The investigation of canonical metrics in Kaehler geometry has led to many monumental works in geometric analysis, such as Yau's solution to the Calabi conjecture and more recently, relations between stability in the sense of algebraic geometry and Kaehler-Einstein metrics together with recent breakthroughs on cscK metrics. The existence of such metrics has numerous applications to topology and algebraic geometry, for example, the uniqueness of projective spaces and Chern number inequalities, or Donaldson's work on instanton moduli spaces using Hermitian-Einstein metrics.
Is it possible and meaningful to generalize canonical metrics to symplectic manifolds? As for a given symplectic structure, there are "abundant" almost complex structures compatible with the symplectic form. Is it possible to extract information about the underlying manifold by generalizing canonical metrics?
Obviously, the situations are quite different as much rigidity in the Kaehler setting comes from the complex structure.