Kählerity of complete almost-Kähler gradient shrinking Ricci solitons

Abstract: In this paper, we prove that any complete, compact or noncompact, almost-Kähler gradient shrinking Ricci soliton is Kähler in arbitrary even dimension. Among other applications, combining our result with the classification of complete Kähler-Ricci shrinker surfaces, we obtain a full classification of complete almost-Kähler gradient shrinking Ricci solitons in real dimension four.