does not depend on continuous changes of $J$ and since the set of $\omega$-tamed almost complex structures is contractible, we usually omit the dependence of $c_1(X)$ on $J$.

Lemma 1.5.1. [*Let $M=\bigcup_{j=1}^d M_j$ be a compact immersed symplectic surface in four-dimensional symplectic manifold $X$. Then $$\sum_{j=1}^d g_j = {[M]^2 - c_1(X)[M]\over2} + d - \delta .\eqno(1.5.1)$$*]{}

Proof. By replacing every $M_j$ by its small perturbation, we can suppose that $M_j$ has only transversal double self-intersection points. Let $N_j$ be a normal bundle to $M_j$ and let $\widetilde M_j$ denote the zero section of $N_j$. Also let $\exp_j $ be the exponential map from a neighborhood $V_j$ of $\widetilde M_j\subset N_j$ onto the neighborhood $W_j$ of $M_j$. Lift $\omega$ and $J$ onto $V_j$. Since $\widetilde M_j$ is embedded to $V_j$, we can apply [*Lemma 1.4.2*]{} to obtain the $\omega$-tame almost complex structure $J_j$ on $V_j$ such that $\widetilde M_j$ is $J_j$-holomorphic.

For every $j$ we now have the following exact sequence of complex bundles:

$$0\longrightarrow TS_j {\buildrel du\over \longrightarrow }
E_j \buildrel \pr \over \longrightarrow
N_j \longrightarrow 0 \eqno(1.5.2)$$

Here $E_j=(u^*TX)\ogran_{S_j}$ is endowed with complex structure given by $J_j$. Since $du$ is nowhere degenerate [*complex*]{} linear morphism, $N_j \deff
E_j/du(TM_j)$ is 