Classification of generalized K\"ahler-Ricci solitons on complex surfaces

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in arXiv:1802.00170. This construction also reveals that these solitons are generalized K"ahler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized K"ahler structures with nonvanishing Poisson structure on non-standard Hopf surfaces, completing the existence question for such structures. Moreover this gives a complete answer to the existence question for generalized K"ahler-Ricci solitons on compact complex surfaces. In the setting of generalized K"ahler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized K"ahler-Ricci flow among metrics with maximal symmetry.