Virtual cycles on projective completions and quantum Lefschetz formula

Abstract

For a compact quasi-smooth derived scheme M with p´1qshifted cotangent bundle N , there are at least two ways to localise the virtual cycle of N to M via torus and cosection localisations, introduced by Jiang-Thomas [JT]. We produce virtual cycles on both the projective completion N :" PpN ' OM q and projectivisation PpN q and show the ones on N push down to Jiang-Thomas cycles and the one on PpN q computes the difference.Using similar ideas we give an expression for the difference of the quintic and t-twisted quintic GW invariants of Guo-Janda-Ruan [GJR].Notation. For a morphism of schemes f : X Ñ Y and a perfect complex E on Y , we often denote by E| X the pullback f ˚E . We use E ˚for the usual dual of E, whereas E _ for the derived dual.A vector bundle E is sometimes thought of as its total space. For a morphism of vector bundles f : E Ñ F , ker f is sometimes thought of as the space Spec `Sym `coker f ˚˘˘.