On Holomorphy

"Mirror symmetry is concerned with counting the number of holomorphic curves on Calabi-Yau manifolds, i.e. compact Kähler manifolds X with trivial canonical bundle KX." (Robbert Dijkgraaf, "Mirror symmetry and elliptic curves" [in Robert Dijkgraaf et al (Eds), "The moduli space of curves", Progress in Mathematics vol. 129] 1995) 

"By definition, a Kähler manifold is one with a complex structure (this means in particular that the coordinates changes are holomorphic for the complex coordinates) together with a Riemannian metric which has with this complex structure the best possible link, namely that multiplication of tangent vectors by unit complex numbers preserves the metric, but moreover the complex structure is invariant under parallel transport. This is equivalent to the condition that the holonomy group be included in the unitary group, hence equivalent also to ask for the existence of a 2-form of maximal rank and of zero covariant derivative."(Marcel Berger, "A Panoramic View of Riemannian Geometry", 2003) 

"Harmonic measure is a device for estimating harmonic functions on a domain. It has become an essential tool in potential theory and in studying the corona problem. It is useful in studying the boundary behavior of conformal mappings, and it tells us a great deal about the boundary behavior of holomorphic functions and solutions of the Dirichlet problem."(Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"One of the most striking facts about the Poincaré metric on the disk is that it turns the disk into a complete metric space. How could this be? The boundary is missing! The reason that the disk is complete in the Poincaré metric is the same as the reason that the plane is complete in the Euclidean metric: the boundary is infinitely far away. [...] One of the important facts about the Poincaré metric is that it can be used to study not just conformal maps but all holomorphic maps of the disk. The key to this assertion is the classical Schwarz lemma." (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"The Schwarz lemma is one of the simplest results in all of complex function theory. A direct application of the maximum principle, it is merely a statement about the rate of growth of holomorphic functions on the unit disk."  (Steven G. Krantz, "Geometric Function Theory: Explorations in complex analysis", 2006)

"The primary aspects of the theory of complex manifolds are the geometric structure itself, its topological structure, coordinate systems, etc., and holomorphic functions and mappings and their properties. Algebraic geometry over the complex number field uses polynomial and rational functions of complex variables as the primary tools, but the underlying topological structures are similar to those that appear in complex manifold theory, and the nature of singularities in both the analytic and algebraic settings is also structurally very similar." (Raymond O Wells Jr, "Differential and Complex Geometry: Origins, Abstractions and Embeddings", 2017)