Hyperbolicity and determinantal representations for higher-codimensional subvarieties

Let $X $ be a real subvariety of codimension $\ell$ in the complex projective space ${\mathbb P}^d$.
We say that $X$ is hyperbolic with respect to a real linear space $V$ of dimension $\ell-1$
if $X \cap V = \emptyset$ and $X$ intersects any real linear space of dimension $\ell$ through $V$
in real points only.
Alternatively, if $Y$ is the associated hypersurface of $X$ in the Grassmanian ${\mathbb G}(\ell-1,d)$
of $\ell-1$-dimensional linear spaces in ${! \mathbb P}^d$, then $V \not\in Y$ and $Y$ intersects any
real one-dimensional Schubert cycle through $V$ in real points only.

In the case $\ell=1$, i.e., $X$ is a hypersurface, this simply means that $X$ is the zero locus of a homogeneous
hyperbolic polynomial.

I will discuss hyperbolic subvarieties of a higher codimension, the analogues of hyperbolicity cones,
and a class of definite Hermitian determinantal representations that witnesses hyperbolicity.
It turns out that the analogue of the Lax conjecture holds --- any real curve in ${\mathbb P}^d$ that is hyperbolic
with respect to some $d-2$-dimensional linear space admits a definite Hermitian, or even real symmetric, determinantal representation.

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