$T^2$-fibered K3 surface with involution - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T04:23:45Z http://mathoverflow.net/feeds/question/106467 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/106467/t2-fibered-k3-surface-with-involution $T^2$-fibered K3 surface with involution Carmen 2012-09-05T23:21:32Z 2012-09-06T00:41:28Z <p>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 $k \in \mathbb{N}$). </p> <p>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? </p> <p><strong>Edit</strong> 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. </p>