Symplectic rigidity of real and complex polydiscs

Abstract

In R2n with its standard symplectic structure, the complex polydisc, DC2n(r), is constructed as the product of n open complex discs of radius r. When n=2, the real polydisc, DR4(r), is constructed as the product of 2 open real/Lagrangian discs of radius r. Sukhov and Tumanov recently showed that DC4(1) and DR4(1) are not symplectically equivalent. We extend this result in two ways. First we give the necessary and sufficient conditions for an orthogonal image of DC4(1) to be symplectically equivalent to DC4(1). Second, we show that for all r≥1 and n≥1, DR4(1)×DC2n−4(r) is not symplectically equivalent to DC4(1)×DC2n−4(r).