I Ask [this question](https://math.stackexchange.com/questions/1292642/symplectic-reversing-diffeomorphisms) in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:


Let $(M,\omega)$ be a compact symplectic manifold.


Is there always a diffeomorphism $f$ on M with $f^{\*}\omega =-\omega$?



Symplectic reversing diffeomorphisms on a compact symplectic manifold
I'd like to mention the work of [Castaño-Bernard-Matessi-Solomon](http://arxiv.org/abs/0908.0966), who proved the existence of an anti-symplectic involution for symplectic manifolds carrying a Lagrangian torus fibration of a certain class. Such a class of Lagrangian fibration is constructed by gluing local models of Lagrangian fibrations to integral affine manifolds with singularities.


More precisely, let $(B,\mathscr{A},\Delta)$ be an integral affine manifold, whose affine structure $\mathscr{A}$ is singular along $\Delta\subset B$. In this case, we have a decomposition


$\Delta=\Delta\_p\cup\Delta\_g\cup\Delta\_n$,


where the subscripts indicate the types of the singularity, namely positive, generic or negative. For simplicity, let's restricts to the case of a symplectic 6-manifold. After taking away the discriminant locus, $B\setminus\Delta$ carries the structure of an integral affine manifold, and we may form a Lagrangian torus bundle $T^\ast(B\setminus\Delta)/\Lambda^\ast$ in the obvious way, where $\Lambda\subset T(B\setminus\Delta)$ is a lattice bundle coming from the affine structure $\mathscr{A}$ on $B\setminus\Delta$ and $\Lambda^\ast$ denotes its dual. The aim is to glue local models of singular Lagrangian fibrations using fiber-preserving symplectomorphism to the torus bundle $T^\ast(B\setminus\Delta)/\Lambda^\ast$ so that we obtain a Lagrangian fibration $\pi:M\rightarrow B$ on the symplectic 6-manifold $M$ which serves as a compactification of the previous torus bundle.


For a small neighborhood $U\_p\subset B$ so that $\Delta\_p\subset U\_p$, one considers the standard Lagrangian fibration $\pi\_{std}:\mathbb{C}^3\setminus\{xyz=-1\}\rightarrow\mathbb{R}^3$ defined by


$(x,y,z)\rightarrow\left(\log|xyz+1|,\left(|x|^2-|y|^2\right)/2,\left(|x|^2-|z|^2\right)/2\right)$


Similarly one has local models of Lagrangian fibrations for $\Delta\_g\subset U\_g$ and $\Delta\_n\subset U\_n$. The outcome is a Lagrangian torus fibration $\pi:M\rightarrow B$ of preferred topology and singularities. In the paper of Castaño-Bernard-Matessi-Solomon, they call such Lagrangian fibrations belong to class $\mathscr{C}$.


For these Lagrangian fibrations, one can analyze their local models explicitly and glue local Lagrangian sections to a global one. One distinguished property of a Lagrangian fibration $\pi$ in $\mathscr{C}$ is that $\pi$ is not smooth near $\Delta\_n$, so one has to impose some restrictions on the Lagrangian sections $\sigma$ of $\pi$ we work with, these restrictions specify a class $\mathfrak{C}$ of Lagrangian sections. With these preliminaries the main result of Castaño-Bernard-Matessi-Solomon can be stated as follows:


**Theorem.** Let $\pi:M\rightarrow B$ be a Lagrangian fibration of class $\mathscr{C}$. Given a Lagrangian section $\sigma$ of $\pi$ in class $\mathfrak{C}$, there is a unique antisymplectic involution $\phi\_{\pi,\sigma}$ of $M$ such that $\phi\_{\pi,\sigma}$ preserves the fibers of $\pi$ and fixes the Lagrangian section $\sigma$.


Notice that the class $\mathscr{C}$ is actually very general, it includes Lagrangian fibrations without singular fibers (cotangent bundles, tori), the so-called almost toric symplectic manifolds introduced by Leung-Symington (which plays an essential role in the recent deep work of Vianna in symplectic topology), and 6-dimensional symplectic Calabi-Yau manifolds homeomorphic to a quintic or Schoen's Calabi-Yau manifolds.


**Speculations.** From the point of view of mirror symmetry, the non-existence of an anti-symplectic involution roughly means the mirror of $M$ does not exist or at least carries a gerbe $\alpha\in H^2\_{et}(M^\vee,\mathscr{O}^\ast)$. This is because the anti-symplectic involutions constructed by Castaño-Bernard-Matessi-Solomon (which are roughly reflections with respect to the Lagrangian section) are in some sense mirror to the functor $R\underline{Hom}(-,\mathscr{O}\_{M^\vee})$ on the bounded derived category of coherent sheaves $D^b(M^\vee)$. It should be interesting to establish this speculation rigorously at least in some specific examples. But I don't know how to do this.


