Infinitesimal Deformations of Double Covers of Smooth Algebraic Varieties

The goal of this paper is to give a method to compute the space of infinitesimal deformations of a double cover of a smooth algebraic variety. This research was inspired by the analysis of Calabi–Yau manifolds that arise as smooth models of double covers of P branched along singular octic surfaces ([4, 3]). It is of considerable interest to determine the Hodge numbers for these manifolds, but the methods to compute these are only available in very special cases. For example, the results of [2] are applicable in the case where the octic has only ordinary double points. Since for a Calabi–Yau 3-fold the Hodge number h equals the dimension of the space of infinitesimal deformations our approach is to study the latter. Let X π −−−−→ Y be a double cover of a non–singular, complete, complex algebraic variety Y branched along a non–singular (reduced) divisor D. In the space HΘX of all infinitesimal deformations of X one can distinguish two subspaces T 1 X→Y – infinitesimal deformations of X , which are double covers of deformations of Y , T 1 X/Y – infinitesimal deformations of X , which are double covers of Y . In Proposition 2.2 we give formulae for the above two subspaces. They have the following geometrical interpretation: the space T 1 X→Y is isomorphic to the space of simultaneous deformations of D ⊂ Y , whereas T 1 X/Y is isomorphic to the space of deformations of D as a subscheme of Y modulo those coming from infinitesimal automorphisms of Y . The space of simultaneous deformations of D ⊂ Y is isomorphic to the cohomology groupH(ΘY (logD)), the so–called logarithmic deformations (cf. [9, § 2]) In the space of all infinitesimal deformations of X we identify (Proposition 2.1) a subspace isomorphic to H(ΘY ⊗L), which is complementary to T 1 X→Y , where L is the line bundle on Y defining the double cover. We call deformations from this subspace transverse because they induce trivial (up to order one) deformations of the branch locus D. The main result of the paper is a description of the effect of imposing singularities in the branch locus. If the divisorD is singular then there exist a sequence of blow– ups σ : Ỹ −→ Y and a non–singular (reduced) divisor D ⊂ Ỹ , s.t. D̃ ≤ D ≤ σD and D is an even element of the Picard group of Ỹ . The double cover X̃ of Ỹ branched along D is a smooth model of X . We prove that the space T 1 X̃−→Ỹ can be interpreted as the space of equisingular deformations of D in deformations of Y , under the additional assumption that Y