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The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces. (English) Zbl 1287.14027

In this article, the authors construct examples of a smooth orbifold \(X\) and a complete intersection \(Y\subset X\) determined by sections of vector bundles \(E=\oplus E_{j}\rightarrow X\) such that \(E_{j}\) is a line bundle with \(c_{1}(E_{j})\cdot d \geq 0\), whenever \(d\) is the degree of genus 0 stable map to \(X\), but there is no cohomology class \(e\) on \(X_{0,n,d}\)(moduli stack of degree d stable maps to \(X\) from genus 0 curves with \(n\) marked points) with \(\sum_{\delta:i_{*}\delta=d}i_{*}[Y_{0,n,\delta}]^{vir}=[X_{0,n,d}]^{vir} \cap e.\) In other words, the quantum Lefschetz hyperplane principle can fail for positive orbifold complete intersections.

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
14D23 Stacks and moduli problems
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