Method for the manufacture of a complex component with following method steps: constructing the component from a metal material in an additive manufacturing method with at least one cavity segment that has a cavity open on at least one side and defined by an interior surface of the component; forming an auxiliary electrode during the construction of the component; forming at least two supporting structures that connect the auxiliary electrode to the interior surface of the component during the construction of the component; electrically insulating the auxiliary electrode from the interior surface by mechanically separating the at least two supporting structures from the interior surface or from the auxiliary electrode, wherein electrically insulating includes: inserting a separating head of a separating tool into an opening, defined by the interior surface, of the cavity along the auxiliary electrode, wherein the separating head has an interior circumferential surface that defines and encloses a longitudinal axis and, oriented opposite to this, an exterior circumferential surface, and cutting through the at least two supporting structures with a separating segment of the separating head which, with reference to the longitudinal axis, forms a front end of the separating head, wherein the separating head is inserted into the opening sufficiently far that the at least two supporting structures lie against an electrically insulating material forming, at least in segments, the interior circumferential surface and/or the exterior circumferential surface with reference to the longitudinal axis; and performing an electro-polishing of the interior surface in an electrolyte bath by connecting the component and the auxiliary electrode to different poles of an electric voltage source in such a way that the auxiliary electrode forms a cathode and the component forms an anode, wherein the separating head remains arranged in the cavity during the electro-polishing. constructing the component from a metal material in an additive manufacturing method with at least one cavity segment that has a cavity open on at least one side and defined by an interior surface of the component; forming an auxiliary electrode during the construction of the component; forming at least two supporting structures that connect the auxiliary electrode to the interior surface of the component during the construction of the component; electrically insulating the auxiliary electrode from the interior surface by mechanically separating the at least two supporting structures from the interior surface or from the auxiliary electrode, wherein electrically insulating includes: inserting a separating head of a separating tool into an opening, defined by the interior surface, of the cavity along the auxiliary electrode, wherein the separating head has an interior circumferential surface that defines and encloses a longitudinal axis and, oriented opposite to this, an exterior circumferential surface, and cutting through the at least two supporting structures with a separating segment of the separating head which, with reference to the longitudinal axis, forms a front end of the separating head, wherein the separating head is inserted into the opening sufficiently far that the at least two supporting structures lie against an electrically insulating material forming, at least in segments, the interior circumferential surface and/or the exterior circumferential surface with reference to the longitudinal axis; and performing an electro-polishing of the interior surface in an electrolyte bath by connecting the component and the auxiliary electrode to different poles of an electric voltage source in such a way that the auxiliary electrode forms a cathode and the component forms an anode, wherein the separating head remains arranged in the cavity during the electro-polishing. inserting a separating head of a separating tool into an opening, defined by the interior surface, of the cavity along the auxiliary electrode, wherein the separating head has an interior circumferential surface that defines and encloses a longitudinal axis and, oriented opposite to this, an exterior circumferential surface, and cutting through the at least two supporting structures with a separating segment of the separating head which, with reference to the longitudinal axis, forms a front end of the separating head, wherein the separating head is inserted into the opening sufficiently far that the at least two supporting structures lie against an electrically insulating material forming, at least in segments, the interior circumferential surface and/or the exterior circumferential surface with reference to the longitudinal axis; and performing an electro-polishing of the interior surface in an electrolyte bath by connecting the component and the auxiliary electrode to different poles of an electric voltage source in such a way that the auxiliary electrode forms a cathode and the component forms an anode, wherein the separating head remains arranged in the cavity during the electro-polishing.