Consider for example any compact connected complex manifold M: any holomorphic function on it is constant by the maximum modulus principle.
Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon.
The set of complex structures on a given orientable surface, modulo biholomorphic equivalence, itself forms a complex algebraic variety called a moduli space, the structure of which remains an area of active research.
This tensor is defined on pairs of vector fields, X, Y by For example, the 6-dimensional sphere S6 has a natural almost complex structure arising from the fact that it is the orthogonal complement of i in the unit sphere of the octonions, but this is not a complex structure.
The existence of holomorphic coordinates is equivalent to saying the manifold is complex (which is what the chart definition says).
Tensoring the tangent bundle with the complex numbers we get the complexified tangent bundle, on which multiplication by complex numbers makes sense (even if we started with a real manifold).
Like a Riemannian metric, a Hermitian metric consists of a smoothly varying, positive definite inner product on the tangent bundle, which is Hermitian with respect to the complex structure on the tangent space at each point.
The quotient is a complex manifold whose first Betti number is one, so by the Hodge theory, it cannot be Kähler.