The Hodge conjecture for certain moduli varieties
Let \(C\) be a smooth irreducible curve of genus \(g\geq 2\) over the complex numbers, and let \(M(n,\xi)\) be the moduli space of stable vector bundles \(V\) of rank \(n\) on \(C\) such that \(\text{det }V\cong \xi\), where \(\xi\) is a line bundle of degree \(d\) such that \(n\) and \(d\) are coprime. In this paper, the author proves the Hodge conjecture Hodge\((p,p)\) for the cases \(n=3\), \(g=2\) and \(n=2\), \(g=3,4\). In the case \(n=2\), results of \textit{M. Thaddeus} [Invent. Math. 117, No. 2, 317-353 (1994)] show that \(M(2,\xi)\) is dominated by a variety that is obtained from a projective space by blowing up and down along smooth centers of dimension at most 3, and the Hodge conjecture follows by an argument of \textit{J. P. Murre} [Nederl. Akad. Wet., Proc., Ser. A 80, 230-232 (1977; Zbl 0352.14006)]. The paper is mainly devoted to the case \(n=3\). The author uses the method of normal functions (due to H. Poincaré and S. Lefschetz, and further developed by P. Griffiths and S. Zucker). In this case the normal functions are not associated to a Lefschetz pencil, but to a fibration over a curve whose fibers are projective bundles over \(M(2,\xi)\). The proof is finished using Hecke correspondences.
scientific article; zbMATH DE number 903372
scientific article