Lemma 1.4.2 the structure $J_1$ can be made complex in the neighborhood of $M$.* ]{}

The proof is left to the reader.

[1.5. Adjunction Formula for Immersed Symplectic Surfaces.]{}

Let us now prove the Adjunction Formula for *immersed symplectic surfaces. Let $u:\bigsqcup_{j=1}^dS_j\to (X,\omega)$ be a reduced compact symplectic surface (see Definition 1.4.1) immersed into a symplectic four-dimensional manifold. Let $g_j$ denote the genus of $S_j$ and $M_j=u(S_j)$. Put $M:=\bigcup_{j=1}^dM_j$ and denote by $[M]^2$ the homological self-intersection number of $M$. Define a geometrical self-intersection number $\delta $ of $M$ in the following way. Perturb $M$ to obtain a symplectic surface $\widetilde M$ with only transversal double points. Then $\delta$ will be the sum of indices of the intersection over those double points. Those indices can be equal to $1$ or $-1$.*

Exercise. Let $P$ be an oriented plane in $\rr^4$. Call $P$ symplectic if $\omega\st (P)>0$. Find two symplectic planes $P_1,P_2\in \rr^4$ which intersect transversally at origin with intersection index $-1$.

Let $J$ be some almost complex structure which is tamed by $\omega$, [*,*]{} $\omega(\xi, J\xi)>0$ for any nonzero $\xi \in TX$. Denote by $c_1(X, J)$ the first Chern class of $X$ with respect to $J$. Since, in fact, $c_1(X, J)$ 