A coincidence theorem for holomorphic maps to G/P

Abstract

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=S²ⁿ⁺¹xS²ⁿ⁺¹ denote a Calabi-Eckmann manifold. If f,g:V_n→Pⁿ 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∈Vn. 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⊂G is a maximal parabolic subgroup, where one of the maps is dominant.