On the lattice of transcendental cycles on an elliptic surface.

*(English)*Zbl 0619.14007Let \(f: X\to B\) 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.]

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 |

##### Keywords:

elliptic surface; lattice of transcendental cycles; Néron-Severi group; functional invariant; homological invariant**OpenURL**

##### 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 |

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