Recent zbMATH articles in MSC 14C17https://zbmath.org/atom/cc/14C172022-09-13T20:28:31.338867ZWerkzeugVirtual cycles on projective completions and quantum Lefschetz formulahttps://zbmath.org/1491.140102022-09-13T20:28:31.338867Z"Oh, Jeongseok"https://zbmath.org/authors/?q=ai:oh.jeongseokSummary: For a compact quasi-smooth derived scheme \(M\) with \((- 1)\)-shifted 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 \textit{Y. Jiang} and \textit{R. P. Thomas} [J. Algebr. Geom. 26, No. 2, 379--397 (2017; Zbl 1401.14221)]. We produce virtual cycles on both the projective completion \(\overline{N} : = \mathbb{P}(N \oplus \mathcal{O}_M)\) and projectivisation \(\mathbb{P}(N)\) and show the ones on \(\overline{N}\) push down to Jiang-Thomas cycles and the one on \(\mathbb{P}(N)\) computes the difference.
Using similar ideas we give an expression for the difference of the quintic and \(t\)-twisted quintic GW invariants of \textit{S. Guo}, \textit{F. Janda} and \textit{Y. Ruan} [``Structure of higher genus Gromov-Witten invariants of quintic 3-folds'', Preprint, \url{arXiv:1812.11908}].