Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question 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 $f^{*}\omega =-\omega$?

• Note that the question has received an answer on MSE. I am sure there is more to be said though
– user35370
Commented Jun 3, 2015 at 2:12
• @PaulPlummer That answer in MSE is very interesting and deep. I need to spend time to underestand its details. However I presented this question in MO befor his answer in MSE. Any way, as you said for manifold without boundary there are some thing to be said. Thanks for your comment on this question. Commented Jun 3, 2015 at 5:43
• @EmilyMaw Thank you so much for your edits. Commented Aug 24, 2017 at 18:54

I'd like to mention the work of Castaño-Bernard-Matessi-Solomon, 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.

• +1 for your deep answer. I will learn a lot from it. Commented Jun 3, 2015 at 21:19