  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.
