Q.E.D. for algebraic varieties

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We introduce a new equivalence relation for complete algebraic varieties with canonical singularities, generated by birational equivalence, by flat algebraic deformations (of varieties with canonical singularities), and by quasi-étale morphisms, i.e., morphisms which are unramified in codimension 1. We denote the above equivalence by A.Q.E.D. : = Algebraic-Quasi-Étale- Deformation. A completely similar equivalence relation, denoted by $\mathbb{C}$-Q.E.D., can be considered for compact complex spaces with canonical singularities. 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 algebraic varieties of the same dimension and with the same Kodaira dimension are Q.E.D.-equivalent (A.Q.E.D., or at least $\mathbb{C}$-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 $\mathbb{C}$-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 Sönke Rollenske shows that the hypothesis of even first Betti number is necessary: he proves that any sur face which is $\mathbb{C}$-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 $\leq$ 1. The answer to the Q.E.D. question is instead negative for surfaces 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.