Algebraic Surfaces with Quotient Singularities - including Some Discussion on Automorphisms and Fundamental Groups

We work over the complex numbers field C. In the present survey, we report some recent progress on the study of varieties with mild singularities like log terminal singularities (which are just quotient singularities in the case of dimension 2; see [KMM]). Singularities appear naturally in many ways. The minimal model program developed by Mori et al shows that a minimal model will inevitably have some terminal singularities [KMo]. Also the degenerate fibres of a family of varieties will have some singularities. We first follow Iitaka’s strategy to divide (singular) varieties Y according to the logarithmic Kodaira dimension κ(Y ) of the smooth locus Y 0 of Y . One key result (2.3) says that for a relatively minimal log terminal surface Y we have either nef KY or dominance of Y 0 by an affine-ruled surface. It is conjectured to be true for any dimension [KMc]. In smooth projective surfaces of general type case, we have Miyaoka-Yau inequality c1 ≤ 3c2 and Noether inequalities: pg ≤ (1/2)c1 + 2, c1 ≥ (1/5)c2 − (36/5). Similar inequalities are given for Y 0 in Section 4; these will give effective restriction on the region where non-complete algebraic surfaces of general type exist. In Kodaira dimension zero case, an interesting conjecture (3.12) (which is certainly true when Y is smooth projective by the classification theory) claims that for a relatively minimal and log terminal surface Y of Kodaira dimension κ(Y ) = 0, one has either π1(Y ) finite, or an etale cover Z → Y 0 where Z is the complement of a finite set in an abelian surface Z. Some partial answers to (3.12) are given in Section 3. The topology of Y 0 is also very interesting. We still do not know whether π1 of the complement of a plane curve is always residually finite or not. Conjecture (2.4) proposed in [Z7] claims that the smooth locus of a log terminal Fano variety has finite topological fundamental group. This is confirmed when the dimension is two and now there are three proofs: [GZ1, 2] (using Lefschetz hyperplane section theorem and Van Kampen theorem), [KMc] (via rational connectivity), [FKL] (geometric).