An extension of the (1,2)-symplectic property for $f$-structures on flag manifolds

N. Cohen ; S. Pinzon

Izvestiya. Mathematics , Tome 72 (2008) no. 3, pp. 479-496

Résumé

The (1,1)-symplectic property for $f$-structures on a complex Riemannian manifold $M$ is a natural extension of the (1,2)-symplectic property for almost-complex structures on $M$, and arises in the analysis of complex harmonic maps with values in $M$. A characterization of this property in combinatorial terms is known only for almost-complex structures or when $M$ is the classical flag manifold $\mathbb{F}(n)$. In this paper, we remove these restrictions by considering an intersection graph defined in terms of the corresponding root system. We prove that the $f$-structure is (1,1)-symplectic exactly when the intersection graph is locally transitive. Our intersection graph construction may be helpful in characterizing many other Kähler-like properties on complex flag manifolds.