Global symplectic coordinates on complex domains

In this paper we deal with complex domains M⊂Cn equipped with a Kähler form ω=i2∂∂¯f, where f:M→R only depends on |zj|2, j=1,…,n for the complex coordinates (z1,…,zn) in Cn. We give an explicit symplectic immersion Φ of (M,ω) into R2n in Section 2. In Section 3 we study when the map Φ is a global symplectomorphism for the case of complete Reinhardt domains in C2.