrong if we try to use tame but not necessarily compatible $J_t$ to define $HF(L_0,L_1)$?

REPLY [9 votes]: As far as I know, nothing goes wrong if one uses tame instead of compatible almost complex structures (if anyone knows better please correct me).  Either one of these conditions implies that the area of a holomorphic curve is bounded in terms of the integral of the symplectic form on it.  This is what one needs to get Gromov compactness and to set up the Novikov ring.  I don't think tameness or compatibility is needed elsewhere in the theory.