We introduce a new equivalence relation for complete alge-
braic varieties with canonical singularities, generated by birational
equivalence, by flat algebraic deformations (of varieties with canon-
ical singularities), and by quasi-´etale morphisms, i.e., morphisms
which are unramified in codimension 1. We denote the above
equivalence by A.Q.E.D. : = Algebraic-Quasi-´ Etale- Deformation.
A completely similar equivalence relation, denoted by C-Q.E.D.,
can be considered for compact complex spaces with canonical sin-
gularities.
By a recent theorem of Siu, dimension and Kodaira dimension
are invariants for A.Q.E.D. of complex varieties.
We address the interesting question whether conversely two al-
gebraic varieties of the same dimension and with the same Ko-
daira dimension are Q.E.D. - equivalent (A.Q.E.D., or at least
C-Q.E.D.), the answer being positive for curves by well known
results.
Using Enriques’ (resp. Kodaira’s) classification we show first
that the answer to the C-Q.E.D. question is positive for special
algebraic surfaces (those with Kodaira dimension at most 1), resp.
for compact complex surfaces with Kodaira dimension 0, 1 and
even first Betti number.
The appendix by S¨onke Rollenske shows that the hypothesis
of even first Betti number is necessary: he proves that any sur-
face which is C-Q.E.D.-equivalent to a Kodaira surface is itself a
Kodaira surface.
We show also that the answer to the A.Q.E.D. question is pos-
itive for complex algebraic surfaces of Kodaira dimension ≤ 1.
The answer to the Q.E.D. question is instead negative for sur-
faces of general type: the other appendix, due to Fritz Grunewald,
is devoted to showing that 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.