2015/09/07 by Kossovskiy, Ilya, Xiao, Ming
#Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1509.01962
In this paper, we provide \em effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, for any N >n ≥ 1, the defining functions φ(z, z,u) of all real-analytic hypersurfaces M=\v=φ(z, z,u)\⊂\mathbb Cn+1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric \mathcal Q⊂\mathbb CN+1 satisfy an \em universal algebraic partial differential equation D(φ)=0, where the algebraic-differential operator D=D(n,N) depends on n, N only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n,N as above there exists μ=μ(n,N) such that a Zariski generic real-analytic hypersurface M⊂\mathbb Cn+1 of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric \mathcal Q⊂\mathbb CN+1. We also provide an explicit upper bound for μ in terms of n,N. To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree.