  We introduce a new equivalence relation, denoted by $A.Q.E.D.$ (=
Algebraic-Quasi-\'Etale- Deformation) for complete algebraic varieties with
canonical singularities: it is generated by birational equivalence, by flat
algebraic deformations, and by quasi-\'etale morphisms, i.e., morphisms which
are unramified in codimension $1$. $\C$-Q.E.D is the similar relation for
compact complex manifolds and spaces. By a recent theorem of Siu dimension and
Kodaira dimension are invariants for $A.Q.E.D.$. The question whether
conversely two algebraic varieties of the same dimension and with the same
Kodaira dimension are Q.E.D. -equivalent has a positive answer for curves and
using Enriques' classification we show that the answer to the $\C$-Q.E.D.
question is positive for special algebraic surfaces (Kodaira dimension at most
$1$) and for compact complex surfaces with Kodaira dimension $0, 1$ and even
first Betti number. The appendix by S\"onke Rollenske shows that the same does
not hold if we allow odd first Betti number( Kodaira surfaces form a single
class). The answer to the A.Q.E.D. question is yes for special complex
algebraic surfaces, while the appendix due to Fritz Grunewald shows that the
answer is no for surfaces of general type. Because the (rigid) Kuga-Shavel type
surfaces of general type obtained as quotients of the bidisk via discrete
groups constructed from quaternion algebras belong to countably many distinct
Q.E.D. equivalence classes.
