## Advanced Studies in Pure Mathematics

Yoichi Miyaoka

#### Abstract

Smooth quadric hypersuraces in $\mathbb{P}^{n+1} (\mathbb{C})$ are numerically characterised as the smooth Fano $n$-folds of length $n$, i.e., a smooth Fano $n$-fold $X$ is isomorphic to a hyperquadric if and only if the minimum of the intersection number $(C, -K_X)$ is $n$, where $C$ runs through the rational curves on $X$.

#### Article information

Dates
First available in Project Euclid: 3 January 2019

https://projecteuclid.org/ euclid.aspm/1546542852

Digital Object Identifier
doi:10.2969/aspm/04210209

Mathematical Reviews number (MathSciNet)
MR2087053

Zentralblatt MATH identifier
1063.14050

#### Citation

Miyaoka, Yoichi. Numerical characterisations of hyperquadrics. Complex Analysis in Several Variables — Memorial Conference of Kiyoshi Oka's Centennial Birthday, Kyoto/Nara 2001, 209--235, Mathematical Society of Japan, Tokyo, Japan, 2004. doi:10.2969/aspm/04210209. https://projecteuclid.org/euclid.aspm/1546542852