Orbifold Gromov-Witten theory of weighted blowups
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https://doi.org/10.1007/s11425-020-1774-x
Orbifold Gromov-Witten theory of weighted blowups Bohui Chen1 , Cheng-Yong Du2,∗ & Rui Wang3 1Department 2School
of Mathematics and Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China; of Mathematical Sciences and V. C. & V. R. Key Lab, Sichuan Normal University, Chengdu 610068, China; 3Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720, USA Email: [email protected], [email protected], [email protected] Received January 13, 2020; accepted September 10, 2020
Abstract
Consider a compact symplectic sub-orbifold groupoid S of a compact symplectic orbifold groupoid
(X, ω). Let Xa be the weight-a blowup of X along S, and Da = PNa be the exceptional divisor, where N is the normal bundle of S in X. In this paper we show that the absolute orbifold Gromov-Witten theory of Xa can be effectively and uniquely reconstructed from the absolute orbifold Gromov-Witten theories of X, S and Da , ∗ (X) → H ∗ (S) and the first Chern class of the tautological line the natural restriction homomorphism HCR CR
bundle over Da . To achieve this we first prove similar results for the relative orbifold Gromov-Witten theories of (Xa | Da ) and (Na | Da ). As applications of these results, we prove an orbifold version of a conjecture of Maulik and Pandharipande (Topology, 2006) on the Gromov-Witten theory of blowups along complete intersections, a conjecture on the Gromov-Witten theory of root constructions and a conjecture on the Leray-Hirsch result for the orbifold Gromov-Witten theory of Tseng and You (J Pure Appl Algebra, 2016). Keywords
orbifold Gromov-Witten theory, Leray-Hirsch result, weighted projective bundle, weighted blowup,
root stack, blowup along complete intersection MSC(2010)
53D45, 14N35
Citation: Chen B H, Du C-Y, Wang R. Orbifold Gromov-Witten theory of weighted blowups. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-020-1774-x
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Introduction
Symplectic birational geometry, proposed by Li and Ruan [32], studies symplectic birational cobordism invariants defined via the Gromov-Witten theory. Two symplectic manifolds are called symplectic birational cobordant (see [22, 23]) if one can be obtained from the other by a sequence of Hamiltonian S 1 symplectic reductions. Guillemin and Sternberg [22] proved that every symplectic birational cobordism can be realized by a finite number of symplectic blowups/blowdowns and Z-linear deformations of symplectic forms1) . With noticing Gromov-Witten invariants are preserved under smooth deformations of symplectic structures (see, for example, [15, 37]), to understand the relation of Gromov-Witten invariants between two symplectic birational corbordant manifolds, we only need to take care of the change of Gromov-Witten invariants after a symplectic blowup/blowdown. * Corresponding author 1) A Z-linear deformation of a symplectic form ω on a manifold X is a path of a symplectic form ω + tκ, t ∈ I, where κ is a closed 2-form representing an integral class and I
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