Donaldson–Thomas invariants of Calabi–Yau orbifolds under flops

Abstract

We study the Donaldson-Thomas type invariants for the Calabi-Yau threefold Deligne-Mumford stacks under flops. A crepant birational morphism between two smooth Calabi-Yau threefold Deligne-Mumford stacks is called an orbifold flop if the flopping locus is the quotient of weighted projective lines by a cyclic group action. We prove that the Donaldson-Thomas invariants are preserved under orbifold flops. CONTENTSYUNFENG JIANG Proof of Theorem 1.3 32 6. Discussion on the Hard Lefschetz condition. 32 33 References 33 DONALDSON-THOMAS INVARIANTS UNDER FLOPS 3 Definition 1.1. ([43]) A stable pair(1) dim(F) ≤ 1 and F is pure;(2) s has zero-dimensional cokernel.The moduli scheme PT n (X, β) of stable pairs with fixing topological data [F] = β ∈ H 2 (X, Z), χ(F) = n is a scheme and the PT-invariant is defined by PT n,β (X) = χ(PT n (X, β), ν PT ), where ν PT is the Behrend function on PT n (X, β). Both DT-invariants and PT-invariants are curve counting invariants of X; The famous DT/PTcorrespondence conjecture in [43] equates these two invariants in terms of partition functions.The conjecture was proved by Bridgeland [9], and Toda [47] using the wall crossing idea, under which the DT-moduli space and the PT moduli space correspond to different (limit) stability conditions in Bridgeland's space of stability conditions.We pay more attention to Bridgeland's method for the proof. In [9] Bridgeland uses some Hall algebra identities in the motivic Hall algebra H(A) of the abelian category of coherent sheaves A = Coh(X); such that the DT-moduli space and the PT-moduli space are both elements in the Hall algebra. Then applying the integration map as in [33], [10] Bridgeland gets the DT/PT-correspondence.The same idea works for threefold flops. Let