On the Embeddability of Real Hypersurfaces into Hyperquadrics

A well known result of Forstnerić [18] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [19] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. Explicit examples of real-analytic hypersurfaces non-embaddable into hyperquadrics were obtained by Zaitsev [38]. In contrast, the classical theorem of Webster [37] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide 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, z̄, u) of all real-analytic hypersurfaces M = {v = φ(z, z̄, u)} ⊂ C containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C satisfy an 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 ⊂ C of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C. 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.