On the lattice of transcendental cycles on an elliptic surface
type: Item
instance of: scholarly article
title: On the lattice of transcendental cycles on an elliptic surface
zbMATH Open document ID: 0619.14007
DOI: 10.1007/BF01163358
author: Mangala Nori
published in: Mathematische Zeitschrift
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173772
review text: 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.]
reviewed by: Takao Fujita
Mathematics Subject Classification ID: 14C99
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J25
zbMATH DE Number: 4004342
zbMATH Keywords: elliptic surface
zbMATH Keywords: lattice of transcendental cycles
zbMATH Keywords: Néron-Severi group
zbMATH Keywords: functional invariant
zbMATH Keywords: homological invariant
MaRDI profile type: Publication
cites work: Q4846449
cites work: Q5510176
cites work: On compact analytic surfaces. III
cites work: On elliptic modular surfaces
OpenAlex ID: W2032455221
description: scientific article; zbMATH DE number 4004342