TL;DR
This paper establishes a comparison formula for Donaldson-Thomas invariants of Calabi-Yau threefolds related by a flop, using derived equivalences and motivic Hall algebras to relate their curve-counting invariants.
Contribution
It introduces a new comparison formula for DT invariants across flops, utilizing derived categories and motivic Hall algebras to connect invariants of related Calabi-Yau threefolds.
Findings
Proves a comparison formula for DT invariants under flops.
Constructs motivic Hall algebras for categories of perverse coherent sheaves.
Shows invariants are related via derived equivalences and Hall algebra structures.
Abstract
We prove a comparison formula for the Donaldson-Thomas curve-counting invariants of two smooth and projective Calabi-Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop.
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