2 Fabrizio Catanese Universität Bayreuth

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-´ etale morphisms, i.e., morphisms which are unramified in codimen-sion 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. (: = Complex-Quasi-´ Etale-Deformation), can be considered for compact complex manifolds and for compact complex spaces with canonical singularities. It is generated by bimeromorphic equivalence, by flat deformations of complex spaces with canonical singularities, and by quasi-´ etale morphisms. By a recent theorem of Siu (conjecturally holding also for compact Kähler manifolds), dimension and Kodaira dimension are invariants for A.Q.E.D. (also for C-Q.E.D.-equivalence of algebraic varieties?). It was an interesting question whether conversely two algebraic varieties of the same dimension and with the same Kodaira dimension are Q.E.D.-equivalent (the question then bifurcates into: A.Q.E.D., or at least C-Q.E.D.). The answer to the A.Q.E.D. question is positive for curves by well known results. Using Enriques' classification we show first that the answer to the C-Q.E.D. question is positive for special algebraic surfaces (i.e., for algebraic surfaces with Kodaira dimension at most 1). More generally, using Kodaira's classification we show that the same result holds for compact complex surfaces with Kodaira dimension 0, 1 and even first Betti number. The appendix by Sönke Rollenske shows that the same does not hold if we consider also surfaces with odd first Betti number. Indeed he proves that any surface which is C-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface. Concerning the A.Q.E.D. question, we show that the answer is positive for complex algebraic surfaces of Kodaira dimension ≤ 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. We leave aside here the A.Q.E.D. question for algebraic surfaces of Ko-daira dimension ≤ 1 defined over an algebraically closed field of positive characteristic; we hope to be able to address this question in the future. We pose then several questions, and point out possible applications of the above ideas, with the aim of showing the importance for classification theory …