A Coincidence Theorem for Holomorphic Maps to $G/P$

http://dx.doi.org/10.4153/CMB-2003-029-4
Canad. Math. Bull. 46(2003), 291-298

  Published:2003-06-01
 Printed: Jun 2003

The purpose of this note is to extend to an arbitrary generalized Hopf and Calabi-Eckmann manifold the following result of Kalyan Mukherjea: Let $V_n = \mathbb{S}^{2n+1} \times \mathbb{S}^{2n+1}$ denote a Calabi-Eckmann manifold. If $f,g \colon V_n \longrightarrow \mathbb{P}^n$ are any two holomorphic maps, at least one of them being non-constant, then there exists a coincidence: $f(x)=g(x)$ for some $x\in V_n$. Our proof involves a coincidence theorem for holomorphic maps to complex projective varieties of the form $G/P$ where $G$ is complex simple algebraic group and $P\subset G$ is a maximal parabolic subgroup, where one of the maps is dominant.