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.

  • $\begingroup$ Kaehler-Einstein metrics involve simultaneously a Riemannian metric and an (integrable) complex structure that are compatible. Are you asking to replace the Riemannian metric by a symplectic form that is compatible with the almost complex structure? In that case, the symplectic form is equivalent to the data of a Kaehler metric on the complex manifold. If this is not what you mean, what precisely do you mean? $\endgroup$ Mar 5 '18 at 12:17
  • $\begingroup$ A Kaehler manifold could be viewed as a smooth manifold equipped with three structures, i.e. a Riemannian metric, a symplectic form and an integrable complex structure. The compatibility condition means that any two will determine the other. In the symplectic case, I am wondering if it is plausible to find out the "best" almost complex structure compatible with the symplectic form, which in turn determines a "best" Riemannian metric compatible with the symplectic structure. Actually, I don't know the right proposal and I am asking for one. $\endgroup$ Mar 5 '18 at 14:55
  • $\begingroup$ For a given symplectic form, many different integrable almost complex structures can all give rise to Kaehler-Einstein manifolds. For instance, the hyperbolic metric on a Riemann surface of genus $g$ is a Kaehler-Einstein metric, there is a moduli space of dimension $3g-3$ for the complex structures, yet there is only one symplectic manifold (up to diffeomorphism and scaling the symplectic form). $\endgroup$ Mar 5 '18 at 15:05
  • $\begingroup$ Well in complex dimension 1 there is actually no "interesting" symplectic geometry happening because one can rescale and apply Moser's trick. It might be interesting to see the "moduli space of special almost complex structures compatible with a symplectic form" given by different symplectic forms, and the word "special" here might relate to some kind of canonical metrics. $\endgroup$ Mar 5 '18 at 15:16
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    $\begingroup$ @ShaoyunBai Were it not for the 4 upvotes, I am tempted to click "flag -> should be closed... -> unclear what you are asking". $\endgroup$
    – Fan Zheng
    Jun 9 '18 at 22:00

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