Hausdorff property of the Zucker extension at the monodromy invariant subspace

We prove the Hausdorff property at the monodromy invariant subspace of the Zucker extension of the family of intermediate Jacobians if the divisor at infinity is smooth. Using a recent theorem of Green, Griffiths and Kerr, it implies in this case the analyticity of the closure of the zero locus of an admissible normal function, generalizing a theorem of Brosnan and Pearlstein in the curve case where the assertion is equivalent to the finiteness.