of $J$-complex curves (locally through any point and any direction) for any almost-complex structure $J$ of smoothness $C^{1,\alpha}$, $\alpha >0$. One more principal result here is the so-called [*first a priori estimate*]{} in §2.4. The third lecture, as well as [*Appendix 2*]{}, is devoted to the deeper study of complex curves in real 4-dimensional manifolds. We shall prove the positivity of intersections and the Adjunction Formula, showing that local properties of complex curves in nonintegrable almost-complex structures are similar to that in integrable ones.

[Lecture 1]{}

[Complex Curves in Almost-Complex Manifolds.]{}

[1.1. Almost-Complex Manifolds, Hermitian Metrics, Associated (1,1)-forms.]{}

Let $X$ be a real manifold, and $TX$ its tangent bundle.

Definition 1.1.1. [*A continuous endomorphism $J:TX\to TX$ such that $J^2=-\id $ is called an almost-complex structure on $X$. A pair $(X,J)$ is called an almost-complex manifold.* ]{}

Exercise. Prove that locally one can always find vector fields $e_1,e_2,...,e_{2n-1},e_{2n}$ such that $Je_{2k-1}=e_{2k}, k=1,...,n$. In particular, $X$ is even-dimensional.

Examples. $1^0.$ $(\rr^{2n},J\st )$. Denote by $e_1,...,e_{2n}$ the standard basis in $\rr^{2n}$. Define a [*standard*]{} complex structure $J\st $ in $\rr^{2n}$ by $J\st (e_{2k-1})=e_{2k}$.

If $(V,J)$ is an $\rr $-linear space endowed with an endomorphism $J$, such that $J^2=-\id $, then a structure of 