# zbMATH — the first resource for mathematics

On the lattice of transcendental cycles on an elliptic surface. (English) Zbl 0619.14007
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.]
Reviewer: T.Fujita

##### MSC:
 14C99 Cycles and subschemes 14J10 Families, moduli, classification: algebraic theory 14J25 Special surfaces
Full Text:
##### References:
 [1] Griffiths, P.A., Harris, J.: Principles of algebraic geometry. New York: John Wiley and Sons, Inc. · Zbl 0836.14001 [2] Kodaira, K.: On compact analytic surfaces, Analytic functions, pp. 121-135. Princeton: University Press 1960. · Zbl 0137.17401 [3] Kodaira, K.: On compact analytic surfaces II, III, Ann. Math.77, (1963); Ann. Math.78, 1-40 (1963). · Zbl 0171.19601 [4] Shioda, T.: On elliptic modular surfaces. J. Math. Soc. Japan24, 20-59 (1972). · Zbl 0226.14013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.