Digital Object Identifier (DOI) 10.1007/s002090100396 Math. Z. 240, 731–743 (2002) Cohomologiesofadoublecovering ofanon-singularalgebraic3-fold Sl ´ awomirCynk Instytut Matematyki UJ, ul. Reymonta 4, 30-059 Krak´ ow, Poland (e-mail: cynk@im.uj.edu.pl) Received: 6 March 2001 / in final form: 4 September 2001/ Published online: 4 April 2002 – c  Springer-Verlag 2002 Abstract. In this paper we study the Hodge numbers of a branched double covering of a smooth, complete algebraic threefold. The involution on the double covering gives a splitting of the Hodge groups into symmetric and skew-symmetric parts. Since the symmetric part is naturally isomorphic to the corresponding Hodge group of the base we study only the skew- symmetric parts and prove that in many cases it can be computed explicitly. Mathematics Subject Classification (1991): 14B05, 14J30, 32B10 1. Introduction Let X π -→ Y be a double covering of a non-singular, complete, complex algebraic variety Y branched along a non-singular (reduced) divisor D. Then X is determined by the branch locus D and a line bundle L on Y that can be described in the following way: the natural involution τ on X (exchanging sheets of the covering) gives the splitting π ∗ O X ∼ = O Y ⊕L -1 , where L is a line bundle on Y such that L ⊗2 ∼ = O Y (D). In this paper we shall study the Hodge numbers of X . In the case of a curve the only non- obvious Hodge number is equal to the genus so it can be computed from the Hurwitz formula. Also the case when dim X =2 is relatively easy (and well studied in literature). In that case we have Partially supported by KBN grant no 2P03A 022 17 Part of this work was done during the Author stay at the Erlangen-N¨ urnberg University supported by DFG project number 436 POL 113/89/0