  A linear map $\Phi$ between matrix spaces is called cross-positive if it is
positive on orthogonal pairs $(U,V)$ of positive semidefinite matrices in the
sense that $\langle U,V\rangle:=\text{Tr}(UV)=0$ implies $\langle
\Phi(U),V\rangle\geq0$, and is completely cross-positive if all its ampliations
$I_n\otimes \Phi$ are cross-positive. (Completely) cross-positive maps arise in
the theory of operator semigroups, where they are sometimes called
exponentially-positive maps, and are also important in the theory of affine
processes on symmetric cones in mathematical finance. To each $\Phi$ as above a
bihomogeneous form is associated by $p_\Phi(x,y)=y^T\Phi(xx^T)y$. Then $\Phi$
is cross-positive if and only if $p_\Phi$ is nonnegative on the variety of
pairs of orthogonal vectors $\{(x,y)\mid x^Ty=0\}$. Moreover, $\Phi$ is shown
to be completely cross-positive if and only if $p_\Phi$ is a sum of squares
modulo the principal ideal $(x^Ty)$. These observations bring the study of
cross-positive maps into the powerful setting of real algebraic geometry. Here
this interplay is exploited to prove quantitative bounds on the fraction of
cross-positive maps that are completely cross-positive. Detailed results about
cross-positive maps $\Phi$ mapping between $3\times 3$ matrices are given.
Finally, an algorithm to produce cross-positive maps that are not completely
cross-positive is presented.
