On 4-Dimensional J-Invariant Shrinking Ricci Solitons

As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are K¨ahler and/or Einstein. In this note, we prove that all four dimensional J-invariant gradient shrinking Ricci solitons satisfy a differential form identity relating K¨ahlerity and Einstein-ness.