Smooth Q$\mathbb {Q}$‐homology planes satisfying the negativity conjecture

Abstract

AbstractA complex algebraic surface is a ‐homology plane if for . The Negativity Conjecture of Palka asserts that , where is a log smooth completion of . We give a complete description of smooth ‐homology planes satisfying the Negativity Conjecture. We restrict our attention to those of log general type, as otherwise their geometry is well‐understood. We show that, as conjectured by tom Dieck and Petrie, they can be arranged in finitely many discrete series, each obtained in a uniform way from an arrangement of lines and conics on . We infer that these surfaces satisfy the Rigidity Conjecture of Flenner and Zaidenberg; and a conjecture of Koras, which asserts that .