Theta functions and cycles on some abelian fourfolds
type: Item
instance of: scholarly article
title: Theta functions and cycles on some abelian fourfolds
zbMATH Open document ID: 0862.14029
DOI: 10.1007/BF02622136
published in: Mathematische Zeitschrift
publication date: 1996-11-05T00:00:00Z
full work available at URL: https://eudml.org/doc/174870
review text: A complex abelian variety of Weil type of dimension \(2n\) is a pair \((X,K)\) with a \(2n\)-dimensional abelian variety \(X\) and an embedding of an imaginary quadratic field \(K\) into \(\text{End} (X) \otimes \mathbb{Q}\) such that every \(\alpha \in K\subset \text{End} (X) \otimes \mathbb{Q}\) has \(n\) eigenvalues \(\alpha\) and \(n\) eigenvalues \(\overline \alpha\) (with respect to a fixed embedding \(K\subset \mathbb{C})\). The group of Hodge classes \(B^p(X): = H^{2p} (X, \mathbb{Q}) \cap H^{p,p} (X)\) contains naturally the subring \(D^p\) of those Hodge classes which are generated by intersections of divisor classes. For a general abelian variety \(X\) the Hodge \((p,p)\)-conjecture holds according to results of Mattuck, so \(B^p(X) = D^p(X) = \mathbb{Q}\) for all \(p\). However for abelian varieties \((X,K)\) of Weil type of dimension \(2n\), \(n>1\), \textit{A. Weil} has shown [in ``Abelian varieties and the Hodge ring''. In: Collected Papers, Vol. III, 421-429 (1980; Zbl 0428.01014)] that \(B^1(X) = \mathbb{Q}\) (and thus \(D^p(X) = \mathbb{Q}\) for all \(p)\) but \(\dim B^n(X) =3\). In fact \(B^n(X)\) is the direct sum of \(D^n(X)\) and the 2-dimensional subspace of Weil-Hodge cycles \(W= \bigwedge_K^{2n} H^1(X, \mathbb{Q})\). For certain abelian varieties \((X,K)\) of Weil type with \(K= \mathbb{Q}(i)\), \(\mathbb{Q} (\omega)\), \(\omega^3= -1\), \textit{C. Schoen} gave in Compos. Math. 65, No. 1, 3-32 (1988; Zbl 0663.14006) a construction for algebraic cycles spanning \(W\). In the paper under consideration theta functions are used to construct algebraic cycles on principally polarized abelian 4-folds \((X, \mathbb{Q} (i))\) of Weil type. In fact it is shown that such abelian varieties allow a rational map (a projection of the Kummer map) onto a smooth quadric \(\overline Q\subset \mathbb{P}^5\) whose rulings yield classes which do not lie in \(D^2\).
Mathematics Subject Classification ID: 14K25
Mathematics Subject Classification ID: 14C30
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14J35
zbMATH DE Number: 922481
zbMATH Keywords: complex abelian variety of Weil type
zbMATH Keywords: Hodge classes
zbMATH Keywords: theta functions
author: Bert van Geemen
reviewed by: Christina Birkenhake
MaRDI profile type: Publication
cites work: Q3795785
cites work: Theta functions on Riemann surfaces
cites work: Q4320865
cites work: Kummer Varieties and the Moduli Spaces of Abelian Varieties
cites work: Q3813897
OpenAlex ID: W4247937634
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description: scientific article; zbMATH DE number 922481