From the University of Tokyo:
"Complex manifolds are high-dimensional geometric objects that have complex numbers as coordinates. They are invisible to the naked eye except for Riemann surfaces, which are one-dimensional complex manifolds. But by using analytical methods such as harmonic integrals and/or algebraic machinery such as sheaf cohomology, Prof. Kodaira found that one could develop a geometry of complex manifolds as rich as that of visible figures, a discovery of great importance… Prof. Kodaira's major achievements include the Kodaira vanishing theorem, deformation theory of complex structure, and the classification of complex surfaces. He was the first Japanese to receive the Fields Medal in 1954 for his series of work on harmonic analysis represented by the Kodaira vanishing theorem."
Fields Medalist, 1954