Some New Surfaces of General Type

In this paper we shall give two series of rare examples of algebraic surfaces of general type.One is a series of surfaces with positive topological indices and another with the geometric genus pg=0 .In Part I, we construct surfaces with positive topological indices.In [V] (1966), A. Van de Ven pointed out that there are not many known examples of algebraic surfaces of general type with positive topological indices.By the index theorem of Hirzebruch, the topological index τ(S) of a surface S is equal to (KS²-2e(S))/3 where KS is the canonical line bundle and $e(S)$ the topological Euler number.Hence the positivity of τ(S) is equivalent to KS²>2e(S) .At that time, the only known examples were the ones due to F. Hirzebruch [H], which are the compact quotients of the 2-dimensional unit ball and which therefore satisfy KS²=3e(S) .Following Van de Ven's remark, K. Kodaira [K] (1967) was the first to construct a series of examples as branch- ed coverings of the product of two algebraic curves.Kodaira's examples satisfy 3 e(s)>KS²>2e(S) and have many interesting properties.Afterwards some new examples were discovered, for instance, by Mostow-Siu (1979) and Y. Miyaoka (1980).But even now the known examples are rare.In Part I, we construct new examples of such surfaces as branched coverings of the so-called elliptic modular surfaces which are investigated in detail in T. Shioda [S] (1972).Our construction depends heavily on some results in [S], which we shall recall in 1.We remark that our examples are discovered independently also by R. Livn\'e.In Part II, we construct surfaces with pg=0 .If S is a minimal surface of general type with pg(S)=0 , then we know generally that $q(S)=0,$ KS²=1,2, ⋯, $9$ .Till 1974, the known and verified example was only the classical one due to L. Godeaux, on which we refer to Y. Miyaoka [M] (1976).Afterwards some classically known examples were verified and furthermore some new examples were discovered by, for instance,