Let $(X, \omega)$ be a closed symplectic 4-manifold. Let $\mathcal{C}=(C_i, mi_i)$ be a holomorphic current in $X$, where $C_i$ is a somewhere injective $J$-holomorphic curve in $X$ and $m_i$ is positive integer. Then we can define the ECH index of $\mathcal{C}$ as follow: $$I(\mathcal{C})= \langle c_1(TX), \mathcal{C}\rangle + \mathcal{C} \cdot \mathcal{C}.$$ This integer comes from the dimension of the moduli space of solutions of he Seiberg Witten equation.

Taubes defines his Gromov invariant by counting $I=0$ holomorphic currents. Fix $A \in H_2(X)$, let $\mathcal{M}(X, \omega)=\{\mathcal{C}=(C_i,m_i) : I(\mathcal{C})=0, [\mathcal{C}]=A\}$ to be moduli space of holomorphic currents with $I=0$. Here $I$ only depends on the homology class $A$. If a multiple cover arises, i.e. $m_i>1$ for some $i$, then $I$ doesn't involve any information about the holomorphic map. In contrast to the usual moduli space of holomorphic curves (considered as holomorphic maps), this moduli space is quite strange; it only consists of currents.

Does make sense to talk about the transversality of $\mathcal{M}(X, \omega)$, or does $\mathcal{M}(X, \omega)$ admit a virtual cycle structure?