What is the Genus? by Patrick Popescu-Pampu

Author:Patrick Popescu-Pampu
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

The surfaces of these infinitely many distinct families were later called K3 surfaces by Weil (see [188]). The fact that they form an infinite number of families comes from another phenomenon which begins to express itself only from dimension 2 onwards: namely, one may deform their complex analytic structures as little as one wants and get a surface which is no longer algebraic!

But in order to contemplate such a phenomenon, one had to wait for the notion of abstract Riemann surfaces, explained by Weyl in [191] (see Chap. 23), to be extended to higher dimensions. Once this was done, Kodaira [122] made a classification of complex analytic surfaces analogous to the classification of complex algebraic ones. In this classification, the K3 surfaces form a single family. One may consult the lecture [15] by Beauville for a modern viewpoint on K3 surfaces and on this property, as well as the collective book [16] for much more details.

Figure 31.1 presents the classification table of algebraic surfaces, as it may be found at the end of Enriques’ treatise [71] from 1949:

Fig. 31.1Enriques’ classification of surfaces

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