  We present results from simulations of core-collapse supernovae in FLASH
using a newly-implemented multidimensional neutrino transport scheme and a
newly-implemented general relativistic (GR) treatment of gravity. We use a
two-moment method with an analytic closure (so-called M1 transport) for the
neutrino transport. This transport is multienergy, multispecies,
velocity-dependent and truly multidimensional, i.e., we do not assume the
commonly used "ray-by-ray" approximation. Our GR gravity is implemented in our
Newtonian hydrodynamics simulations via an effective relativistic potential
(GREP) that closely reproduces the GR structure of neutron stars and has been
shown to match GR simulations of core collapse quite well. In axisymmetry, we
simulate core-collapse supernovae with four different progenitor models in both
Newtonian and GR gravity. We find that the more compact protoneutron star
structure realized in simulations with GR gravity gives higher neutrino
luminosities and higher neutrino energies. These differences in turn give
higher neutrino heating rates (upwards of $\sim$20-30% over the corresponding
Newtonian gravity simulations) that increase the efficacy of the neutrino
mechanism. Three of the four models successfully explode in the simulations
assuming GREP gravity. In our Newtonian gravity simulations, two of the four
models explode, but at times much later than observed in our GR gravity
simulations. Our results, both in Newtonian and GR gravity, compare well with
several other studies in the literature. These results conclusively show that
the approximation of Newtonian gravity for simulating the core-collapse
supernova central engine is not acceptable. We also simulate four additional
models in GR gravity to highlight the growing disparity between parameterized
1D models of core-collapse supernovae and the current generation of 2D models.
