Hitchin–Thorpe inequality and Kaehler metrics for compact almost Ricci soliton

The purpose of this paper is to prove a Hitchin–Thorpe inequality for a fourdimensional compact almost Ricci soliton. Moreover, we prove that under a suitable integral condition, a four-dimensional compact almost Ricci soliton is isometric to standard sphere. Finally, we prove that under a simple condition, a four-dimensional compact Ricci soliton with harmonic self-dual part of Weyl tensor is either isometric to a standard sphere S^4 or is Kaehler–Einstein.