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The cohomological crepant resolution conjecture for \(\mathbb{P}(1,3,4,4)\). (English) Zbl 1190.14053
The authors prove Ruan’s cohomological crepant resolution conjecture for the weighted projective space \(X=\mathbb P(1,3,4,4)\) (Theorem 4.1). The cohomological crepant resolution conjecture predicts the existence of an isomorphism between Chern–Ruan cohomology ring of a complex orbifold and some specification of a small quantum cohomology ring of its crepant resolution. The authors compute the Chern–Ruan cohomology ring of \(X\) using the known technic of such computations for weighted projective spaces. The crepant resolution \(Z\) for \(X\) is given by a subdivision of its fan. The resolution has 4 exceptional divisors. Thus the quantum corrected cohomology ring of \(Z\) depends on 4 parameters; 3 of them correspond to a non-isolated singularity of \(X\) and one corresponds to an isolated one. The authors specify these parameters to \((i,i,i,q)\) or \((-i,-i,-i,q)\), where \(i\) is an arbitrary natural number and \(q\) is a complex parameter. The quantum corrected cohomology ring of \(Z\) after such specification depends on a particular power series in \(q\). This series depends on a local neighborhood of an isolated singularity of \(X\). The convergence of this series at 1 follows from a computation for \(\mathbb P(1,1,1,3)\) (which is locally isomorphic to \(X\) in the neighborhoods of isolated singularities) done in T. Coates and H. Iritani and H.-H. Tseng [Geom. Topol. 13, No. 5, 2675–2744 (2009; Zbl 1184.53086)]. The authors computes the analytic continuation of the power series at 1 and give a particular isomorphism of a specified quantum cohomology ring of \(Z\) and a Chern–Ruan cohomology ring for \(X\).

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Full Text: DOI
[1] DOI: 10.1080/00927870802248902 · Zbl 1178.14056
[2] DOI: 10.1090/S0894-0347-04-00471-0 · Zbl 1178.14057
[3] DOI: 10.1090/S1056-3911-07-00467-5 · Zbl 1129.14075
[4] DOI: 10.1090/surv/068
[5] DOI: 10.5802/aif.2008 · Zbl 1063.14018
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[7] DOI: 10.1090/S1056-3911-07-00465-1 · Zbl 1146.14029
[8] DOI: 10.1142/S0129167X07004436 · Zbl 1149.14010
[9] DOI: 10.1090/conm/403/07597
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