Classification of Generalized Kähler‐Ricci Solitons on Complex Surfaces

Abstract

Using toric geometry we give an explicit construction of the compact steady solitons for pluriclosed flow first constructed in by the first author in 2019. This construction also reveals that these solitons are generalized Kähler in two distinct ways, with vanishing and nonvanishing Poisson structure. This gives the first examples of generalized Kähler structures with nonvanishing Poisson structure on nonstandard Hopf surfaces, completing the existence question for such structures. Moreover, this gives a complete answer to the existence question for generalized Kähler-Ricci solitons on compact complex surfaces. In the setting of generalized Kähler geometry with vanishing Poisson structure, we show that these solitons are unique. We show that these solitons are global attractors for the generalized Kähler-Ricci flow among metrics with maximal symmetry.

Generalized Kähler-Ricci flow on toric Fano varieties

“…Proposition 6.2 shows that the only self-similar solutions of the normalized GKRF with symplectic-type initial data are actually Kähler-Ricci solitons. In contrast to the shrinking case, there exist non-Kähler steady solitons for GKRF [68,77,78].…”

On the dynamical behaviour of the generalized Ricci flow

“…Remark 2.3. Notice that the hypothesis µ(g, b) ≤ 0 in Lemma 2.2 is necessary, due to the existence of non-trivial (i.e., with non-constant f g,H ) solitons for the gRF [18,23].…”

Bismut Ricci flat manifolds with symmetries

“…for some vector field X ∈ Γ(T M ). The existence of non-trivial solitons on compact (complex) 4-manifolds has been recently proved in [23,27]. Remarkably, the flow (1.2) can be seen as a generalization of Hamilton's Ricci flow to Bismut connections with closed torsion form [20] and as a flow of generalized metrics on exact Courant algebroids [13,21].…”