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Definability of mixed period maps

We equip integral graded-polarized mixed period spaces with a natural $\mathbb{R}_{alg}$-definable analytic structure, and prove that any period map associated to an admissible variation of integral graded-polarized mixed Hodge structures is definable in $\mathbb{R}_{an,exp}$ with respect to this structure. As a consequence we reprove that the zero loci of admissible normal functions are algebraic.