  Strong asymptotics on the whole complex plane of a sequence of monic Jacobi
polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$
\lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta
_n}{n}=B, $$ with $A$ and $B$ satisfying $ A > -1$, $ B>-1$, $A+B < -1$. The
asymptotic analysis is based on the non-Hermitian orthogonality of these
polynomials, and uses the Deift/Zhou steepest descent analysis for matrix
Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived.
We show that in a generic case the zeros distribute on the set of critical
trajectories $\Gamma$ of a certain quadratic differential according to the
equilibrium measure on $\Gamma$ in an external field. However, when either
$\alpha_n$, $\beta_n$ or $\alpha_n+\beta_n$ are geometrically close to $\Z$,
part of the zeros accumulate along a different trajectory of the same quadratic
differential.
