Some remarks on the Hodge conjecture for Abelian varieties of Fermat type (Q2712563)

Let \(X\) be a smooth complex projective variety of dimension \(n\), and denote by \(B^i(X): =H^{2i}(X,\mathbb{Q}) \cap H^{i,i}(X)\), \(0\leq i\leq n\), the space of Hodge cycles of codimension \(n\) on \(X\). Then the famous Hodge conjecture states that \(B^i(X)\) is generated by fundamental classes of algebraic cycles of codimension \(i\) in \(X\). The Hodge conjecture has been intensely studied for various kinds of complex projective varieties, over the past decades, in particular for special types of complex abelian varieties.NEWLINENEWLINENEWLINEThe paper under review is devoted to the Hodge conjecture for abelian varieties of Fermat type of degree \(m>2\). An abelian variety over \(\mathbb{C}\) is said to be of Fermat type of degree \(m>2\) if it is isogenous to a factor of a power of \(J_m\), where \(J_m\) denotes the Jacobian variety of the Fermat curve defined by the equation \(x^m+y^m +z^m=0\). Generalizing an earlier result of \textit{T. Shioda} [Math. Ann. 258, 65-80 (1981; Zbl 0515.14005)], the author proves that the Hodge conjecture indeed holds for abelian varieties of Fermat type of degree \(m\), whenever the number \(m\) is of the formNEWLINENEWLINENEWLINE\(m=2^a\cdot 3^b\cdot 5^c\cdot 7^d\), with either \(c=0\) or \(d=0\), orNEWLINENEWLINENEWLINE\(m=p^e\) or \(m=2p^e\), where \(p\) is an odd prime. NEWLINENEWLINENEWLINEApart from this main result of the paper, the author gives some more affirmative answer to the Hodge conjecture for particular complex abelian varieties. He shows that certain abelian varieties of cyclotomic type satisfy the Hodge conjecture, and he derives some criteria for the validity of the Hodge conjecture for all abelian varieties of Fermat type \(m>2\) in terms of the Hodge conjecture for certain Jacobian varieties of curves.NEWLINENEWLINENEWLINEWithout any doubt, these various special results enhance the common knowledge in this area of research and shed some new light on the general problem of the Hodge conjecture for abelian varieties.