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Donaldson–Thomas theory of \([\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1\)

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Abstract

We study the relative orbifold Donaldson–Thomas theory of \([\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1\). A correspondence is established between the DT theory relative to disjoint union of vertical fibers to quantum multiplication by divisors for the Hilbert scheme of points on \([\mathbb {C}^2/\mathbb {Z}_{n+1}]\). This determines a correspondence between the whole theories if a further nondegeneracy condition is assumed. The result can also be viewed as a crepant resolution correspondence to the DT theory of \(\mathcal {A}_n\times \mathbb {P}^1\).

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Acknowledgements

The author would like to thank Davesh Maulik, Hiraku Nakajima, Andrei Okounkov, Amdrey Smirnov, Changjian Su, Richard Thomas and Jingyu Zhao for helpful discussions and suggestions. Moreover, the author would like to express his acknowledgements to Professor Chiu–Chu Melissa Liu, for useful conversations and suggestions. The project would not have been possible without her guidance and encouragement.

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  1. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    Zijun Zhou

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  1. Zijun Zhou
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Zhou, Z. Donaldson–Thomas theory of \([\mathbb {C}^2/\mathbb {Z}_{n+1}]\times \mathbb {P}^1\). Sel. Math. New Ser. 24, 3663–3722 (2018). https://doi.org/10.1007/s00029-017-0384-9

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