Global symplectic coordinates on complex domains

Abstract

In this paper we deal with complex domains M ⊂ C n equipped with a Kähler form ω = i 2 ∂ ∂ ¯ f , where f : M → R only depends on | z j | 2 , j = 1 , … , n for the complex coordinates ( z 1 , … , z n ) in C n . We give an explicit symplectic immersion Φ of ( M , ω ) into R 2 n in Section 2. In Section 3 we study when the map Φ is a global symplectomorphism for the case of complete Reinhardt domains in C 2 .