  Let $(X,\omega)$ be a symplectic manifold, $J$ be an $\omega$-tame almost
complex structure, and $L$ be a Lagrangian submanifold. The stable
compactification of the moduli space of parametrized $J$-holomorphic curves in
$X$ with boundary in $L$ (with prescribed topological data) is compact and
Hausdorff in Gromov's $C^\infty$-topology. We construct a Kuranishi structure
with corners in the sense of Fukaya and Ono. This Kuranishi structure is
orientable if $L$ is spin. In the special case where the expected dimension of
the moduli space is zero, and there is an $S^1$ action on the pair $(X,L)$
which preserves $J$ and acts freely on $L$, we define the Euler number for this
$S^1$ equivariant pair and the prescribed topological data. We conjecture that
this rational number is the one computed by localization techniques using the
given $S^1$ action.
