Theta functions and cycles on some abelian fourfolds
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 \(D^2\).
scientific article; zbMATH DE number 922481
scientific article