Polynomial periodicity for Betti numbers of covering surfaces
Let \(Y\) be a complex projective surface and let \(B\subseteq Y\) be a reduced divisor. For \(n\in\mathbb{N}\), \(n\geq 2\), consider the map \(\varphi_ n:H^ 1(Y-B;\mathbb{Z})\to H^ 1(Y-B;\mathbb{Z}/n)\): \(\varphi_ n\) determines an unbranched Galois covering \(W_ n\to Y-B\) and this in turn determines uniquely a normal covering \(X_ n\to Y\) branched along \(B\). The author studies here the behaviour of the first Betti number \(b(n)\) of a desingularization \(\tilde X_ n\) of \(X_ n\) as a function of \(n\). (Remark that \(b(n)\) does not depend on the chosen desingularization.) The main result of the paper can be summarized as follows: under some topological conditions on \(B\), \(b(n)\) is a polynomial periodic function of \(n\), namely there exist an integer \(N\) and polynomials \(q_ 0,\dots,q_{N-1}\) such that if \(n\equiv i\pmod N\) then \(b(n)=q_ i(n)\). Polynomial periodicity for the first Betti number \(b'(n)\) of the unbranched covering \(W_ n\to Y-B\) had already been proven by \textit{Sarnak} [``Betti numbers of congruence groups'' (The Austral. Nat. Univ. Res. Rep., Canberra/Australia 1989)]. The author uses a generalization of Sarnak's method to show that the difference \(b(n)-b'(n)\) is polynomial periodic, too.
polynomial periodicity for Betti numbers
scientific article; zbMATH DE number 149675