Symplectic maps of complex domains into complex space forms

Let M⊂CnM⊂Cn be a complex domain of CnCn endowed with a rotation invariant Kähler form ωΦ=i2∂∂̄Φ. In this paper we describe sufficient conditions on the Kähler potential ΦΦ for (M,ωΦ)(M,ωΦ) to admit a symplectic embedding (explicitly described in terms of ΦΦ) into a complex space form of the same dimension of MM. In particular we also provide conditions on ΦΦ for (M,ωΦ)(M,ωΦ) to admit global symplectic coordinates. As an application of our results we prove that each of the Ricci-flat (but not flat) Kähler forms on C2C2 constructed by LeBrun in [C. LeBrun, Complete Ricci-flat Kähler metrics on CnCn need not be flat, in: Proceedings of Symposia in Pure Mathematics, vol. 52, 1991, pp. 297–304. Part 2] admits explicitly computable global symplectic coordinates.