## 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⊂P^{d} of an arbitrary codimension ℓ with respect to a real ℓ−1-dimensional linear subspace V⊂P^{d} 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 P^{d} 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 language | English |
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Pages (from-to) | 487-522 |

Number of pages | 36 |

Journal | Advances in Mathematics |

Volume | 329 |

DOIs | |

State | Published - 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