ALGEBRAIC SURFACES WITH MINIMAL BETTI NUMBERS

Abstract

These are algebraic surfaces with the Betti numbers of the complex projective plane, and are called Q-homology projective planes. Fake projective planes and the complex projective plane are smooth examples. We describe recent progress in the study of such surfaces, singular ones and fake projective planes. We also discuss open questions. 1 Q-homology Projective Planes and Montgomery-Yang problem A normal projective surface with the Betti numbers of the complex projective plane CP 2 is called a rational homology projective plane or a Q-homology CP . When a normal projective surface S has only rational singularities, S is a Q-homology CP 2 if its second Betti number b2(S) = 1. This can be seen easily by considering the Albanese fibration on a resolution of S . It is known that a Q-homology CP 2 with quotient singularities (and no worse singularities) has at most 5 singular points (cf. Hwang and Keum [2011b, Corollary 3.4]). The Q-homology projective planes with 5 quotient singularities were classified in Hwang and Keum [ibid.]. In this section we summarize progress on the Algebraic Montgomery-Yang problem, which was formulated by J. Kollár. Conjecture 1.1 (Algebraic Montgomery–Yang Problem Kollár [2008]). Let S be a Qhomology projective plane with quotient singularities. Assume that S := SnSing(S) is simply connected. Then S has at most 3 singular points. This is the algebraic version ofMontgomery–Yang ProblemFintushel and Stern [1987], which was originated from pseudofree circle group actions on higher dimensional sphere. MSC2010: primary 14J29; secondary 14F05, 14J17, 14J26, 32Q40, 32N15.