On the locus of Hodge classes

Source abstract

Let S S be a nonsingular complex algebraic variety and V \mathcal {V} a polarized variation of Hodge structure of weight 2 p 2p with polarization form Q Q . Given an integer K K , let S ( K ) {S^{(K)}} be the space of pairs ( s , u ) (s,u) with s ∈ S s \in S , u ∈ V s u \in {\mathcal {V}_s} integral of type ( p , p ) (p,p) , and Q ( u , u ) ≤ K Q(u,u) \leq K . We show in Theorem 1.1 that S ( K ) {S^{(K)}} is an algebraic variety, finite over S S . When V \mathcal {V} is the local system H 2 p ( X s , Z ) {H^{2p}}({X_s},\mathbb {Z}) /torsion associated with a family of nonsingular projective varieties parametrized by S S , the result implies that the locus where a given integral class of type ( p , p ) (p,p) remains of type ( p , p ) (p,p) is algebraic.