What is 'Transcendental algebraic geometry'?

Question

What is 'Transcendental algebraic geometry'? Could you give me some good references in this field? Thanks.

Answer 1

Transcendental Algebraic Geometry is the study of the algebraic geometry of a variety defined over the complex numbers $\mathbb C$ by concentrating on its undelying structure as a holomorphic manifold or variety.
This allows one to study the variety through the powerful tools of topology, analysis and differential geometry like : characteristic classes, elliptic partial differential equations, Kähler structure, ...
Serre wrote a remarkable article (always quoted as GAGA!) giving a very precise functorial correspondence between a projective algebraic variety $X$ over $\mathbb C$ and the corresponding complex analytic variety $X^{an}$: here it is.

Voisin's book mentioned by sweetjazz is indeed masterful but rather advanced.
Two good introductory, fairly reader friendly, books to that area are Taylor's textbook
and Wells's classic Differential Analysis on Complex Manifolds .
The latter book culminates in the profound solution by Kodaira of Hodge's problem of characterizing the Kähler complex manifolds underlying a projective algebraic variety.

Answer 2

Transcendental Algebraic Geometry generally refers to algebraic geometry studied using techniques from the theory of complex variables, so that the results generally only apply to varieties defined over $\mathbb{C}$. One of the basic references for this area is Claire Voisin's two volume series Hodge Theory and Complex Algebraic Geometry.

Answer 3

Griffith & Harris - "Principles of algebraic geometry" is also very good reference.