Transcendental methods of algebraic geometry have been extensively studied since a very long time, starting with the work of Abel, Jacobi and Riemann in the nineteenth century. More recently, in the period 1940-1970, the work of Hodge, Hirzebruch.
Kodaira, Atiyah revealed still deeper relations between complex analysis, topology, PDE theory and algebraic geometry. In the last ten years, gauge theory has proved to be a very efficient tool for the study of many important questions: moduli spaces, stable sheaves, non abclian Hodge theory, low dimensional topology...