Virtual classes of $$\mathbb {G}_\text {m}$$ G m -gerbes
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© Springer-Verlag GmbH Germany, part of Springer Nature 2020
F. Qu
Virtual classes of Gm -gerbes Received: 8 December 2019 / Accepted: 9 September 2020 Abstract. We show that a perfect obstruction theory for a Gm -gerbe determines a semiperfect obstruction theory for its base, which is perfect if the gerbe is quasi-compact and affine-pointed. These results streamline the construction of a semi-perfect obstruction theory for the base, and allow us to relate virtual classes of the gerbe and its base.
1. Introduction We discuss virtual classes of Gm -gerbes with perfect obstruction theories. These gerbes appear naturally in the Donaldson–Thomas (DT) theory of smooth projective threefolds. As moduli stacks of Bridgeland stable objects, Gm -gerbes are used to extract invariants when there are no strictly semi-stable objects. See e.g., [4] and references therein. In the presence of strictly semi-stable objects, the approach using virtual cycles to define generalized DT invariants is to associate some Gm -gerbe intrinsically to a (derived) moduli stack of Bridgeland semi-stable objects. See e.g., [17]. Recently virtual classes of Artin stacks are defined unconditionally [3,13] using higher categorical ingredients. Gm -gerbes are probably the simplest Artin stacks, and their virtual classes can also be treated using the more classical approach of [26], thus examining their virtual cycles is a natural step to take towards understanding examples. Consider a Gm -gerbe G with an absolute perfect obstruction theory over a DM stack B. The main observation is that after truncating its perfect obstruction theory from [−1, 1] to [−1, 0], we can decompose the truncation into moving and fixed parts. The moving part is given by a locally free sheaf of finite rank H in degree −1, and the fixed part determines a semi-perfect obstruction theory for B. The virtual class of G is obtained by pulling back the virtual class of B then cap with the Euler class of the vector bundle associated to H . When B is quasi-compact and affine-pointed, the semi-perfect obstruction theory for B is actually a perfect obstruction theory. The author is supported by NSFC Young Scientists Fund 11801185. F. Qu (B): Northeast Normal University, Changchun, China e-mail: [email protected] Mathematics Subject Classification: Primary:14C17 · Secondary:14N35
https://doi.org/10.1007/s00229-020-01245-8
F. Qu
Obstruction theories for B have been constructed in [8,11], and our results come from efforts to formulate those constructions using the perfect obstruction theory for G. These results are not hard to prove, and details can be found in Sect. 3. The two key ingredients recalled in Sect. 2.2 are decomposition of quasi-coherent sheaves on G into direct summands indexed by the characters of Gm and the equivalence between the derived category of complexes of OG -modules with quasi-coherent cohomology sheaves and the derived category of quasi-coherent sheaves. As an application, at the end of Sect. 3 we remark that two choices of fixing determinant of perfect com
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