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**Q.E.D. for algebraic varieties.**
*(English)*
Zbl 1128.14026

The author introduces a new equivalence relation, denoted by A.Q.E.D. (Algebraic-Quasi-Étale- Deformation), for complete algebraic varieties with canonical singularities: it is generated by birational equivalence, flat proper algebraic deformation with base a connected algebraic variety, and with all the fibres having canonical singularities, and by quasi-étale morphisms, i.e., morphisms which are unramified in codimension \(1\). \(\mathbb{C}\)-Q.E.D. is the similar relation for compact complex manifolds and spaces.

By Y. T. Siu’s recent Theorem [in: Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. 223–277 (2002; Zbl 1007.32010)], not only dimension, but also the Kodaira dimension is an invariant for A.Q.E.D.-equivalence of projective varieties with canonical singularities (defined over the complex numbers). It is conjectured [ibidem, cf. also Y. T. Siu, Sci. China, Ser. A 48, suppl., 1–31 (2005; Zbl 1131.32010)] that the deformation invariance of plurigenera should be true more generally for Kähler complex spaces (with canonical singularities).

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 the author shows that the answer to the \(\mathbb{C}\)-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 Sönke Rollenske shows that the same does not hold if one allows odd first Betti number: he proves that any surface which is \(\mathbb{C}\)-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface.

The author shows also that the answer to the A.Q.E.D. question is positive 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. The paper ends with some interesting remarks, and open problems.

By Y. T. Siu’s recent Theorem [in: Complex geometry. Collection of papers dedicated to Hans Grauert on the occasion of his 70th birthday. 223–277 (2002; Zbl 1007.32010)], not only dimension, but also the Kodaira dimension is an invariant for A.Q.E.D.-equivalence of projective varieties with canonical singularities (defined over the complex numbers). It is conjectured [ibidem, cf. also Y. T. Siu, Sci. China, Ser. A 48, suppl., 1–31 (2005; Zbl 1131.32010)] that the deformation invariance of plurigenera should be true more generally for Kähler complex spaces (with canonical singularities).

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 the author shows that the answer to the \(\mathbb{C}\)-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 Sönke Rollenske shows that the same does not hold if one allows odd first Betti number: he proves that any surface which is \(\mathbb{C}\)-Q.E.D.-equivalent to a Kodaira surface is itself a Kodaira surface.

The author shows also that the answer to the A.Q.E.D. question is positive 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. The paper ends with some interesting remarks, and open problems.

Reviewer: Paolo A. Oliverio (Rende)