I know the following different concepts of complex structures on manifolds:
1) complex manifold $M$, which means that there is an holomorphic atlas for $M$
2) manifold $M$ with almost complex structure $J$, which means a complex structure on the bundle $TM \rightarrow M$, $J$ is called integrable, if the exists an atlas of charts $\varphi:U \rightarrow \mathbb{C}^n$, s.t. $J_{st} \circ d\varphi= d \varphi \circ J$ ,where $J_{st}$ is the standard complex structure
Now the definition for a Kähler manifold from wikipedia is:
$(M, \omega)$ symplectic manifold with integrable almost complex structure $J$, which is compatible with $\omega$
In my lecture I have the following definition:
A Kähler manifold is a triple $(M, \omega,J)$, where $\omega$ is a symplectic form on $M$ and $(M,J)$ is a complex manifold and $(\omega, J)$ is a competible pair.
We didn't define the term "complex manifold" in my lecture, so I just assume it is meant in the sense of 1).
My question is: Are 1) and 2) equivalent?
How does a complex manifold in the sense of 1) admit a complex structure (at all and then, for the equivalence, in the sense of 2)?