Livsic-type determinantal representations and hyperbolicity

Abstract

Hyperbolic homogeneous polynomials with real coefficients, i.e., hyperbolic real projective hypersurfaces, and their determinantal representations, play a key role in the emerging field of convex algebraic geometry. In this paper we consider a natural notion of hyperbolicity for a real subvariety X⊂Pd of an arbitrary codimension ℓ with respect to a real ℓ−1-dimensional linear subspace V⊂Pd and study its basic properties. We also consider a class of determinantal representations that we call Livsic-type and a nice subclass of these that we call very reasonable. Much like in the case of hypersurfaces (ℓ=1), the existence of a definite Hermitian very reasonable Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a very reasonable Livsic-type determinantal representation. Our basic tools are Cauchy kernels for line bundles and the notion of the Bezoutian for two meromorphic functions on a compact Riemann surface that we introduce. We then proceed to show that every real curve in Pd hyperbolic with respect to some real d−2-dimensional linear subspace admits a definite Hermitian, or even definite real symmetric, very reasonable Livsic-type determinantal representation.