Quotient of an abelian surface by an antisymplectic involution What can we say about the quotient of an abelian surface by an antisymplectic involution?
 A: Denote by $\sigma$ the involution on the abelian surface $A$ and set $X:=A/\sigma$. The eigenvalues of the action of $\sigma$ on $H^0(\Omega^1_A)$ are $+1$ and $-1$.
So $\sigma$ has no isolated fixed points and it follows that $X$ is smooth with $h^1(\mathcal O_X)=1$.
There are two possibilities:
1) $\sigma$ has no fixed point. Then the quotient map $A\to X$ is \'etale and $X$ is a bielliptic surface almost by definition. 
2) $\sigma$ has a fixed point. Then, up to a translation,  we may assume that $\Sigma$ is an endomorphism. The fixed locus is a union of elliptic curves and $X$ is ruled. This can be seen either by classification of surfaces or observing that $A$ contains a family of translates of elliptic curves on which $\sigma$ acts as multiplcation by $-1$. 
A: Averaging a Kaehler class over the involution, and taking
the corresponding Ricci-flat metric, we may assume that
the involution preserves a flat metric on a torus. 
At each fixed point, the eigenvalues of the involution
are +1, -1, and the fixed point set is a subtorus $T_0\subset T$.
This means that the involution acts as -1 on $T_1 :=T/T_0$,
hence the quotient is a product $T_0\times T_1/\{\pm 1\}$,
that is, $T_0 \times {\Bbb C} P^1$.
