That is, is there a Kähler manifold $X$ on which there is no map
$$
\tau:X\to X
$$
such that
$$
d\tau\circ I=-I\circ d\tau
$$
and
$$
\tau\circ \tau=\mathrm{Id}\_X?
$$



Is there a Kähler manifold with no anti-holomophic involution?
The moduli space $\mathcal{M}\_1^\mathbb{R}$ of real algebraic curves of genus $1$ equals the real part of the moduli space $\mathcal{M}\_1=\mathbb{C}$. In particular, the general elliptic curve has *no* anti-holomorphic involution, as suggested by P. Achinger in his comment.


Furthermore, similar statements hold also for real curves of higher genera. See


*Seppälä, Mika*, [**Real algebraic curves in the moduli space of complex curves**](http://www.numdam.org/item?id=CM_1990__74_3_259_0), Compos. Math. 74, No. 3, 259-283 (1990). [ZBL0725.14019](https://zbmath.org/?q=an:0725.14019).


