Action of orientation-preserving isometric involution on complex structure Let $(M, J, \omega)$ be a compact Kähler manifold. Let $\phi:M\to M$ is an orientation-preserving isometric involution.
Given a point $p\in M$ must there exist a decomposition $T_pM=\oplus_i W_i$ with each $W_i$ being preserved by both $J$ and $\phi^*J$ such that $J|_{W_i}=\pm \phi^*J|_{W_i}$?
 A: No.  Here is a construction.
It is not hard to see that there is an orientation-preserving isometry $L:\mathbb{H}\to\mathbb{H}$ (where $\mathbb{H}\simeq\mathbb{R}^4$ is the ring of quaternions) such that $L^2=1$, namely,
$$
L(x) = \tfrac12 (j+k)\,x\,(j+k).
$$
This isometry satisfies $L(jx)=kL(x)$.
Now let $\Lambda\subset\mathbb{H}$ be a lattice that is preserved by $L$ (for example, it could be generated by lattices in the two $2$-dimensional eigenspaces of $L$), and let $M = \mathbb{H}/\Lambda$.  Then $L$ induces an orientation-preserving isometric involution $\phi$ of $M$.
Moreover, left multliplication by $j$ (respectively, $k$) defines a translation-invariant orthogonal complex structure $J$ (respectively, $K$) on $\mathbb{H}$ that descends to $M$, and, by the equation $L(jx)=kL(x)$, we have $\phi^*(J) = K$.
Note that, with the quotient flat metric $g$ on $M$, we have that $(M,J,g)$ and $(M,K,g)$ are Kähler manifolds and $\phi$ is an orientation-preserving isometric involution of $g$.
For any tangent vector $v$, the vectors $v$, $Jv$, and $Kv$ are linearly independent, hence there are no $2$-dimensional subspaces of the tangent space at any point that are both $J$ and $K$ invariant.  Thus, the only possibility would be to take $W_i = T_pM$, but $J\not=\pm K$ on $T_pM$ for any $p\in M$.
