Livsic-type Determinantal Representations and Hyperbolicity

TL;DR

This paper explores Livsic-type determinantal representations for real subvarieties, establishing their connection to hyperbolicity and demonstrating their existence for all curves, with implications for convex algebraic geometry.

Contribution

It introduces Livsic-type determinantal representations for arbitrary codimension subvarieties and proves their existence for all curves, linking them to hyperbolicity.

Findings

Existence of Livsic-type determinantal representations for all curves.

Connection between definite Hermitian representations and hyperbolicity.

Development of tools like Cauchy kernels and Bezoutians for the study.

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 $X \subset \mathbb{P}^d$ of an arbitrary codimension $\ell$ with respect to a real $\ell - 1$-dimensional linear subspace $V \subset \mathbb{P}^d$ and study its basic properties. We also consider a special kind of determinantal representations that we call Livsic-type and a nice subclass of these that we call \vr{}. Much like in the case of hypersurfaces ($\ell=1$), the existence of a definite Hermitian \vr{} Livsic-type determinantal representation implies hyperbolicity. We show that every curve admits a \vr{} 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 $\mathbb{P}^d$ hyperbolic with respect to some real $d-2$-dimensional linear subspace admits a definite Hermitian, or even real symmetric, \vr{} Livsic-type determinantal representation.