Hodge classes of moduli spaces of parabolic bundles over the general curve. (English) Zbl 0891.14002

Let \(X\) be a smooth projective curve of genus \(g\geq 2\) over \(\mathbb{C}\) and let \( M\) be the moduli space of stable bundles over \(X\) of rank \(n\) and degree \(d\) with \((n,d) =1\). The main theorem of the present article is that when \(X\) is a general curve, the Hodge conjecture holds for \(M\). The proof is by a monodromy argument. Consider the monodromy action on \(H^*(M,\mathbb{Q})\) given by the variation of \(X\). The authors show
(i) that this action is an action of \(Sp(g,\mathbb{Z})\) on \(H^* (M,\mathbb{Q})\),
(ii) that the algebra of invariants in \(H^* (M,\mathbb{Q})\) consists of algebraic cycles constructed explicitly, and
(iii) that any linear subspace of \(H^* (M,\mathbb{C})\) stable under the action of \(Sp(g,\mathbb{Z})\) and consisting entirely of Hodge classes must be contained in the space of invariants in \(H^* (M,\mathbb{C})\).
It follows from these three facts that there exists a no-where dense subset \(B\) of the base space \(U\) of the variation of \(X\) such that the Hodge conjecture is true for the varieties \(M_u\), \(u\in U/B\).


14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14H10 Families, moduli of curves (algebraic)
14H60 Vector bundles on curves and their moduli