We apply geometric techniques from representation theory to the study of homologically finite differential graded (DG) modules M over a finite dimensional, positively graded, commutative DG algebra U. In particular, in this setting we prove a version of a theorem of Voigt by exhibiting an isomorphism between the Yoneda Ext group YExt1U(M,M) and a quotient of tangent spaces coming from an algebraic group action on an algebraic variety. As an application, we answer a question of Vasconcelos from 1974 by showing that a local ring has only finitely many semidualizing complexes up to shift-isomorphism in the derived category D(R).