### Question


The question asked is:



> 
> On a manifold $M$ equipped with a Riemann metric $g$ and a symplectic structure $\omega$, with $\ast$ the Hodge star and $\ast\_s$ the symplectic star, does $\ast=\ast\_s$ iff $(M,g,\omega)$ is Kähler?
> 
> 
> 


### Answer: no


For the reason [posted below](https://mathoverflow.net/questions/97160/does-equality-of-hodge-star-and-symplectic-star-imply-kahler-structure/97264#97264).


### Definitions


Here the star operators $\ast$ and $\ast\_s$ are defined in the usual way (see for example, equation 2.9 of Tseng and Yau *[Cohomology and Hodge Theory on Symplectic Manifolds: II](http://arxiv.org/abs/1011.1250 "review by Li-Sheng Tseng and Shing-Tung Yau")*):


$$ A\wedge \ast B := \langle A,B\rangle\_{g^{-1}} dV\_g$$
$$A\wedge \ast\_{s} B := \langle A,B\rangle\_{\omega^{-1}} dV\_\omega $$


where $A$ and $B$ are $k$-forms, $\langle\cdot,\cdot\rangle\_{g^{-1}}$ is the inner product associated to $g$, $dV\_g$ is the volume form associated to $g$, and $\langle\cdot,\cdot\rangle\_{\omega^{-1}}$ and $dV\_\omega$ are defined analogously with respect to $\omega$.


### Motivation


In seeking a mathematically natural link between Hilbert-space expositions of quantum mechanics and [product space expositions](http://arxiv.org/abs/0907.2796 "review of matrix product states"), both the Hodge star operator and the symplectic star operator enter naturally (for example, in the context of [Onsager theory](http://faculty.washington.edu/sidles/ENC_2011/Onsager_transport.pdf "summary of pullback dynamics")). Moreover, in cases of practical quantum systems engineering interest it is commonly observed that:


* the two star operations are identical, and
* the dynamical state-manifold is Kählerian.


Does each of these observations mathematically imply the other? An engineer-friendly reference for this fact (if it is a fact) would be very welcome---it's not easy to find discussion of the practical relevance of the symplectic star operation.



Does equality of Hodge star and symplectic star imply Kähler structure?
Dear John, 


The symplectic Hodge star and Riemannian Hodge star (associated to a Kahler metric) are never identical to each other.


In fact, it is proved in Brylinski's paper on symplectic Hodge theory that on a Kahler manifold, for a $(p,q)$ form $\alpha$ we have that


$$\* \alpha =(-1)^{p-q}\*\_s \alpha.$$
One can also find a detailed proof of the above identity in Guillemin's lecture notes on Hodge theory, which is available at


<http://math.mit.edu/~vwg/shlomo-notes.pdf>


