Ancestor vanishing conjecture for ADE surface singularities

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Let GG be a finite group of ADE type, let ρ\rho be an irreducible two-dimensional representation of GG, and let Fρ1F_{\rho}^{1} be the associated Hodge bundle on M‾g,n(BG;[ ⁣[a1] ⁣],…,[ ⁣[an] ⁣])\overline{\mathcal{M}}_{g,n}(BG;\lbrack\!\lbrack a_{1}\rbrack\!\rbrack,\ldots,\lbrack\!\lbrack a_{n}\rbrack\!\rbrack). Let ψˉ[ ⁣[ak] ⁣]\bar{\psi}_{\lbrack\!\lbrack a_{k}\rbrack\!\rbrack} denote the corresponding ancestor cotangent-line class, and let c2g−2+n(Fρ1)c_{2g-2+n}(F_{\rho}^{1}) be the indicated Chern class. Ancestor vanishing conjecture. If ([ ⁣[a1] ⁣],…,[ ⁣[an] ⁣])≠[ ⁣[1] ⁣]n(\lbrack\!\lbrack a_{1}\rbrack\!\rbrack,\ldots,\lbrack\!\lbrack a_{n}\rbrack\!\rbrack)\neq\lbrack\!\lbrack 1\rbrack\!\rbrack^{n} and ∑k=1nlk=g−1\sum_{k=1}^{n}l_{k}=g-1, then

∫M‾g,n(BG;[ ⁣[a1] ⁣],…,[ ⁣[an] ⁣])c2g−2+n(Fρ1)∏k=1nψˉ[ ⁣[ak] ⁣]lk=0.\int_{\overline{\mathcal{M}}_{g,n}(BG;\lbrack\!\lbrack a_{1}\rbrack\!\rbrack,\ldots,\lbrack\!\lbrack a_{n}\rbrack\!\rbrack)}c_{2g-2+n}(F_{\rho}^{1})\prod_{k=1}^{n}\bar{\psi}_{\lbrack\!\lbrack a_{k}\rbrack\!\rbrack}^{l_{k}}=0.

This is proposed as the ancestor analogue of the corresponding primary vanishing statement and is intended to support the crepant resolution conjecture for gravitational ancestors. The supplied text does not state whether it has been proved in full generality.

References

Primary source

Xiaowen Hu, “On the Crepant Resolution Conjecture for Gromov-Witten Gravitational Ancestors in All Genera for Surface Singularities”, arXiv:1308.0997 (2013).

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