On the lattice of transcendental cycles on an elliptic surface (Q1089396)

Let \(f: X\to B\) be a smooth projective elliptic surface over \({\mathbb{C}}\) and let T(X) be the lattice of transcendental cycles of X, that is, the orthogonal complement of the Néron-Severi group NS(X) in \(H^ 2(X,{\mathbb{Z}})\) with respect to the cup product. This paper shows that T(X)\(\otimes {\mathbb{Q}}\) is determined up to isomorphism by the functional invariant and the homological invariant of f. The proof is roughly as follows: Let \(f': X'\to B\) be the unique elliptic surface having a section and with the same invariants as f. The author shows that there exist a Galois covering \(C\to B\), an elliptic surface \(Y\to C\) and actions \(\rho\), \(\rho\) ' of \(G=Gal(C/B)\) on Y such that \(\rho\), \(\rho\) ' are compatible with the action of G on C and that Y/\(\rho\cong X\) and Y/\(\rho\) '\(\cong X'\). These actions are proved to induce the same action on T(Y)\(\otimes {\mathbb{Q}}\). So T(X)\(\otimes {\mathbb{Q}}\cong T(X')\otimes {\mathbb{Q}}\) since they are isomorphic to the invariant parts of the above actions. [Reviewer's remark: On page 108, the author claims that D is unramified over the singular locus. This is false if a singular fiber has a component of multiplicity \(>1\). Thus the fiber product of X and C over B is singular in general. However, taking desingularizations and modifying the argument suitably, one can overcome this trouble.]