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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 $$X$$.

##### MSC:
 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
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