Arrangements of Lines and Algebraic Surfaces

Abstract

In recent years the Chern numbers c ₁ ² and c ₂ of algebraic surfaces have aroused special interest. For a minimal surface of general type they are positive and satisfy the inequality c ₁ ² ≤ 3c ₂ (see Miyaoka [22] anal Yau [31]) where the equality sign holds if and only if the universal cover of the surface is the unit ball EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaaca % WG6bWaaSbaaSqaaiaaigdaaeqaaaGccaGLhWUaayjcSdWaaWbaaSqa % beaacaaIYaaaaOGaey4kaSYaaqWaaeaacaWG6bWaaSbaaSqaaiaaik % daaeqaaaGccaGLhWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaeyip % aWJaaGymaaaa!44A1!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$${\left| {{z_1}} \right|^2} + {\left| {{z_2}} \right|^2} < 1$$ (see Yau [31] and [23] §2 for the difficult and [11] for the easy direction of this equivalence). It is interesting to know which positive rational numbers ≤ 3 occur as c ₁ ²/c ₂ for a minimal surface of general type. For a long time (before 1955) it was believed that c ₁ ²/c ₂ ≤ 2, in other words that the signature of a surface of general type is nonpositive. It is interesting to find surfaces with 2 < c ₁ ²/c ₂ ≤ 3. Some were found by Kodaira [19]. Other authors constructed more surfaces of this kind (Holzapfel [13], [14], Inoue [16], Livné [21], Mostow-Siu [25], Miyaoka [23]). Recently Mostow [24] and Deligne-Mostow [5] used a paper of E. Picard of 1885 to construct discrete groups of automorphisms of the unit ball leading to interesting surfaces. In fact, the author was stimulated by Mostow’s lecture at the Arbeitstagung 1981 to study the surface Y ₁ (see §3.2) which is related to one of the surfaces of Mostow.