  We study the relation between $J$-anti-invariant $2$-forms and
pseudoholomorphic curves in this paper. We show the zero set of a closed
$J$-anti-invariant $2$-form on an almost complex $4$-manifold supports a
$J$-holomorphic subvariety in the canonical class. This confirms a conjecture
of Draghici-Li-Zhang. A higher dimensional analogue is established. We also
show the dimension of closed $J$-anti-invariant $2$-forms on an almost complex
$4$-manifold is a birational invariant, in the sense that it is invariant under
degree one pseudoholomorphic maps.
