Cutting-edge issues of core-collapse supernova theory

Based on multi-dimensional neutrino-radiation hydrodynamic simulations, we report several cutting-edge issues about the long-veiled explosion mechanism of core-collapse supernovae (CCSNe). In this contribution, we pay particular attention to whether three-dimensional (3D) hydrodynamics and/or general relativity (GR) would or would not help the onset of explosions. By performing 3D simulations with spectral neutrino transport, we show that it is more difficult to obtain an explosion in 3D than in 2D. In addition, our results from the first generation of full general relativistic 3D simulations including approximate neutrino transport indicate that GR can foster the onset of neutrino-driven explosions. Based on our recent parametric studies using a light-bulb scheme, we discuss impacts of nuclear energy deposition behind the supernova shock and stellar rotation on the neutrino-driven mechanism, both of which have yet to be included in the self-consistent 3D supernova models. Finally we give an outlook with a summary of the most urgent tasks to extract the information about the explosion mechanisms from multi-messenger CCSN observables.