Celebratio Mathematica

F. Hirzebruch: “Algebraic surfaces with extremal Chern numbers (report on the thesis of T. Höfer, Bonn, 1984),” Uspekhi Mat. Nauk 40 : 4(244) (1985), pp. 121–129. An English translation was published in Russ. Math. Surv. 40:4 (1985). MR 807793 article

F. Hirzebruch: “Algebraic surfaces with extreme Chern numbers (report on the thesis of T. Höfer, Bonn 1984),” Russ. Math. Surv. 40 : 4 (1985), pp. 135–145. English translation of article published in Uspekhi Mat. Nauk 40:4(244). Zbl 0593.14023 article

For a smooth algebraic surface \( X \) the Chern numbers \( c_1^2(X) \) and \( c_2(X) \) are defined. Here \( c_2(X) \) is equal to the Euler–Poincaré characteristic of \( X \) and \( c_1^2(X) \) is the self-intersection number of a canonical divisor of \( X \). For a surface of general type they satisfy the Miyaoka–Yau inequality \( c_1^2 \leq 3c_2 \) (see [Miyaoka 1977] and [Yau 1977]), where the equality sign holds if and only if the universal cover of the surface is the unit ball \( |z_1|^2 + |z_2|^2 \) \( < 1 \) (see [Yau 1977] and [Miyaoka 1983, §2] for the difficult, and [Hirzebruch 1958] for the easy direction of this equivalence.) If \( c_1^2 = 3c_2 \), then automatically \( c_2 > 0 \), in fact \( c_2 \) is the volume of the surface (normalized by the Gauss–Bonnet form) with respect to the complex-hyperbolic metric induced from the ball.

We wish to construct surfaces of general type with extreme Chern numbers (\( c_1^2 = 3c_2 \)) that are ramified covers of the complex projective plane branched along lines. Thus, we continue the investigation of the paper [Hirzebruch 1983]. However, the much better developed theory and the new examples are due to Höfer [1985]. This note is a report on his work.

We have to omit many things. For example, we only consider the case when the surface is the quotient of the ball by a discrete group \( \Gamma \) of automorphisms, \( \Gamma \) operating freely with compact quotient. Höfer includes the case when \( \Gamma \) may have torsion and cusps (at infinity) and constructs many such ball quotients as ramified covers of the plane. His formulae (with suitable modifications) hold for this more general case.

G. Barthel, F. Hirzebruch, and T. Höfer: Geradenkonfigurationen und Algebraische Flächen [Configurations of lines and algebraic surfaces]. Aspekte der Mathematik D4. Vieweg & Sohn (Braunschweig), 1987. MR 912097 Zbl 0645.14016 book

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F. Hirzebruch and T. Höfer: “On the Euler number of an orbifold,” Math. Ann. 286 : 1–3 (1990), pp. 255–260. Dedicated to Hans Grauert on his sixtieth birthday. MR 1032933 Zbl 0679.14006 article

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Friedrich E. P. Hirzebruch

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