A Meromorphic Pairwise-Trace Criterion for the Algebraicity of Hodge Classes

We give a criterion for the algebraicity of rational cohomology classes on a smooth complex projective variety in terms of pairwise traces of a finite family of sections spanning an endomorphism bundle. We consider a holomorphic vector bundle, with normalized lower Chern-character components, on the complement of a divisor in an auxiliary space determined by a projective embedding. In even codimension, meromorphic extendibility of the pairwise traces to a neighborhood of the zero section implies algebraicity of the prescribed class. The proof uses the coherent image of the trace Gram matrix, its reflexive hull, derived restriction to the zero section, and Serre's algebraic--analytic comparison theorem (GAGA). Failure of generation is allowed in codimension at least two; neither extension of the original bundle nor extension of multiplication is required. Conversely, an algebraic class gives rise to a bundle and a generating family satisfying the criterion. Odd codimension is treated by passing to a new class on the product with the projective line. The criterion does not assert that the general Hodge condition supplies the required meromorphic extensions.