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The aim of this paper is to investigate uniqueness of conic constant scalarcurvature Kaehler (cscK) metrics, when the cone angle is less than $\pi$. Weintroduce a new H\"older space called $\cC^{4,\a,\b}$ to study the regularitiesof this fourth order elliptic equation, and prove that any $\cC^{2,\a,\b}$conic cscK metric is indeed of class $\cC^{4,\a,\b}$. Finally, the reductivityis established by a careful study of the conic Lichnerowicz operator.

Uniqueness of constant scalar curvature K\"ahler metrics with cone singularities, I: Reductivity