Livsic-type determinantal representations and hyperbolicity

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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.

Original languageEnglish
Pages (from-to)487-522
Number of pages36
JournalAdvances in Mathematics
Volume329
DOIs
StatePublished - 30 Apr 2018

Keywords

  • Bezoutian on a compact Riemann surface
  • Cauchy kernels on a compact Riemann surface
  • Convexity in the Grassmanian
  • Determinantal representations
  • Hyperbolic polynomials
  • Hyperbolic subvarieties in the projective space
  • Hyperbolicity cones
  • Real Riemann surfaces of dividing type

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