Shigeru Iitaka (飯高 茂, born 1941) is a prominent Japanese mathematician known for his significant contributions to the field of algebraic geometry, particularly in the classification theory of algebraic varieties. He is a professor emeritus at Gakushuin University.
Career and Contributions
Iitaka studied mathematics at the University of Tokyo, where he also completed his doctoral degree. He subsequently held positions at various institutions, including the University of Tokyo and Gakushuin University, where he spent a substantial part of his career.
His most notable contributions include:
- Iitaka Dimension (飯高次元): Iitaka introduced and extensively developed the concept of the Iitaka dimension (also sometimes referred to as the Kodaira dimension in certain contexts, though Iitaka's work provided a unified framework). This invariant, denoted by $\kappa(X)$, plays a crucial role in the birational classification of algebraic varieties. It measures the "size" or "complexity" of a variety and ranges from $-\infty$ to $n$ (the dimension of the variety).
- Iitaka Fibration (飯高ファイブレーション): Central to his work is the Iitaka fibration, a canonical map associated with an algebraic variety $X$. This fibration, roughly speaking, decomposes $X$ into "fibers" that are Fano varieties (or varieties with trivial canonical bundle) over a "base" variety whose dimension is the Iitaka dimension $\kappa(X)$. It provides a powerful tool for understanding the structure of algebraic varieties.
- Iitaka Conjecture (飯高予想): He formulated the Iitaka conjecture, which posits that for a fibration $f: X \to Y$ (a proper surjective map with connected fibers between smooth projective varieties), the Iitaka dimension of $X$ is greater than or equal to the sum of the Iitaka dimension of $Y$ and the Iitaka dimension of a generic fiber $F$: $\kappa(X) \ge \kappa(Y) + \kappa(F)$. This conjecture is fundamental to the classification program and has been proven in many important cases, though it remains an active area of research.
Iitaka's work, especially on the Iitaka dimension and fibration, has been instrumental in the development of the Minimal Model Program (also known as the Mori program) and the general classification theory of higher-dimensional algebraic varieties. His books and articles have significantly influenced researchers worldwide.
Awards and Recognition
Iitaka's pioneering work in algebraic geometry has earned him recognition, including the Saki Award (日本数学会幾何学賞) from the Mathematical Society of Japan in 1982 for his contributions to algebraic geometry.
Selected Publications
- Algebraic Geometry: An Introduction to Birational Geometry of Algebraic Varieties (Graduate Texts in Mathematics, Springer-Verlag, 1982)
- Numerous research papers in leading mathematical journals on topics such as algebraic varieties, classification theory, and Kodaira dimension.