We define a subset of an almost complex manifold
(
M
,
J
)
(M,J)
to be a holomorphic shadow if it is the image of a
J
J
-holomorphic map from a compact complex manifold. Notice that a
J
J
-holomorphic curve is a holomorphic shadow, and so is a complex subvariety of a compact complex manifold.
We show that under some conditions on an almost complex structure
J
J
on a manifold
M
M
, the holomorphic shadows in the Cartesian products of
(
M
,
J
)
(M,J)
form a Zariski-type structure. Checking this leads to non-trivial geometric questions and results. We then apply the work of Hrushovski and Zilber on Zariski-type structures.
We also restate results of Gromov and McDuff on
J
J
-holomorphic curves in symplectic geometry in the language of shadows structures.