  Motivated by a recent paper by the Potsdam numerical relativity group, we
have constructed a new numerical code for hydrodynamic simulation of
axisymmetric systems in full general relativity. In this code, we solve the
Einstein field equation using Cartesian coordinates with appropriate boundary
conditions. On the other hand, the hydrodynamic equations are solved in
cylindrical coordinates. Using this code, we perform simulations to study
axisymmetric collapse of rotating stars, which thereby become black holes or
new compact stars, in full general relativity. To investigate the effects of
rotation on the criterion for prompt collapse to black holes, we first adopt a
polytropic equation of state, $P=K\rho^{\Gamma}$, where $P$, $\rho$, and $K$
are the pressure, rest mass density, and polytropic constant, with $\Gamma=2$.
In this case, the collapse is adiabatic (i.e., no change in entropy), and we
can focus on the bare effect of rotation. As the initial conditions, we prepare
rigidly and differentially rotating stars in equilibrium and then decrease the
pressure to induce collapse. In this paper, we consider cases in which $q
\equiv J/M_g^2 < 1$, where $J$ and $M_g$ are the angular momentum and the
gravitational mass. It is found that the criterion of black hole formation is
strongly dependent on the angular momentum parameter $q$. For $q < 0.5$, the
criterion is not strongly sensitive to $q$; more precisely, if the rest mass is
slightly larger than the maximum allowed value of spherical stars, a black hole
is formed. However, for $q \alt 1$, it changes significantly: For $q \simeq
0.9$, the maximum allowed rest mass becomes $\sim 70$ - 80% larger than that
for spherical stars.
