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Livsic-type determinantal representations and hyperbolicity. (English) Zbl 1391.32009
Summary: 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 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 \((\ell = 1)\), 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 \(\mathbb{P}^d\) 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.

32C05 Real-analytic manifolds, real-analytic spaces
14M12 Determinantal varieties
30F10 Compact Riemann surfaces and uniformization
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