Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration. Assume there is a $k$-section, then a fiber and the $k$-section generate a sublattice $L\subset H^{2}(S,\mathbb{Z})$, which is isomorphic to $U(k)$ (hyperbolic lattice multiplied by $k \in \mathbb{N}$).

Assume also that there is an holomorphic involution $\sigma$ of $S$ such that induced action $\sigma^{*}$ acts as $-id$ on $L$ (especially preserves $L\subset H^{2}(S,\mathbb{Z})$). Is it true that $\sigma$ preserves the fibration? If so, could one tell how $\sigma$ acts on each smooth fiber of $f$? If not so, what additional condition is required?

Edit The original question does not mach much sense. The fibration is NOT holomorphic. the following is the motivation of my question. I have a K3 surface $S$ with an anti-symplectic involution $\sigma$. Assume that, by using another complex structure, we can construct an elliptic fibration $f:S\rightarrow \mathbb{P}^{1}$ (possibly with no section). I hope this map to be a special Lagrangian $T^2$-fibration with respect to the original complex structure. I now want to understnad how $\sigma$ and the map $f$ are related.

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration (not necessarily holomorphic, I have a special Langrangian fibration in mind). Assume there is a $k$-section, then a fiber and the $k$-section generate a sublattice $L\subset H^{2}(S,\mathbb{Z})$, which is isomorphic to $U(k)$ (hyperbolic lattice multiplied by $k \in \mathbb{N}$).

Assume also that there is an holomorphic involution $\sigma$ of $S$ such that induced action $\sigma^{*}$ acts as $-id$ on $L$ (especially preserves $L\subset H^{2}(S,\mathbb{Z})$). Is it true that $\sigma$ preserves the fibration? If so, could one tell how $\sigma$ acts on each smooth fiber of $f$? If not so, what additional condition is required?

Edit The original question does not mach much sense. The fibration is NOT holomorphic. the following is the motivation of my question. I have a K3 surface $S$ with an anti-symplectic involution $\sigma$. Assume that, by using another complex structure, we can construct an elliptic fibration $f:S\rightarrow \mathbb{P}^{1}$ (possibly with no section). I hope this map to be a special Lagrangian $T^2$-fibration with respect to the original complex structure. I now want to understnad how $\sigma$ and the map $f$ are related.

Let $f:S\rightarrow \mathbb{P}^1$ be an elliptic fibration (with possibly no section). Assume that there is a $k$-section of $f$, then a fiber and the $k$-section generate a sublattice $L\subset H^{2}(S,\mathbb{Z})$, which is isomorphic to $U(k)$ (hyperbolic lattice multiplied by $k \in \mathbb{N}$).

Assume also that there is an involution $\sigma$ of $S$ such that induced action $\sigma^{*}$ acts as $-id$ on $L$ (especially preserves $L$). Is it true that $\sigma$ preserves the fibration and acts as $-id$ (of elliptic curve) on each smooth fiber of $f$?

  If this is not true, what additional condition is required?

Thanks in advance,

Let $S$ be a K3 surface and $f:S\rightarrow \mathbb{P}^1$ a $T^2$-fibration. Assume there is a $k$-section, then a fiber and the $k$-section generate a sublattice $L\subset H^{2}(S,\mathbb{Z})$, which is isomorphic to $U(k)$ (hyperbolic lattice multiplied by $k \in \mathbb{N}$).

Assume also that there is an holomorphic involution $\sigma$ of $S$ such that induced action $\sigma^{*}$ acts as $-id$ on $L$ (especially preserves $L\subset H^{2}(S,\mathbb{Z})$). Is it true that $\sigma$ preserves the fibration? If so, could one tell how $\sigma$ acts on each smooth fiber of $f$? If not so, what additional condition is required?

Edit The original question does not mach much sense. The fibration is NOT holomorphic. the following is the motivation of my question. I have a K3 surface $S$ with an anti-symplectic involution $\sigma$. Assume that, by using another complex structure, we can construct an elliptic fibration $f:S\rightarrow \mathbb{P}^{1}$ (possibly with no section). I hope this map to be a special Lagrangian $T^2$-fibration with respect to the original complex structure. I now want to understnad how $\sigma$ and the map $f$ are related.