Reduced Donaldson-Thomas invariants and the ring of dual numbers

Abstract

Let $A$ be an abelian variety. We introduce $A$-equivariant Grothendieck rings and $A$-equivariant motivic Hall algebras, and endow them with natural integration maps to the ring of dual numbers. The construction allows a systematic treatment of reduced Donaldson-Thomas invariants by Hall algebra techniques. We calculate reduced Donaldson-Thomas invariants for $K3\times E$ and abelian threefolds for several imprimitive curve classes. This verifies (in special cases) multiple cover formulas conjectured by Oberdieck-Pandharipande and Bryan-Oberdieck-Pandharipande-Yin.

35 pages

References in corpus (3)

• 13/2 ways of counting curves
• Stability conditions and curve counting invariants on Calabi-Yau 3-folds
• Donaldson-Thomas invariants for complexes on abelian threefolds

Cited by in corpus (2)

• Multiple cover formulas for K3 geometries, wall-crossing, and Quot schemes
• Stable pairs and Gopakumar-Vafa type invariants on holomorphic symplectic 4-folds