Holomorphic symplectic geometry of elliptic surfaces

When a complex surface $X$ admits a nowhere vanishing holomorphic 2-form, it determines a (holomorphic) symplectic structure on $X$. We study the symplectic geometry of such a symplectic structure when $X$ is an elliptic surface. When the elliptic fibration is nonisotrivial, we define a factorization of Kodaira's functional invariant, called the symplecto-functional invariant and prove that the symplecto-functional invariant determines the symplectic geometry of a nonisotrivial elliptic fibration. This leads to a classification of isogenies of nonisotrivial symplectic elliptic fibrations with a fixed source. We also classify isogenies of symplectic elliptic fibrations with a fixed target by studying symplectic automorphisms of germs of singular fibers. As an application, we prove that a symplecto-biholomorphic map between germs of fibers of nonisotrivial elliptic K3 surfaces can be extended to compositions of isogenies of K3 surfaces.